SYMMETRIZED PERTURBATION DETERMINANTS AND APPLICATIONS TO BOUNDARY DATA MAPS AND KREIN-TYPE RESOLVENT FORMULAS FRITZ GESZTESY AND MAXIM ZINCHENKO Dedicated to the memory of Pierre Duclos (1948–2010 )
Abstract. The aim of this paper is twofold: On one hand we discuss an abstract approach to symmetrized Fredholm perturbation determinants and an associated trace formula for a pair of operators of positive-type, extending a classical trace formula. On the other hand, we continue a recent systematic study of boundary data maps, that is, 2 × 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schr¨ odinger operators on a compact interval [0, R] with separated boundary conditions at 0 and R. One of the principal new results in this paper reduces an appropriately symmetrized (Fredholm) perturbation determinant to the 2 × 2 determinant of the underlying boundary data map. In addition, as a concrete application of the abstract approach in the first part of this paper, we establish the trace formula for resolvent differences of self-adjoint Schr¨ odinger operators corresponding to different (separated) boundary conditions in terms of boundary data maps.
1. Introduction In his joint 1983 paper [17] with Jean-Michel Combes and Ruedi Seiler, Pierre Duclos considered various one-dimensional Dirichlet and Neumann Schr¨odinger operators and associated Krein-type resolvent formulas to study the classical limit of discrete eigenvalues in a multiple-well potential. One of the principal aims of the present paper is to consider related Krein-type resolvent formulas for general separated boundary conditions on a compact interval and establish connections with recently established boundary data maps in [16], perturbation determinants, and trace formulas. In addition, we discuss an abstract approach to symmetrized (Fredholm) perturbation determinants and an associated trace formula for a pair of operators of positive-type, extending a classical trace formula for perturbation determinants described by Gohberg and Krein [35, Sect. IV.3]. Date: September 11, 2011. 2000 Mathematics Subject Classification. Primary: 34B05, 34B27, 34B40, 34L40; Secondary: 34B20, 34L05, 47A10, 47E05. Key words and phrases. (non-self-adjoint) Schr¨ odinger operators on a compact interval, separated boundary conditions, boundary data maps, Robin-to-Robin maps, Krein-type resolvent formulas, perturbation determinants, trace formulas. Based upon work partially supported by the US National Science Foundation under Grant No. DMS-0965411. Proc. London Math. Soc. (to appear). 1
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F. GESZTESY AND M. ZINCHENKO
In Section 2 we depart from our consideration of Schr¨odinger operators on a compact interval and turn our attention to an abstract result on symmetrized (Fredholm) determinants of the form detH (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 (1.1) associated with a pair of operators (A, A0 ) of positive-type (and z in appropriate sectors of the complex plane). In particular, this permits a discussion of sectorial (and hence non-self-adjoint) operators. It also naturally permits a study of selfadjoint operators (A, A0 ), where A is a small form perturbation of A0 , extending the traditional case in which A is a small (Kato–Rellich-type) operator perturbation of A0 . Our principal result in Section 2 then concerns a proof of the trace formula d − ln detH (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 dz (1.2) = trH (A − zIH )−1 − (A0 − zIH )−1 , an extension of the well-known operator perturbation case in which the symmetrized expression (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 (1.3) is replaced by the traditional expression (A − zIH )(A0 − zIH )−1
(1.4)
on the left-hand side of (1.2) (cf. Gohberg and Krein [35, Sect. IV.3]). The generalized trace formula (1.2) appears to be without precedent under our general hypothesis that A and A0 are operators of positive-type and hence seems to be of independent interest. Returning to the second principal aim of this paper, the discussion of boundary data maps for Schr¨ odinger operators on a compact interval with separated boundary conditions, let R > 0, introduce the strip S2π = {z ∈ C | 0 ≤ Re(z) < 2π}, and consider the boundary trace map 1 2 C ([0, " R]) → C , # γθ0 ,θR : θ0 , θR ∈ S2π , (1.5) cos(θ0 )u(0) + sin(θ0 )u0 (0) , u 7→ 0 cos(θR )u(R) − sin(θR )u (R) where “prime” denotes d/dx. In addition, assuming that V ∈ L1 ((0, R); dx)
(1.6)
(V is not assumed to be real-valued in Sections 1 and 3), one can introduce the family of one-dimensional Schr¨odinger operators Hθ0 ,θR in L2 ((0, R); dx) by Hθ0 ,θR f = −f 00 + V f, θ0 , θR ∈ S2π , f ∈ dom(Hθ0 ,θR ) = g ∈ L2 ((0, R); dx) g, g 0 ∈ AC([0, R]); γθ0 ,θR (g) = 0; (1.7) (−g 00 + V g) ∈ L2 ((0, R); dx) , where AC([0, R]) denotes the set of absolutely continuous functions on [0, R]. Assuming that z ∈ C\σ(Hθ0 ,θR ) (with σ(T ) denoting the spectrum of T ) and θ0 , θR ∈ S2π , we recall that the boundary value problem given by −u00 + V u = zu,
u, u0 ∈ AC([0, R]),
(1.8)
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
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γθ0 ,θR (u) =
c0 ∈ C2 , cR
(1.9)
has a unique solution denoted by u(z, ·) = u(z, · ; (θ0 , c0 ), (θR , cR )) for each c0 , cR ∈ C. To each boundary value problem (1.8), (1.9), we now associate a family of θ 0 ,θ 0 0 general boundary data maps, Λθ00 ,θR (z) : C2 → C2 , for θ0 , θR , θ00 , θR ∈ S2π , where R c θ 0 ,θ 0 θ 0 ,θ 0 Λθ00 ,θR (z) 0 = Λθ00 ,θR (z) γθ0 ,θR (u(z, · ; (θ0 , c0 ), (θR , cR ))) R R cR (1.10) = γθ00 ,θR0 (u(z, · ; (θ0 , c0 ), (θR , cR ))). θ 0 ,θ 0
With u(z, ·) = u(z, · ; (θ0 , c0 ), (θR , cR )), Λθ00 ,θR (z) can be represented as a 2 × 2 R complex matrix, where " # 0 0 cos(θ0 )u(z, 0) + sin(θ0 )u0 (z, 0) c0 θ00 ,θR θ00 ,θR Λθ0 ,θR (z) = Λθ0 ,θR (z) cR cos(θR )u(z, R) − sin(θR )u0 (z, R) # " cos(θ00 )u(z, 0) + sin(θ00 )u0 (z, 0) . (1.11) = 0 0 )u0 (z, R) )u(z, R) − sin(θR cos(θR θ 0 ,θ 0
The map Λθ00 ,θR (z), z ∈ C\σ(Hθ0 ,θR ), was the principal object studied in the R recent paper [16]. In Section 3 we recall the principal results of [16] most relevant to the present θ 0 ,θ 0 investigation. More precisely, we review the basic properties of Λθ00 ,θR (z), and deR θ 0 ,θ 0
tail the explicit representation of the boundary data maps Λθ00 ,θR (z) in terms of R the resolvent of the underlying Schr¨odinger operator Hθ0 ,θR . We discuss the associated boundary trace maps, associated linear fractional transformations relating the 0 θ 0 ,θ 0 δ 0 ,δR θ 0 ,θ 0 boundary data maps Λθ00 ,θR (z) and Λδ00 ,δR (z) and mention the fact that Λθ00 ,θR (·) R R is a matrix-valued Herglotz function (i.e., analytic on C+ , the open complex upper half-plane, with a nonnegative imaginary part) in the special case where Hθ0 ,θR is self-adjoint. We conclude our review of [16] with Krein-type resolvent formulas explicitely relating the resolvents of Hθ0 ,θR and Hθ00 ,θR0 . In Section 4, we focus on the second group of new results in this paper and θ 0 ,θ 0 relate Λθ00 ,θR (z) with the trace formula for the difference of resolvents of Hθ0 ,θR and R Hθ00 ,θR0 and the underlying perturbation determinants. In this context we will be assuming self-adjointness of Hθ0 ,θR and Hθ00 ,θR0 . More precisely, we will prove the following facts: detL2 ((0,R);dx) (Hθ00 ,θR0 − zI)1/2 (Hθ0 ,θR − zI)−1 (Hθ00 ,θR0 − zI)1/2 0 0 sin(θ0 ) sin(θR ) θ0 ,θR 2 Λ det (z) , (1.12) = C θ0 ,θR 0 ) sin(θ00 ) sin(θR 0 θ0 , θR ∈ [0, 2π), θ00 , θR ∈ (0, 2π)\{π}, z ∈ ρ(Hθ0 ,θR ), and trL2 ((0,R);dx) (Hθ00 ,θR0 − zI)−1 − (Hθ0 ,θR − zI)−1 0 0 (1.13) d θ ,θ = − ln detC2 Λθ00 ,θR (z) , z ∈ ρ(Hθ0 ,θR ) ∩ ρ(Hθ00 ,θR0 ). R dz For classical as well as recent fundamental literature on Weyl–Titchmarsh operators (i.e., spectral parameter dependent Dirichlet-to-Neumann maps, or more
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F. GESZTESY AND M. ZINCHENKO
generally, Robin-to-Robin maps, resp., Poincar´e–Steklov operators), relevant in the context of boundary value spaces (boundary triples, etc.), we refer, for instance, to [3]– [20], [28]–[31], [36], [37, Ch. 13], [66]–[70], and especially, to the extensive bibliography in [16]. Finally, we briefly summarize some of the notation used in this paper: Let H be a separable complex Hilbert space, (·, ·)H the scalar product in H (linear in the second argument), and IH the identity operator in H. Next, let T be a linear operator mapping (a subspace of) a Banach space into another, with dom(T ) and ker(T ) denoting the domain and kernel (i.e., null space) of T . The closure of a closable operator S is denoted by S. The spectrum essential spectrum, discrete spectrum, and resolvent set of a closed linear operator in H will be denoted by σ(·), σess (·), σd (·), and ρ(·), respectively. The Banach space of bounded linear operators on H is denoted by B(H), the analogous notation B(X1 , X2 ), will be used for bounded operators between two Banach spaces X1 and X2 . The Banach space of compact operators defined on H is denoted by B∞ (H) and the `p -based trace ideals are denoted by Bp (H), p ≥ 1. The Fredholm determinant for trace class perturbations of the identity in H is denoted by detH (·), the trace for trace class operators in H will be denoted by trH (·). 2. Symmetrized Perturbation Determinants and Trace Formulas: An Abstract Approach In this section we present our first group of new results, the connection between appropriate perturbation determinants and trace formulas in an abstract setting. Throughout this section, H denotes a complex, separable Hilbert space with inner product (·, ·)H , and IH represents the identity operator in H. For basic facts on trace ideals and infinite determinants we refer, for instance, to [33]–[35], [72], and [73]. We start with the following classical result: Theorem 2.1 ([35], p. 163). Let T (·) be analytic in the B1 (H)-norm on some open set Ω ⊆ C. Then detH (IH + T (·)) is analytic in Ω and d ln(detH (IH + T (z))) = trH (IH + T (z))−1 T 0 (z) , dz (2.1) z ∈ ζ ∈ Ω (IH + T (ζ))−1 ∈ B(H) . Next, we recall a classical special case in connection with standard perturbation determinants (cf. [35, Ch. IV]): Theorem 2.2 ([35], Sect. IV.3, [49]). Assume that A and A0 are densely defined, closed, linear operators in H satisfying dom(A0 ) ⊆ dom(A), (A − zIH ) (A − zIH )−1 − (A0 − zIH )−1 ∈ B1 (H) for some (and hence for all ) z ∈ ρ(A) ∩ ρ(A0 ) .
(2.2) (2.3)
Then −
d ln detH (A − zIH )(A0 − zIH )−1 = trH (A − zIH )−1 − (A0 − zIH )−1 , dz z ∈ ρ(A) ∩ ρ(A0 ). (2.4)
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
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Proof. For completeness, and since we intend to extend this type of result to certain quadratic form perturbations, we briefly sketch the proof of (2.4). Pick z ∈ ρ(A) ∩ ρ(A0 ). Since assumption (2.3) is equivalent to − (A − zIH ) (A − zIH )−1 − (A0 − zIH )−1 = (A − A0 )(A0 − zIH )−1 ∈ B1 (H), (2.5) the identity (A − zIH )(A0 − zIH )−1 = IH + (A − A0 )(A0 − zIH )−1 (2.6) −1 shows that detH (A−zIH )(A0 −zIH ) is well-defined and analytic for z ∈ ρ(A0 ). Incidentally, (2.5) also yields that if (2.3) is satisfied for some z ∈ ρ(A) ∩ ρ(A0 ), then it is satisfied for all z ∈ ρ(A) ∩ ρ(A0 ). An application of (2.1) and cyclicity of the trace (i.e., trH (ST ) = trH (T S) whenever S, T ∈ B(H) with ST, T S ∈ B1 (H), cf. [73, Corollary 3.8]) imply d − ln detH (A − zIH )(A0 − zIH )−1 dz −1 0 = −trH (A − zIH )(A0 − zIH )−1 (A − A0 )(A0 − zIH )−1 −1 = −trH (A − zIH )(A0 − zIH )−1 (A − A0 )(A0 − zIH )−2 −1 = −trH (A0 − zIH )−1 (A − zIH )(A0 − zIH )−1 (A − A0 )(A0 − zIH )−1 −1 = trH (A0 − zIH )−1 (A − zIH )(A0 − zIH )−1 × (A − zIH ) (A − zIH )−1 − (A0 − zIH )−1 −1 = trH (A0 − zIH )−1 (A − zIH )(A0 − zIH )−1 (A − zIH )(A0 − zIH )−1 × (A0 − zIH ) (A − zIH )−1 − (A0 − zIH )−1 = trH (A − zIH )−1 − (A0 − zIH )−1 . (2.7) For an extension of Theorem 2.2, applicable, in particular, to suitable quadratic form perturbations, we briefly recall a few basic facts on operators of positivetype and their fractional powers. While this theory has been fully developed in connection with complex Banach spaces, we continue to restrict ourselves here to the case of complex, separable Hilbert spaces. For details on this theory we refer, for instance, to [38, Chs. 2, 3, 7], [44, Ch. 4], [55, Ch. 4], [60, Chs. 1, 3–5], and [79, Chs. 2, 16]. Definition 2.3. Let A be a densely defined, closed, linear operator in H and denote by Sω ⊂ C, ω ∈ [0, π), the open sector ( {z ∈ C | z 6= 0, | arg(z)| < ω}, ω ∈ (0, π), Sω = (2.8) (0, ∞), ω = 0, with vertex at z = 0 along the positive real axis and opening angle 2ω. (i) A is said to be of nonnegative-type if (α) (−∞, 0) ⊂ ρ(A),
(β) M (A) = sup t(A + tIH )−1 t>0
B(H)
< ∞.
(2.9)
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F. GESZTESY AND M. ZINCHENKO
(ii) A is said to be of positive-type if (α) (−∞, 0] ⊂ ρ(A),
(β) MA = sup (1 + t)(A + tIH )−1
B(H)
t≥0
< ∞.
(2.10)
(iii) A is called sectorial of angle ω ∈ [0, π), denoted by A ∈ Sect(ω), if (α) σ(A) ⊆ Sω , (β) For all ω 0 ∈ (ω, π), M (A, ω 0 ) =
sup z(A − zIH )−1 B(H) < ∞.
(2.11)
z∈C\Sω0
(iv) A is called quasi-sectorial of angle ω ∈ [0, π) if there exists t0 ∈ R such that A + t0 IH is sectorial of angle ω ∈ [0, π). In this context we introduce the shifted sector −t0 + Sω , where ( {z ∈ C | z 6= −t0 , | arg(−t0 + z)| < ω}, ω ∈ (0, π), − t0 + Sω = (2.12) (−t0 , ∞), ω = 0. Next, we recall a number of useful facts: (I) If A is of nonnegative-type, then (cf., e.g., [38, Proposition 2.1.1 a)]) M (A) ≥ 1 and A ∈ Sect(π − arcsin(1/M (A))).
(2.13)
Moreover, if A is of nonnegative-type (resp., of positive-type) then A + tIH is of nonnegative-type (resp., of positive-type) for all t > 0.
(2.14)
If A is of positive-type, then (cf., e.g., [55, Lemma 4.2]) {z ∈ C | Re(z) ≤ 0, |Im(z)| < (|Re(z)| + 1)/MA } ∪ {z ∈ C | |z| < 1/MA } ⊂ ρ(A), (2.15) and for every ω0 ∈ (0, arctan(1/MA )), r0 ∈ (0, 1/MA ), there exists M0 (A, ω0 , r0 ) > 0 such that M0 (A, ω0 , r0 ) , 1 + |z| z ∈ {ζ ∈ C | Re(ζ) < 0, |Im(ζ)|/|Re(ζ)| ≤ tan(ω0 )} ∪ {ζ ∈ C | |ζ| ≤ r0 }. k(A − zIH )−1 kB(H) ≤
(2.16)
(II) If A ∈ Sect(ω) for some ω ∈ [0, π) and ker(A) = {0}, then (cf., e.g., [38, Proposition 2.1.1 b)]) A−1 ∈ Sect(ω) and M A−1 , ω 0 ≤ M (A, ω 0 ) + 1, ω 0 ∈ (ω, π). (2.17) (III) If A ∈ Sect(ω) for some ω ∈ [0, π), then (cf., e.g., [38, Proposition 2.1.1 j)]) A∗ ∈ Sect(ω) and M (A∗ , ω 0 = M (A, ω 0 ), ω 0 ∈ (ω, π). (2.18) (IV) Suppose A is of positive-type then (cf., e.g., [44, p. 280]) Z sin(πα) ∞ −α A = dt t−α (A + tIH )−1 ∈ B(H), 0 < Re(α) < 1. π 0
(2.19)
(In this context of bounded operators A−α , 0 < α < 1, and integrands bounded in norm by a Lebesgue integrable function, the integral in (2.19) and in analogous
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
7
situations in this section, is viewed as a norm convergent Bochner integral.) Moreover, A−α has an analytic continuation to the strip 0 < Re(α) < n + 1, n ∈ N, given by Z ∞ sin(πα) n! −α dt tn−α (A + tIH )−n−1 ∈ B(H), A = π (1 − α)(2 − α) · · · (n − α) 0 0 < Re(α) < n + 1. (2.20) In particular, A−α =
sin(πα) π(1 − α)
∞
Z
dt t1−α (A + tIH )−2 ∈ B(H),
0 < Re(α) < 2.
(2.21)
0
We also note that if A ∈ Sect(ω) and α ∈ (0, 1), then (cf., e.g., [38, Remark 3.1.16]) Aα ∈ Sect(αω), M (Aα ) ≤ M (A), and Z tα sin(πα) ∞ α −1 dt (A + tIH )−1 , (A − zIH ) = π (z − tα eiπα )(z − tα e−iπα ) (2.22) 0 | arg(z)| > απ. (V) Suppose A is of positive-type and 0 < Re(α) < n for some n ∈ N, then (cf., e.g., [55, Definition 4.5]) Aα f = An Aα−n f,
f ∈ dom(Aα ) = {g ∈ H | Aα−n g ∈ dom(An )}.
(2.23)
Moreover, dom(Aα ) = ran(A−α ) and Aα = (A−α )−1 , In particular, since A
−α
Re(α) > 0.
(2.24)
∈ B(H),
Aα is closed in H for all Re(α) > 0.
(2.25)
(VI) Suppose A is of positive-type and Re(α1 ) > 0, Re(α2 ) > 0, then (cf., e.g., [55, Proposition 4.4 (iv)]) A−α1 A−α2 = A−α1 −α2 . (2.26) (VII) Suppose A and B are of positive-type and resolvent commuting, that is, (A + sIH )−1 (B + tIH )−1 = (B + tIH )−1 (A + sIH )−1 for some (and hence for all) s > 0, t > 0.
(2.27)
Then (cf., e.g., [55, p. 95]) (AB)α f = Aα B α f = B α Aα f = (BA)α f, f ∈ dom((AB)α ) = {g ∈ dom(B α ) | B α g ∈ dom(Aα )} α
α
α
(2.28) α
= {g ∈ dom(A ) | A g ∈ dom(B )} = dom((BA) ),
α ∈ C, Re(α) 6= 0.
(VIII) In the special case where A is self-adjoint and strictly positive in H (i.e., A ≥ εIH for some ε > 0), Aα , α ∈ C\{0}, defined on one hand as operators of positive-type above, and on the other by the spectral theorem, coincide (cf., e.g., [55, Sect. 4.3.1], [74, Sect. 1.18.10]). In particular, Z dom(Aα ) = f ∈ H kAα f k2H = λ2Re(α) dkEA (λ)f k2H < ∞ , α ∈ C\{0}, [ε,∞]
(2.29) in this case. Here {EA (λ)}λ∈R denotes the family of spectral projections of A. (It is possible to extend some of these formulas to Re(α) = 0, but we omit the
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F. GESZTESY AND M. ZINCHENKO
details since this will play no role in this manuscript.) For the remainder of this section the basic assumptions on A and A0 , extending (2.2) and (2.3), then read as follows: Hypothesis 2.4. Let A and A0 be densely defined, closed, linear operators in H. (i) Suppose there exists t0 ∈ R such that A+t0 IH and A0 +t0 IH are of positive-type and (A + t0 IH ) ∈ Sect(ω0 ), (A0 + t0 IH ) ∈ Sect(ω0 ) for some ω0 ∈ [0, π). (ii) In addition, assume that for some t1 ≥ t0 , dom (A0 + t1 IH )1/2 ⊆ dom (A + t1 IH )1/2 , (2.30) ∗ 1/2 ∗ 1/2 dom (A0 + t1 IH ) ⊆ dom (A + t1 IH ) , (2.31) (A + t1 IH )1/2 (A + t1 IH )−1 − (A0 + t1 IH )−1 (A + t1 IH )1/2 ∈ B1 (H). (2.32) One observes by item (I), there always exists ω0 ∈ [0, π) as in Hypothesis 2.4 (i) as long as A + t0 IH and A0 + t0 IH are of nonnegative-type. Our next results will show that if (2.30)–(2.32) hold for some t1 ≥ t0 , then they actually extend to −t1 = z ∈ C\ −t0 + Sω0 : Lemma 2.5. Assume that A satisfies Hypothesis 2.4 (i). Then (A + tIH )−1/2 ∈ −1/2 B(H) (resp., (A∗ +tIH )−1/2 ∈ B(H)), t > t0 , analytically extends ∈ to (A−zIH ) ∗ −1/2 B(H) (resp., (A − zIH ) ∈ B(H)) for z ∈ C\ −t0 + Sω0 . In addition, 1/2 dom (A − zIH ) = dom (A + t0 IH )1/2 , z ∈ C\ −t0 + Sω0 , (2.33) ∗ 1/2 ∗ 1/2 dom (A − zIH ) = dom (A + t0 IH ) , z ∈ C\ −t0 + Sω0 . (2.34) Proof. Applying (2.19) with α = 1/2 and A replaced by (A + sIH ), s > t0 , one obtains Z 1 ∞ (A + sIH )−1/2 = dt t−1/2 (A + (s + t)IH )−1 , s > t0 . (2.35) π 0 The resolvent estimates in (2.10) and (2.11) then prove that (A + sIH )−1/2 , s > t0 , analytically extends to (A − zIH )−1/2 ∈ B(H), z ∈ C\ −t0 + Sω0 , with the result Z 1 ∞ −1/2 (A − zIH ) = dt t−1/2 (A + (−z + t)IH )−1 , z ∈ C\ −t0 + Sω0 . (2.36) π 0
In the following we choose z, z1 ∈ C\ −t0 + Sω0 such that |z1 − z| < (A −
−1 z1 IH )−1 B(H) and consider the resolvent identity (A − zIH ) = (A − z1 IH ) IH + (z1 − z)(A − z1 IH )−1 . (2.37) It follows from (A + t0 IH ) ∈ Sect(ω0 ) and (2.10), (2.11) that A − zIH , z ∈ C\ −t0 + Sω0 , is of positive-type.
(2.38)
To prove the claim (2.38) we first note that z ∈ C\ −t0 + Sω0 implies that 0 (−∞, 0] ⊂ ρ(A − zIH ). Next, one chooses ω0 ∈ (ω, π) such that actually, z ∈ C\ −t0 + Sω00 . Then
sup (1 + t)(A + (t − z)IH )−1 B(H) t≥0
(2.39) = sup |(1 + t)/ζ| ζ(A + (t0 − z)IH )−1 B(H) ≤ C(z) < ∞ ζ=z+t0 −t, t≥0
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
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since supt≥0 |(1 + t)/(z + t0 − t)| < ∞, the estimate (2.10) (β) becomes a special case of (2.11) (β).
In addition, (z1 − z)(A − z1 IH )−1 B(H) < 1 implies that B = IH + (z1 − z)(A − z1 IH )−1 is of positive-type as well since
−1 −1
(1 + t)(B + tIH )−1 + t) (1 + t)I + (z − z)(A − z I ) =
(1 H 1 1 H B(H) B(H)
h
i−1
z − z 1 −1
=
IH + 1 + t (A − z1 IH )
B(H) i−1 h
< ∞, t ≥ 0. (2.40) ≤ 1 − (z1 − z)(A − z1 IH )−1 B(H) Thus (V II) applied to the resolvent identity (2.37) yields −1/2 (A − zIH )−1/2 = (A − z1 IH )−1/2 IH + (z1 − z)(A − z1 IH )−1 ,
−1 z, z1 ∈ C\ −t0 + Sω0 , |z − z1 | < (A − z1 IH )−1 B(H) .
(2.41)
−1/2 Bounded invertibility of IH +(−z+z1 )(A−z1 IH )−1 for z, z1 ∈ C\ −t0 + Sω0 ,
−1 |z − z1 | < (A − z1 IH )−1 B(H) then implies that ran (A − zIH )−1/2 is locally constant in z ∈ C\ −t0 + Sω0 , ran (A − zIH )−1/2 = ran (A − z1 IH )−1/2 , (2.42)
−1 z, z1 ∈ C\ −t0 + Sω0 , |z − z1 | < (A − z1 IH )−1 B(H) , and hence that ran (A − zIH )−1/2 = ran (A − z1 IH )−1/2 ,
z, z1 ∈ C\ −t0 + Sω0 .
(2.43)
An application of (2.24) then gives (2.33). Equation (2.34) is proved analogously with the help of (III). Lemma 2.6. Assume that A and A0 satisfy Hypothesis 2.4 (i) and suppose that (2.30) and (2.31) hold for some t1 ≥ t0 . Then (2.30) and (2.31) extend to (2.44) dom (A0 − zIH )1/2 ⊆ dom (A − zIH )1/2 , z ∈ C\ −t0 + Sω0 , dom (A∗0 − zIH )1/2 ⊆ dom (A∗ − zIH )1/2 , z ∈ C\ −t0 + Sω0 . (2.45) Moreover, (A − zIH )1/2 (A0 − zIH )−1/2 ∈ B(H) and (A∗ − zIH )1/2 (A∗0 − zIH )−1/2 ∈ B(H), are analytic for z ∈ C\ −t0 + Sω0 with respect to the B(H)-norm. (2.46) Proof. By items (I) and (III) it again suffices to just focus on the proof of (2.44). Since by (2.33) and (2.34) the domains of (A − zIH )1/2 and (A∗ − zIH )1/2 are z independent for z ∈ C\ −t0 + Sω0 , (2.30) and (2.31) extend to z ∈ C\ −t0 + Sω0 . To prove the analyticity statement involving A and A0 in (2.46) we write (A − zIH )1/2 (A0 − zIH )−1/2 = (A − zIH )1/2 (A − z0 IH )−1/2 × (A − z0 IH )1/2 (A0 − z0 IH )−1/2 (A0 − z0 IH )1/2 (A0 − zIH )−1/2 , (2.47) z, z0 ∈ C\ −t0 + Sω0 ,
10
F. GESZTESY AND M. ZINCHENKO
and separately investigate each of the three factors in (2.47). Since by hypothesis (2.33) holds for A and A0 , (2.44) yields that (A − z0 IH )1/2 (A0 − z0 IH )−1/2 ∈ B(H).
(2.48)
Next, applying (2.41) with A replaced by A0 yields (A0 − z0 IH )1/2 (A0 − zIH )−1/2 −1/2 = (A0 − z0 IH )1/2 (A0 − z1 IH )−1/2 IH + (z1 − z)(A0 − z1 IH )−1 , (2.49)
−1 −1 z, z0 , z1 ∈ C\ −t0 + Sω0 , |z − z1 | < (A − z1 IH ) . B(H)
Since by (2.40) (with A replaced by A0 ) B = IH + (z1 − z)(A0 − z1 IH )−1 is of positive-type, it follows from (2.19) (with α = 1/2 and A replaced by B) and a geometric series expansion that −1/2 B −1/2 = IH + (z1 − z)(A0 − z1 IH )−1 Z −1 1 ∞ dt t−1/2 (1 + t)IH + (z1 − z)(A0 − z1 IH )−1 = π 0 Z ∞ X 1 ∞ dt t−1/2 = (−1)m (z1 − z)m (A0 − z1 IH )−m π 0 (1 + t)m+1 m=0 ∞ X Γ(m + (1/2)) (−1)m (z1 − z)m (A0 − z1 IH )−m . (2.50) Γ(m + 1)Γ(1/2) m=0 −1/2 Thus IH + (z1 − z)(A0 − z1 IH )−1 is analytic with respect to z for z, z1 ∈
−1
C\ −t0 + Sω0 , |z − z1 | < (A0 − z1 IH )−1 B(H) . Moreover, by (2.33) (with A
=
replaced by A0 ), (A0 − z0 IH )1/2 (A0 − z1 IH )−1/2 ∈ B(H), and hence one concludes that the left-hand side of (2.49) is analytic with respect to z ∈ C\ −t0 + Sω0 . Finally, writing (A − zIH )1/2 (A − z0 IH )−1/2 = (A − zIH )(A − zIH )−1/2 (A − z0 IH )−1/2 = A(A − zIH )−1/2 (A − z0 IH )−1/2 − z(A − zIH )−1/2 (A − z0 IH )−1/2
(2.51)
it suffices to focus on the term A(A − zIH )−1/2 (A − z0 IH )−1/2 . Writing A(A − zIH )−1/2 (A − z0 IH )−1/2 = A(A − z0 IH )−1 (A − z0 IH )1/2 (A − zIH )−1/2 ,
(2.52)
employing the obvious fact that A(A − z0 IH )−1 ∈ B(H), the analyticity of (A − z0 IH )1/2 (A − zIH )−1/2 (and hence that of the left-hand sides in (2.51) and (2.52)) with respect to z ∈ C\ −t0 + Sω0 then follows as in (2.49) above with A0 replaced by A. Lemma 2.7. Assume that A and A0 satisfy Hypothesis 2.4. Then (2.32) extends to (A − zIH )1/2 (A − zIH )−1 − (A0 − zIH )−1 (A − zIH )1/2 ∈ B1 (H), (2.53) z ∈ C\ −t0 + Sω0 . In addition, (A − zIH )1/2 (A − zIH )−1 − (A0 − zIH )−1 (A − zIH )1/2 is analytic for z ∈ C\ −t0 + Sω0 with respect to the B1 (H)-norm.
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
11
Proof. Let z ∈ C\ −t0 + Sω0 and t1 ≥ t0 as in (2.32). Using the fact (A − zIH )−1 − (A0 − zIH )−1 = (A + t1 IH )(A − zIH )−1 × (A + t1 IH )−1 − (A0 + t1 IH )−1 (A0 + t1 IH )(A0 − zIH )−1 ,
(2.54)
one obtains (A − zIH )1/2 (A − zIH )−1 − (A0 − zIH )−1 (A − zIH )1/2 = (A − zIH )1/2 (A + t1 IH )−1/2 (A + t1 IH )(A − zIH )−1 × cl (A + t1 IH )1/2 (A + t1 IH )−1 − (A0 + t1 IH )−1 (A + t1 IH )1/2 × (A + t1 IH )−1/2 (A0 + t1 IH )(A0 − zIH )−1 (A − zIH )1/2 = (A − zIH )1/2 (A + t1 IH )−1/2 (A + t1 IH )(A − zIH )−1 × (A + t1 IH )1/2 (A + t1 IH )−1 − (A0 + t1 IH )−1 (A + t1 IH )1/2 × (A − zIH )1/2 (A + t1 IH )−1/2 (2.55) ∗ −1/2 −1/2 1/2 ∗ −1/2 ∗ + (z + t)(A + t1 IH ) (A − zIH ) (A − zIH ) (A0 − zIH ) , where we employed the identity (A0 + t1 IH )(A0 − zIH )−1 = IH + (z + t1 )(A0 − zIH )−1
(2.56)
and used the symbol cl{. . . } to denote the operator closure (in addition to our usual bar symbol) as the latter extends over two lines. By Lemmas 2.6 and 2.7, all square brackets [· · · ] in (2.55) lie in B(H). Thus, the trace class property in assumption (2.32) proves that in (2.53). Finally, the analyticity statements in (2.46) (see also the one in (2.51)) employed in (2.55) prove the B1 (H)-analyticity of the operator in (2.53). Theorem 2.8. Assume that A and A0 satisfy Hypothesis 2.4.Then d − ln detH (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 dz (2.57) = trH (A − zIH )−1 − (A0 − zIH )−1 , for all z ∈ C\ −t0 + Sω0 such that (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 is boundedly invertible. Proof. Let z ∈ C\ −t0 + Sω0 . We note that by (2.46) one has (A − zIH )1/2 (A − zIH )−1 − (A0 − zIH )−1 (A − zIH )1/2 = IH − (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 (2.58) ∗ 1/2 −1/2 1/2 ∗ −1/2 ∗ = IH − (A − zIH ) (A0 − zIH ) (A − zIH ) (A0 − zIH ) and hence detH (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 is well-defined. Next, we consider T1 (z) = (A − zIH )1/2 (A0 − zIH )−1/2 , z ∈ C\ −t0 + Sω0 , (2.59) and compute for ε ∈ C\{0}, |ε| sufficiently small such that z, z+ε ∈ C\ −t0 + Sω0 , [T1 (z + ε) − T1 (z)] = (A − (z + ε)IH )(A − (z + ε)IH )−1/2 (A0 − (z + ε)IH )−1/2
12
F. GESZTESY AND M. ZINCHENKO
− (A − zIH )(A − zIH )−1/2 (A0 − zIH )−1/2 = (A − (z + ε)IH )(A − (z + ε)IH )−1/2 (A0 − (z + ε)IH )−1/2 − (A − zIH )(A − (z + ε)IH )−1/2 (A0 − (z + ε)IH )−1/2 + (A − zIH )(A − (z + ε)IH )−1/2 (A0 − (z + ε)IH )−1/2 − (A − zIH )(A − zIH )−1/2 (A0 − zIH )−1/2 = −ε(A − (z + ε)IH )−1/2 (A0 − (z + ε)IH )−1/2 + (A − zIH )(A − (z + ε)IH )−1/2 (A0 − (z + ε)IH )−1/2 − (A − zIH )(A − zIH )−1/2 (A0 − (z + ε)IH )−1/2 + (A − zIH )(A − zIH )−1/2 (A0 − (z + ε)IH )−1/2 − (A − zIH )(A − zIH )−1/2 (A0 − zIH )−1/2 = −ε(A − (z + ε)IH )−1/2 (A0 − (z + ε)IH )−1/2 + (A − zIH ) (A − (z + ε)IH )−1/2 − (A − zIH )−1/2 × (A0 − (z + ε)IH )−1/2
(2.60) + (A − zIH )(A − zIH )−1/2 (A0 − (z + ε)IH )−1/2 − (A0 − zIH )−1/2 . Using (A − (z + ε)IH )−1/2 − (A − zIH )−1/2 Z 1 ∞ = dt t−1/2 (A + (t − z − ε)IH )−1 − (A + (t − z)IH )−1 π 0 Z ε ∞ = dt t−1/2 (A + (t − z − ε)IH )−1 (A + (t − z)IH )−1 π 0
(2.61)
in (2.60) yields 1 [T1 (z + ε) − T1 (z)] = −(A − (z + ε)IH )−1/2 (A0 − (z + ε)IH )−1/2 ε Z ∞ 1 −1/2 −1 −1 dt t (A + (t − z − ε)IH ) (A + (t − z)IH ) + (A − zIH ) π 0 × (A0 − (z + ε)IH )−1/2 + (A − zIH )(A − zIH )−1/2 Z ∞ 1 × dt t−1/2 (A0 + (t − z − ε)IH )−1 (A0 + (t − z)IH )−1 π 0 −→ −(A − zIH )−1/2 (A0 − zIH )−1/2 ε→0 Z ∞ 1 + (A − zIH ) dt t−1/2 (A + (t − z)IH )−2 (A0 − zIH )−1/2 π 0 Z ∞ 1 + (A − zIH )(A − zIH )−1/2 dt t−1/2 (A0 + (t − z)IH )−2 π 0 = −(A − zIH )−1/2 (A0 − zIH )−1/2 1 + (A − zIH )(A − zIH )−3/2 (A0 − zIH )−1/2 2
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
13
1 + (A − zIH )1/2 (A0 − zIH )−3/2 2 1 = − (A − zIH )−1/2 (A0 − zIH )−1/2 2 1 + (A − zIH )1/2 (A0 − zIH )−3/2 , 2 (2.62) where the limit ε → 0 is valid in the B(H)-norm. Here we used (cf. (2.21) in the case α = 3/2 and with A replaced by (A − zIH )) Z 1 1 ∞ dt t−1/2 (A + (t − z)IH )−2 = (A − zIH )−3/2 , z ∈ C\ −t0 + Sω0 . (2.63) π 0 2 Thus, 1 1 T10 (z) = − (A − zIH )−1/2 (A0 − zIH )−1/2 + (A − zIH )1/2 (A0 − zIH )−3/2 , 2 2 z ∈ C\ −t0 + Sω0 . (2.64) Similarly, introducing ∗ T2 (z) = (A∗ − zIH )1/2 (A∗0 − zIH )−1/2 ,
z ∈ C\ −t0 + Sω0 ,
(2.65)
one obtains ∗ 1 ∗ (A − zIH )−1/2 (A0 − zIH )−1/2 2 (2.66) ∗ 1 + (A∗ − zIH )1/2 (A∗0 − zIH )−3/2 , z ∈ C\ −t0 + Sω0 . 2 Consequently, one computes i dh (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 dz = [T1 (z)T2 (z)]0 = T10 (z)T2 (z) + T1 (z)T20 (z) i h 1 1 = − (A − zIH )−1/2 (A0 − zIH )−1/2 + (A − zIH )1/2 (A0 − zIH )−3/2 2 2 ∗ × (A∗ − zIH )1/2 (A∗0 − zIH )−1/2 ∗ h 1 + (A − zIH )1/2 (A0 − zIH )−1/2 − (A∗ − zIH )−1/2 (A∗0 − zIH )−1/2 2 ∗ i 1 + (A∗ − zIH )1/2 (A∗0 − zIH )−3/2 2 1 = − (A − zIH )−1/2 (A0 − zIH )−1 (A − zIH )1/2 2 1 − (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )−1/2 2 + (A − zIH )1/2 (A0 − zIH )−2 (A − zIH )1/2 , z ∈ C\ −t0 + Sω0 . (2.67) T20 (z) = −
Due to the B1 (H)-analyticity of the left-hand side of (2.58) according to Lemma 2.7, one can apply (2.1), and using the result (2.67) one finally obtains d − ln detH (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 dz −1 = −trH (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2
14
F. GESZTESY AND M. ZINCHENKO
0 × (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 =
−1 1 trH (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 2 × (A − zIH )−1/2 (A0 − zIH )−1 (A − zIH )1/2 − 2(A − zIH )1/2 (A0 − zIH )−2 (A − zIH )1/2 + (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )−1/2
−1 1 trH (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 2 × (A − zIH )1/2 (A − zIH )−1 − (A0 − zIH )−1 × (A0 − zIH )−1 (A − zIH )1/2 −1 1 + trH (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 2 × (A − zIH )1/2 (A0 − zIH )−1 × (A − zIH )−1 − (A0 − zIH )−1 (A − zIH )1/2 −1 1 = trH (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 2 × (A − zIH )1/2 (A − zIH )−1 − (A0 − zIH )−1 × (A − zIH )−1/2 (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 −1 1 + trH (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 2 × (A − zIH )1/2 (A0 − zIH )−1 (A − zIH )1/2 (A − zIH )−1/2 × (A − zIH )−1 − (A0 − zIH )−1 (A − zIH )1/2 1 = trH (A − zIH )1/2 (A − zIH )−1 − (A0 − zIH )−1 (A − zIH )−1/2 2 1 + trH (A − zIH )−1/2 (A − zIH )−1 − (A0 − zIH )−1 (A − zIH )1/2 2 1 1 = trH (A − zIH )−1 − (A0 − zIH )−1 + trH (A − zIH )−1 − (A0 − zIH )−1 2 2 = trH (A − zIH )−1 − (A0 − zIH )−1 , z ∈ C\ −t0 + Sω0 . (2.68) =
Here we repeatedly used cyclicity of the trace. Remark 2.9. (i) Extensions of the standard perturbation determinant detH (A − zIH )(A0 − zIH )−1 = detH IH + (A − A0 )(A0 − zIH )−1
(2.69)
associated with the pair (A, A0 ) to certain symmetrized (sometimes called, modified) versions involving factorizations of A − A0 have been considered in [48] and [76, Sect. 8.1.4]. However, Theorem 2.8 appears to be of a more general nature and of independent interest. (ii) We emphasize the general nature of the hypotheses on A, A0 in Theorem 2.8. In particular, it covers the frequently encountered special case of self-adjoint operators A, A0 with A0 bounded from below and A a quadratic form perturbation of A0
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
15
with relative bound strictly less than one,in addition to the trace class requirement (2.32). In this case one has dom |A0 |1/2 = dom |A|1/2 . Actually, Theorem 2.8 permits the more general situation where the latter equality of form domains is replaced by dom |A0 |1/2 ⊆ dom |A|1/2 . The latter fact will have to be used in our application to one-dimensional Schr¨odinger operators on a compact interval in Section 4 in the case where the separated boundary conditions involve a Dirichlet boundary condition at one or both interval endpoints. (iii) Going beyond item (V III), we also note that Theorem 2.8 applies when A, A0 are (Dunford) spectral operators of scalar type [22, Ch. XVIII] (in the sense that their resolvent is similar to the resolvent of a self-adjoint operator) with real spectrum bounded from below. 3. Boundary Data Maps and Their Basic Properties This section is devoted to a brief review of boundary data maps as recently introduced in [16]. The results taken from [16] are presented without proof (for detailed proofs and for an extensive bibliography we refer to [16]). We will also present a few new results of boundary data maps in this section (and then of course supply proofs). Taking R > 0, and fixing θ0 , θR ∈ S2π , with S2π the strip S2π = {z ∈ C | 0 ≤ Re(z) < 2π},
(3.1)
we introduce the linear map γθ0 ,θR , the trace map associated with the boundary {0, R} of (0, R) and the parameters θ0 , θR , by 1 2 C ([0, " R]) → C , # θ0 , θR ∈ S2π , (3.2) γθ0 ,θR : cos(θ0 )u(0) + sin(θ0 )u0 (0) , u 7→ 0 cos(θR )u(R) − sin(θR )u (R) where “prime” denotes d/dx. We note, in particular, that the Dirichlet trace γD , and the Neumnann trace γN (in connection with the outward pointing unit normal vector at ∂(0, R) = {0, R}), are given by γD = γ0,0 = −γπ,π ,
γN = γ3π/2,3π/2 = −γπ/2,π/2 .
(3.3)
Next, assuming V ∈ L1 ((0, R); dx),
(3.4)
we introduce the following family of densely defined closed linear operators Hθ0 ,θR in L2 ((0, R); dx), Hθ0 ,θR f = −f 00 + V f, θ0 , θR ∈ S2π , f ∈ dom(Hθ0 ,θR ) = g ∈ L2 ((0, R); dx) g, g 0 ∈ AC([0, R]); γθ0 ,θR (g) = 0; (3.5) (−g 00 + V g) ∈ L2 ((0, R); dx) . Here AC([0, R]) denotes the set of absolutely continuous functions on [0, R]. We remark that V is not assumed to be real-valued in this section. It is well-known that the spectrum of Hθ0 ,θR , σ(Hθ0 ,θR ) is purely discrete, σ(Hθ0 ,θR ) = σd (Hθ0 ,θR ),
θ0 , θR ∈ S2π .
(3.6)
16
F. GESZTESY AND M. ZINCHENKO
In addition, the resolvent of Hθ0 ,θR is a Hilbert–Schmidt operator in L2 ((0, R); dx) and the eigenvalues Eθ0 ,θR ,n of Hθ0 ,θR , in the case of the separated boundary conditions at hand, are of the form Eθ0 ,θR ,n = [(nπ/R) + (an /n)]2 with {an }n∈N ∈ `∞ (N), n→∞
(3.7)
as shown in [59, Lemma 1.3.3]. Moreover, Hθ0 ,θR is known to be m-sectorial (cf. [23, Sect. III.6], [41, Sect. VI.2.4]). One notices that γ(θ0 +π) mod(2π),(θR +π) mod(2π) = −γθ0 ,θR ,
θ0 , θR ∈ S2π ,
(3.8)
θ0 , θR ∈ S2π ,
(3.9)
and, on the other hand, H(θ0 +π) mod(2π),(θR +π) mod(2π) = Hθ0 ,θR ,
hence it suffices to consider θ0 , θR ∈ Sπ = {z ∈ C | 0 ≤ Re(z) < π} rather than θ0 , θR ∈ S2π in connection with Hθ0 ,θR , but for simplicity of notation we will keep using the strip S2π throughout this manuscript. The adjoint of Hθ0 ,θR is given by (Hθ0 ,θR )∗ f = −f 00 + V f, θ0 , θR ∈ S2π , f ∈ dom (Hθ0 ,θR )∗ = g ∈ L2 ((0, R); dx) g, g 0 ∈ AC([0, R]); γθ0 ,θR (g) = 0; (−g 00 + V g) ∈ L2 ((0, R); dx) . (3.10) Having described the operator Hθ0 ,θR in some detail, still assuming (3.4), we now briefly recall the corresponding closed, sectorial, and densely defined sesquilinear form, denoted by QHθ0 ,θR , associated with Hθ0 ,θR (cf. [41, p. 312, 321, 327–328]): Z R QHθ0 ,θR (f, g) = dx f 0 (x)g 0 (x) + V (x)f (x)g(x) 0
− cot(θ0 )f (0)g(0) − cot(θR )f (R)g(R),
(3.11)
1
f, g ∈ dom(QHθ0 ,θR ) = H ((0, R)) = h ∈ L2 ((0, R); dx) | h ∈ AC([0, R]); h0 ∈ L2 ((0, R); dx) , θ0 , θR ∈ S2π \{0, π}, R
Z QH0,θR (f, g) =
dx f 0 (x)g 0 (x) + V (x)f (x)g(x) − cot(θR )f (R)g(R),
(3.12)
0
f, g ∈ dom(QH0,θR ) = h ∈ L2 ((0, R); dx) | h ∈ AC([0, R]); h(0) = 0; h0 ∈ L2 ((0, R); dx) , θR ∈ S2π \{0, π}, R
Z QHθ0 ,0 (f, g) =
dx f 0 (x)g 0 (x) + V (x)f (x)g(x) − cot(θ0 )f (0)g(0),
(3.13)
0
f, g ∈ dom(QHθ0 ,0 ) = h ∈ L2 ((0, R); dx) | h ∈ AC([0, R]); h(R) = 0; h0 ∈ L2 ((0, R); dx) , θ0 ∈ S2π \{0, π}, Z QH0,0 (f, g) = 0
R
dx f 0 (x)g 0 (x) + V (x)f (x)g(x) ,
(3.14)
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
17
f, g ∈ dom(QH0,0 ) = dom |H0,0 |1/2 = H01 ((0, R)) = h ∈ L2 ((0, R); dx) | h ∈ AC([0, R]); h(0) = 0, h(R) = 0; h0 ∈ L2 ((0, R); dx) . Next, we recall the following elementary, yet fundamental, fact: Lemma 3.1. Suppose that V ∈ L1 ((0, R); dx), fix θ0 , θR ∈ S2π , and assume that z ∈ C\σ(Hθ0 ,θR ). Then the boundary value problem given by −u00 + V u = zu, u, u0 ∈ AC([0, R]), c γθ0 ,θR (u) = 0 ∈ C2 , cR
(3.15) (3.16)
has a unique solution u(z, ·) = u(z, · ; (θ0 , c0 ), (θR , cR )) for each c0 , cR ∈ C. Assuming z ∈ ρ(Hθ0 ,θR ), a basis for the solutions of (3.15) is given by u−,θ0 (z, ·) = u(z, · ; (θ0 , 0), (0, 1)),
(3.17)
u+,θR (z, ·) = u(z, · ; (0, 1), (θR , 0)). Explicitly, one then has u−,θ0 (z, R) = 1, u+,θR (z, 0) = 1,
cos(θ0 )u−,θ0 (z, 0) + sin(θ0 )u0−,θ0 (z, 0) = 0, cos(θR )u+,θR (z, R) −
sin(θR )u0+,θR (z, R)
= 0.
(3.18) (3.19)
Recalling the Wronskian of two functions f and g, W (f, g)(x) = f (x)g 0 (x) − f 0 (x)g(x),
f, g ∈ C 1 ([0, R]),
(3.20)
one then computes W (u+,θR (z, ·), u−,θ0 (z, ·)) = u0−,θ0 (z, 0) − u0+,θR (z, 0)u−,θ0 (z, 0) =
u+,θR (z, R)u0−,θ0 (z, R)
−
u0+,θR (z, R).
(3.21) (3.22)
To each boundary value problem (3.15), (3.16), we now associate a family of θ 0 ,θ 0 0 general boundary data maps, Λθ00 ,θR ∈ S2π , where (z) : C2 → C2 , for θ0 , θR , θ00 , θR R c θ 0 ,θ 0 θ 0 ,θ 0 Λθ00 ,θR (z) 0 = Λθ00 ,θR (z) γθ0 ,θR (u(z, · ; (θ0 , c0 ), (θR , cR ))) R R cR (3.23) = γθ00 ,θR0 (u(z, · ; (θ0 , c0 ), (θR , cR ))). θ 0 ,θ 0
With u(z, ·) = u(z, · ; (θ0 , c0 ), (θR , cR )), then Λθ00 ,θR (z) can be represented as a 2 × 2 R complex matrix, where " # 0 0 cos(θ0 )u(z, 0) + sin(θ0 )u0 (z, 0) c0 θ00 ,θR θ00 ,θR Λθ0 ,θR (z) = Λθ0 ,θR (z) cR cos(θR )u(z, R) − sin(θR )u0 (z, R) # " cos(θ00 )u(z, 0) + sin(θ00 )u0 (z, 0) . (3.24) = 0 0 cos(θR )u(z, R) − sin(θR )u0 (z, R) θ 0 ,θ 0
One can show that Λθ00 ,θR is well-defined for z ∈ ρ(Hθ0 ,θR ), that is, it is invariant R with respect to a change of basis of solutions of (3.15) (cf. [16, Theorem 2.3]). 0 00 Moreover, one has the following facts: Let θ0 , θR , θ00 , θR , θ000 , θR ∈ S2π . Then, with 2 I2 denoting the identity matrix in C , R Λθθ00 ,θ ,θR (z) = I2 ,
z ∈ ρ(Hθ0 ,θR ),
(3.25)
18
F. GESZTESY AND M. ZINCHENKO θ 00 ,θ 00
θ 0 ,θ 0
θ 00 ,θ 00
Λθ00 ,θ0R (z)Λθ00 ,θR (z) = Λθ00 ,θRR (z), z ∈ ρ(Hθ0 ,θR ) ∩ ρ(Hθ00 ,θR0 ), R 0 R h 0 0 i−1 θ0 ,θR R Λθθ00 ,θ (z) = Λ (z) , z ∈ ρ(Hθ0 ,θR ) ∩ ρ(Hθ00 ,θR0 ). 0 θ0 ,θR ,θ 0
(3.26) (3.27)
R
θ 0 ,θ 0
Remark 3.2. Even though Λθ00 ,θR is invariant with respect to a change of basis for R θ 0 ,θ 0
the solutions of (3.15), the representation of Λθ00 ,θR with respect to a specific basis R can be simplified considerably with an appropriate choice of basis. For example, by choosing the basis given in (3.17) one obtains, h 0 0 i θ 0 ,θ 0 θ0 ,θR Λθ00 ,θR (z) = Λ (z) , z ∈ ρ(Hθ0 ,θR ), (3.28) θ0 ,θR R j,k 1≤j,k≤2
0 0 θ ,θ Λθ00 ,θR (z) R
θ 0 ,θ 0 Λθ00 ,θR (z) R
θ 0 ,θ 0 Λθ00 ,θR (z) R
0 θ00 ,θR θ0 ,θR
Λ
= 1,1
cos(θ00 )
+ sin(θ00 )u0+,θR (z, 0) , cos(θ0 ) + sin(θ0 )u0+,θR (z, 0)
1,2
cos(θ00 )u−,θ0 (z, 0) + sin(θ00 )u0−,θ0 (z, 0) , cos(θR ) − sin(θR )u0−,θ0 (z, R)
2,1
0 0 )u0+,θR (z, R) )u+,θR (z, R) − sin(θR cos(θR , = cos(θ0 ) + sin(θ0 )u0+,θR (z, 0)
=
(z)
= 2,2
(3.29)
0 0 )u0−,θ0 (z, R) ) − sin(θR cos(θR . cos(θR ) − sin(θR )u0−,θ0 (z, R)
In particular, by (3.18) and (3.19), θ ,θ 0 Λθ00 ,θR (z) = 0, R 1,2
0 θ ,θ Λθ00 ,θR (z) R
= 0.
(3.30)
2,1
π π
,
2 2 Remark 3.3. We note that Λ0,0 (z) represents the Dirichlet-to-Neumann map, ΛD,N (z), for the boundary value problem (3.15), (3.16); that is, when θ0 = θR = 0, 0 = π/2, then (3.24) becomes θ00 = θR π π u(z, 0) u(z, 0) u0 (z, 0) 2,2 ΛD,N (z) = Λ0,0 (z) = , z ∈ ρ(H0,0 ), (3.31) u(z, R) u(z, R) −u0 (z, R)
with u(z, ·) = u(z, · ; (0, c0 ), (0, cR )), u(z, 0) = c0 , u(z, R) = cR . The Dirichlet-toNeumann map in the case V = 0 has recently been considered in [67, Example 5.1]. −1 The Neumann-to-Dirichlet map ΛN,D (z) = Λπ,π in the case π/2,π/2 (z) = −[ΛD,N (z)] V = 0 has earlier been computed in [19, Example 4.1]. We also refer to [8], [12], [18] in the intimately related context of Q and M -functions. We continue with an elementary result needed in the proof of Lemma 3.4, but first we introduce a convenient basis of solutions associated with the Schr¨odinger equation (3.15): Fix z ∈ C and let θ(z, ·), θ0 (z, ·), φ(z, ·), φ0 (z, ·) ∈ AC([0, R]), and such that θ(z, ·) and φ(z, ·) are solutions of − u00 + V u = zu,
(3.32)
uniquely determined by their initial values at x = 0, θ(z, 0) = φ0 (z, 0) = 1,
θ0 (z, 0) = φ(z, 0) = 0.
(3.33)
In particular, θ(z, ·) and φ(z, ·) are entire with respect to z. Introducing ψ(z, ·) = Aθ(z, ·) + Bφ(z, ·),
A, B ∈ C,
(3.34)
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
19
it follows that "
# cos(θ0 )ψ(z, 0) + sin(θ0 )ψ 0 (z, 0) γθ0 ,θR (ψ) = = 0 ∈ C2 cos(θR )ψ(z, R) − sin(θR )ψ 0 (z, R) is equivalent to "
cos(θ0 ) 0= cos(θR )θ(z, R) − sin(θR )θ0 (z, R) A = U(z, R, θ0 , θR ) . B
(3.35)
#" # sin(θ0 ) A 0 cos(θR )φ(z, R) − sin(θR )φ (z, R) B (3.36)
Consequently, introducing the determinant ∆ defined by ∆(z, R, θ0 , θR ) = det U(z, R, θ0 , θR ) ,
(3.37)
one concludes that z0 is an eigenvalue of Hθ0 ,θR ⇐⇒ z0 is a zero of the determinant ∆(·, R, θ0 , θR ). (3.38) Moreover, ∆ is an entire function with respect to z, and an explicit computation reveals that ∆(z, R, θ0 , θR ) = cos(θ0 ) cos(θR )φ(z, R) − cos(θ0 ) sin(θR )φ0 (z, R) (3.39) − sin(θ0 ) cos(θR )θ(z, R) + sin(θ0 ) sin(θR )θ0 (z, R). In addition, we point out that the function ∆ is closely related to the usual Wronskian of two solution u±,θ0 ,θR of (3.32) satisfying the boundary conditions cos(θ0 )u+,θ0 ,θR (z, 0) + sin(θ0 )u0+,θ0 ,θR (z, 0) = 1, cos(θR )u+,θ0 ,θR (z, R) − sin(θR )u0+,θ0 ,θR (z, R) cos(θ0 )u−,θ0 ,θR (z, 0) + sin(θ0 )u0−,θ0 ,θR (z, 0) cos(θR )u−,θ0 ,θR (z, R) − sin(θR )u0−,θ0 ,θR (z, R) In vector form, these boundary conditions correspond to γθ0 ,θR (u+,θ0 ,θR ) γθ0 ,θR (u−,θ0 ,θR ) = I2 . Since γθ0 ,θR is a linear map, it follows from (3.34) that A γθ0 ,θR (ψ) = γθ0 ,θR (θ) γθ0 ,θR (φ) , B
(3.40)
= 0,
(3.41)
= 0,
(3.42)
= 1.
(3.43) (3.44)
(3.45)
hence by (3.36) U(z, R, θ0 , θR ) = γθ0 ,θR (θ) γθ0 ,θR (φ) .
(3.46)
Using (3.44) and the linearity of γθ0 ,θR once again one concludes that −1 u+,θ0 ,θR (z, x) u−,θ0 ,θR (z, x) = θ(z, x) φ(z, x) γθ0 ,θR (θ) γθ0 ,θR (φ) (3.47) since both sides solve (3.32) and satisfy the same boundary condition. Thus, (3.46) and (3.47) yield W (u+,θ0 ,θR (z, ·), u−,θ0 ,θR (z, ·)) = W (θ(z, ·), φ(z, ·)) det(U(z, R, θ0 , θR )−1 ) = ∆(z, R, θ0 , θR )−1 ,
(3.48)
where W (·, ·) denotes the Wronskian of two functions as introduced in (3.20).
20
F. GESZTESY AND M. ZINCHENKO
0 Lemma 3.4. Assume that θ0 , θR , θ00 , θR ∈ S2π , and let Hθ0 ,θR and Hθ00 ,θR0 be defined as in (3.5). Then, with ∆(·, R, θ0 , θR ) introduced in (3.37), 0 0 ∆(z, R, θ0 , θ0 ) θ ,θ 0 R detC2 Λθ00 ,θR (z) = , z ∈ ρ(Hθ0 ,θR ). (3.49) R ∆(z, R, θ0 , θR )
Proof. We recall the formula −1 θ 0 ,θ 0 Λθ00 ,θR , (z) = γθ00 ,θR0 (θ(z, ·)) γθ00 ,θR0 (φ(z, ·)) γθ0 ,θR (θ(z, ·)) γθ0 ,θR (φ(z, ·) R (3.50) established in the proof of Theorem 2.3 in [16]. Since γθ0 ,θR (θ(z, ·)) γθ0 ,θR (φ(z, ·) (3.51) cos(θ0 ) sin(θ0 ) = , cos(θR )θ(z, R) − sin(θR )θ0 (z, R) cos(θR )φ(z, R) − sin(θR )φ0 (z, R) one verifies detC2
γθ0 ,θR (θ(z, ·)) γθ0 ,θR (φ(z, ·) = ∆(z, R, θ0 , θR ),
z ∈ C,
and hence (3.49).
(3.52)
The following asymptotic expansion results will be used in the proof of Theorem 5.3: 0 Lemma 3.5. Assume that θ0 , θR ∈ S2π , θ00 , θR ∈ S2π \{0, π}, and let Hθ0 ,θR and 0 0 Hθ0 ,θR be defined as in (3.5). Then, 0 0 θ ,θ detC2 Λθ00 ,θR (z) R sin(θ0 ) sin(θ0 ) −1/2 0 R ), θ0 , θR ∈ S2π \{0, π}, sin(θ0 ) sin(θR ) + O(|z| 0 0 sin(θ ) sin(θ ) − 0 R |z|1/2 + O(1), θ0 = 0, θR ∈ S2π \{0, π}, sin(θR ) (3.53) = 0 0 sin(θ0 ) sin(θR ) z↓−∞ 1/2 − sin(θ0 ) |z| + O(1), θ0 ∈ S2π \{0, π}, θR = 0, 0 )|z| + O(|z|1/2 ), θ0 = θR = 0. sin(θ00 ) sin(θR
Proof. The standard Volterra integral equations Z x sin(z 1/2 (x − x0 )) θ(z, x) = cos(z 1/2 x) + dx0 V (x0 )θ(z, x0 ), z 1/2 0 Z x sin(z 1/2 x) sin(z 1/2 (x − x0 )) φ(z, x) = + dx0 V (x0 )φ(z, x0 ), 1/2 z z 1/2 0
(3.54) (3.55)
z ∈ C, Im(z 1/2 ) ≥ 0, x ∈ [0, R], readily imply that 1/2 cos(z 1/2 x) + O |z|−1/2 eIm(z )x , |z|→∞ 1/2 θ0 (z, x) = −z 1/2 sin(z 1/2 x) + O eIm(z )x , θ(z, x)
=
|z|→∞
1/2 sin(z 1/2 x) + O |z|−1 eIm(z )x , 1/2 |z|→∞ z 1/2 φ0 (z, x) = cos(z 1/2 x) + O |z|−1/2 eIm(z )x . φ(z, x)
=
|z|→∞
(3.56)
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
21
An insertion of (3.56) into (3.39) then yields 1/2 1/2 2−1 sin(θ0 ) sin(θR )|z|1/2 eIm(z ) + O eIm(z R) , θ0 , θR ∈ S2π \{0, π}, 1/2 −1 Im(z ) −1/2 Im(z 1/2 R) −2 sin(θ )e + O |z| e , R θ0 = 0, θR ∈ S2π \{0, π}, ∆(z, R, θ0 , θR ) = 1/2 1/2 z↓−∞ −2−1 sin(θ0 )eIm(z ) + O |z|−1/2 eIm(z R) , θ0 ∈ S2π \{0, π}, θR = 0, 1/2 1/2 2−1 |z|−1/2 eIm(z ) + O |z|−1 eIm(z )R , θ = θ = 0. 0
R
(3.57) Finally, combining (3.49) and (3.57) proves (3.53).
θ 0 ,θ 0
Next, we recall an explicit formula for Λθ00 ,θR (z) in terms of the resolvent (Hθ0 ,θR − R −1 zI of Hθ0 ,θR and the boundary traces γθ00 ,θR0 . We start with the Green’s function for the operator Hθ0 ,θR in (3.5), Gθ0 ,θR (z, x, x0 ) = (Hθ0 ,θR − zI)−1 (x, x0 ) ( u−,θ0 (z, x0 )u+,θR (z, x), 0 6 x0 6 x, 1 = W (u+,θR (z, ·), u−,θ0 (z, ·)) u−,θ0 (z, x)u+,θR (z, x0 ), 0 6 x 6 x0 ,
(3.58)
z ∈ ρ(Hθ0 ,θR ), x, x0 ∈ [0, R]. Here u+,θR (z, ·), u−,θ0 (z, ·) is a basis for solutions of (3.15) as described in (3.17) and we denote by I = IL2 ((0,R);dx) the identity operator in L2 ((0, R); dx). Thus, one obtains Z R (Hθ0 ,θR − zI)−1 g (x) = dx0 Gθ0 ,θR (z, x, x0 )g(x0 ), (3.59) 0 2 g ∈ L ((0, R); dx), z ∈ ρ(Hθ0 ,θR ), x ∈ (0, R). For future purposes we now introduce the following 2 × 2 matrix sin(θ0 ) 0 Sθ0 ,θR = . 0 sin(θR )
(3.60)
0 Theorem 3.6. Assume that θ0 , θR , θ00 , θR ∈ S2π and let Hθ0 ,θR be defined as in (3.5). Then ∗ θ 0 ,θ 0 Λθ00 ,θR (z)Sθ00 −θ0 ,θR0 −θR = γθ00 ,θR0 γθ0 ,θ0 ((Hθ0 ,θR )∗ − zI)−1 , z ∈ ρ(Hθ0 ,θR ). R 0 R (3.61) θ 0 ,θ 0
δ 0 ,δ 0
The fact that Λθ00 ,θR (z) and Λδ00 ,δR (z) are related by a linear fractional transforR R mation is recalled next: 0 0 Theorem 3.7. Assume that θ0 , θR , θ00 , θR , δ0 , δR , δ00 , δR ∈ S2π , δ00 − δ0 6= 0 mod(π), 0 δR − δR 6= 0 mod(π), and that z ∈ ρ(Hθ0 ,θR ) ∩ ρ(Hδ0 ,δR ). Then, with Sθ0 ,θR defined
22
F. GESZTESY AND M. ZINCHENKO
as in (3.60), −1 θ 0 ,θ 0 δ 0 ,δ 0 Λθ00 ,θR (z) = Sδ00 −δ0 ,δR0 −δR Sδ00 −θ00 ,δR0 −θR0 + Sθ00 −δ0 ,θR0 −δR Λδ00 ,δR (z) R R −1 δ 0 ,δ 0 × Sδ00 −θ0 ,δR0 −θR + Sθ0 −δ0 ,θR −δR Λδ00 ,δR (z) Sδ00 −δ0 ,δR0 −δR . R
(3.62)
0 If in addition, θR − θR 6= 0 mod(π), δ00 − δ0 6= 0 mod(π), then θ 0 ,θ 0
Λθ00 ,θR (z)Sθ00 −θ0 ,θR0 −θR R h i −1 δ 0 ,δ 0 = Sδ00 −θ00 ,δR0 −θR0 + Sδ00 −δ0 ,δR0 −δR Sθ00 −δ0 ,θR0 −δR Λδ00 ,δR (z)Sδ00 −δ0 ,δR0 −δR R h −1 −1 −1 × Sθ00 −θ0 ,θR0 −θR Sδ00 −θ0 ,δR0 −θR + Sθ00 −θ0 ,θR0 −θR Sδ00 −δ0 ,δR0 −δR i−1 δ 0 ,δ 0 0 −δ ,δ 0 −δ × Sθ0 −δ0 ,θR −δR Λδ00 ,δR (z)S . (3.63) δ 0 R R 0 R We denote by C+ the open complex upper half-plane and abbreviate Im(L) = (L − L∗ )/(2i) for L ∈ Cn×n , n ∈ N. In addition, dkΣkC2×2 will denote the total variation of the 2 × 2 matrix-valued measure dΣ below in (3.65). The matrix M (·) is called an n × n matrix-valued Herglotz function if it is analytic on C+ and Im(M (z)) ≥ 0 for all z ∈ C+ . Now we are in position to recall θ 0 ,θ 0 the fundamental Herglotz property of the matrix Λθ00 ,θR (·)Sθ00 −θ0 ,θR0 −θR in the case R where Hθ0 ,θR is self-adjoint: 0 0 −θR 6= 0 mod(π), ∈ [0, 2π), θ00 −θ0 6= 0 mod(π), θR Theorem 3.8. Let θ0 , θR , θ00 , θR z ∈ ρ(Hθ0 ,θR ), and Hθ0 ,θR be defined as in (3.5). In addition, suppose that V is θ 0 ,θ 0
real-valued (and hence Hθ0 ,θR is self-adjoint ). Then Λθ00 ,θR (·)Sθ00 −θ0 ,θR0 −θR is a 2×2 R matrix-valued Herglotz function admitting the representation Z 0 0 0 λ 1 θ00 ,θR θ00 ,θR θ00 ,θR − , (3.64) Λθ0 ,θR (z)Sθ00 −θ0 ,θR0 −θR = Ξθ0 ,θR + dΣθ0 ,θR (λ) λ−z 1 + λ2 R z ∈ ρ(Hθ0 ,θR ),
0 0 Z θ0 ,θR 0 0 ∗ d Σθ0 ,θR (λ) C2×2 θ 0 ,θ 0 θ0 ,θR 2×2 < ∞, (3.65) Ξθ00 ,θR = Ξ ∈ C , θ0 ,θR R 1 + λ2 R where θ 0 ,θ 0 Σθ00 ,θR ((λ1 , λ2 ]) R
1 = lim lim π δ↓0 ε↓0
Z
λ2 +δ
λ1 +δ
0 0 θ ,θ 0 −θ ,θ 0 −θ dλ Im Λθ00 ,θR (λ + iε)S , θ 0 R 0 R R
(3.66)
λ1 , λ2 ∈ R, λ1 < λ2 . In addition, 0 0 θ ,θ 0 −θ ,θ 0 −θ Im Λθ00 ,θR (z)S > 0, θ 0 R R 0 R
z ∈ C+ ,
(3.67)
θ 0 ,θ 0 supp dΣθ00 ,θR ⊆ σ(Hθ0 ,θR ) ∪ σ(Hθ00 ,θR0 ) , R
(3.68)
and θ 0 ,θ 0
in particular, Λθ00 ,θR (·)Sθ00 −θ0 ,θR0 −θR is self-adjoint on R ∩ ρ(Hθ0 ,θR ) ∩ ρ(Hθ00 ,θR0 ). R We note that relation (3.68) is a consequence of (3.49) and of the fact that is a meromorphic Herglotz matrix.
θ 0 ,θ 0 Λθ00 ,θR (·)Sθ00 −θ0 ,θR0 −θR R
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
23
0 0 Remark 3.9. If θ00 − θ0 = 0 mod(π) and θR − θR 6= 0 mod(π) (resp., θR − θR = 0 θ ,θ 0 R 0 0 mod(π) and θ0 − θ0 6= 0 mod(π)) then (3.30) shows that Λθ0 ,θR (·)S0,θR0 −θR (resp., θ 0 ,θ
Λθ00 ,θR (·)Sθ00 −θ0 ,0 ) is a diagonal matrix of the form R 0 0 mθ0 (·) 0 θ ,θ 0 θ00 ,θR 0 0 Λθ00 ,θR (·)S = resp., Λ (·)S = , 0,θR −θR θ0 −θ0 ,0 θ0 ,θR R 0 mθR (·) 0 0 (3.69) with mθR (·) (resp., mθ0 (·)) a scalar Herglotz function. Finally, we briefly turn to a discussion of Krein-type resolvent formulas for the difference of resolvents of Hθ00 ,θR0 and Hθ0 ,θR : 0 Lemma 3.10. Assume that θ0 , θR , θ00 , θR ∈ S2π , let Hθ0 ,θR be defined as in (3.5), and suppose that z ∈ ρ(Hθ0 ,θR ). Then, assuming f ∈ L2 ((0, R); dx), and writing # " γθ00 ,θR0 (Hθ0 ,θR − zI)−1 1 f γθ00 ,θR0 (Hθ0 ,θR − zI)−1 f = ∈ C2 , (3.70) γθ00 ,θR0 (Hθ0 ,θR − zI)−1 2 f
one has γθ00 ,θR0 (Hθ0 ,θR − zI)−1 1 f =
sin(θ00 − θ0 ) u+,θR (z, ·), f L2 ((0,R);dx) W (u+,θR (z, ·), u−,θ0 (z, ·)) − u−,θ0 (z,0) , θ0 ∈ S2π \{0, π}, 0) × u0−,θsin(θ (3.71) (z,0) 0 , θ0 ∈ S2π \{π/2, 3π/2}, cos(θ0 )
γθ00 ,θR0 (Hθ0 ,θR − zI)
−1 2
f=
0 − θR ) − sin(θR u−,θ0 (z, ·), f L2 ((0,R);dx) W (u+,θR (z, ·), u−,θ0 (z, ·)) u+,θR (z,R) , θR ∈ S2π \{0, π}, sin(θR ) × u0+,θ (3.72) (z,R) R , θR ∈ S2π \{π/2, 3π/2}, cos(θR )
in addition, γθ0 ,θR (Hθ0 ,θR − zI)−1 = 0 in B L2 ((0, R); dx), C2 , γθ0 ,θR (Hθ0 ,θR − zI)−1 k = 0 in B L2 ((0, R); dx), C , k = 1, 2. Introducing the orthogonal projections in C2 , 1 0 0 P1 = , P2 = 0 0 0
0 , 1
(3.73) (3.74)
(3.75)
one obtains the following Krein-type resolvent formulas (cf. [1, Ch. 8], [3], [9]–[16], [25], [27]–[30], [36], [45]–[47], [57], [58], [64], [67], [68], [71]): 0 Theorem 3.11. Assume that θ0 , θR , θ00 , θR ∈ S2π , let Hθ0 ,θR and Hθ00 ,θR0 be defined θ 0 ,θ 0
as in (3.5), and suppose that z ∈ ρ(Hθ0 ,θR ) ∩ ρ(Hθ00 ,θR0 ). Then, with Λθ00 ,θR (·) R introduced in (3.23), and with Sθ0 ,θR defined as in (3.60), (Hθ00 ,θR0 − zI)−1 = (Hθ0 ,θR − zI)−1 ∗ − γθ0 ,θ0 ((Hθ0 ,θR )∗ − zI)−1 Sθ−1 0 0 0 −θ0 ,θR −θR 0 R h 0 0 i−1 θ ,θ × Λθ00 ,θR (z) γθ00 ,θR0 (Hθ0 ,θR − zI)−1 , R
(3.76) 0 θ0 6= θ00 mod(π), θR 6= θR mod(π),
24
F. GESZTESY AND M. ZINCHENKO
(Hθ0 ,θR0 − zI)−1 = (Hθ0 ,θR − zI)−1 ∗ 0 − γθ0 ,θ0 ((Hθ0 ,θR )∗ − zI)−1 [sin(θR − θR )]−1 P2 (3.77) R h i −1 θ ,θ 0 0 × Λθ00 ,θR (z) P2 γθ0 ,θR0 (Hθ0 ,θR − zI)−1 , θR 6= θR mod(π), θ0 = θ00 , R (Hθ00 ,θR − zI)−1 = (Hθ0 ,θR − zI)−1 ∗ (3.78) − γθ0 ,θR ((Hθ0 ,θR )∗ − zI)−1 [sin(θ00 − θ0 )]−1 P1 0 h 0 i−1 θ ,θ 0 × Λθ00 ,θR (z) P1 γθ00 ,θR (Hθ0 ,θR − zI)−1 , θ0 6= θ00 mod(π), θR = θR . R 4. Boundary Data Maps, Perturbation Determinants and Trace ¨ dinger Operators Formulas for Schro In this section we present our second group of new results, the connection between boundary data maps, appropriate perturbation determinants, and trace formulas in the context of self-adjoint one-dimensional Schr¨odinger operators. While Theorem 2.8 appears to be an interesting extension of the classical result, Theorem 2.1, it is in general, that is, in the context of non-self-adjoint operators, not a simple task to verify the hypotheses (2.30)–(2.32) as they involve square root domains. In particular, it appears to be unknown whether or not dom (Hθ0 ,θR + λI)1/2 ) and dom (Hθ∗0 ,θR + λI)1/2 coincide for λ > 0 sufficiently large, and hence coincide with dom(QHθ0 ,θR ), the form domain of Hθ0 ,θR (assuming Hθ0 ,θR to be nonself-adjoint): This question amounts to solving “Kato’s problem” in the special case of the non-self-adjoint Schr¨ odinger operator Hθ0 ,θR (cf., e.g., [2], [6], [40], [54], [61], [62], and [63]), a topic we will return to elsewhere1. To be on safe ground, we now confine ourselves to the special case of self-adjoint operators Hθ0 ,θR for the remainder of this section: Necessary and sufficient conditions for Hθ0 ,θR to be self-adjoint are the conditions V ∈ L1 ((0, R); dx) is real-valued,
(4.1)
and θ0 , θR ∈ [0, 2π), (4.2) assumed from now on. Then the 2nd representation theorem for densely defined, semibounded, closed quadratic forms (cf. [41, Sect. 6.2.6]) yields that dom (Hθ0 ,θR − zI)1/2 = dom |Hθ0 ,θR |1/2 = dom(QHθ0 ,θR ), (4.3) θ0 , θR ∈ [0, 2π), z ∈ C\[eθ0 ,θR , ∞), where we abbreviated eθ0 ,θR = inf(σ(Hθ0 ,θR )),
θ0 , θR ∈ [0, 2π).
(4.4)
1/2
Here (Hθ0 ,θR −zI) is defined with the help of the spectral theorem and a choice of a branch cut along [eθ0 ,θR , ∞). A comparison with (3.11)–(3.14), employing the fact that dom (Hθ00 ,θR0 − zI)1/2 = dom |Hθ00 ,θR0 |1/2 = H 1 ((0, R)), (4.5) 0 θ00 , θR ∈ [0, 2π)\{0, π}, z ∈ C\[eθ00 ,θR0 , ∞), 1An affirmative answer to this problem is known in the case where V ∈ L∞ ([0, R]; dx) (cf. [6]) and in the case of self-adjoint boundary conditions θ0 , θR ∈ [0, 2π) (cf. [78]).
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
dom (Hθ0 ,θR − zI)1/2 = dom |Hθ0 ,θR |1/2 ⊆ H 1 ((0, R)),
25
(4.6)
θ0 , θR ∈ [0, 2π), z ∈ C\[eθ0 ,θR , ∞), then shows that (Hθ00 ,θR0 − zI)1/2 (Hθ0 ,θR − zI)−1 (Hθ00 ,θR0 − zI)1/2 = (Hθ00 ,θR0 − zI)1/2 (Hθ0 ,θR − zI)−1/2 ∗ × (Hθ00 ,θR0 − zI)1/2 (Hθ0 ,θR − zI)−1/2 ∈ B L2 ((0, R); dx) , 0 θ00 , θR
(4.7)
∈ [0, 2π)\{0, π}, θ0 , θR ∈ [0, 2π), z ∈ C\[e0 , ∞),
where we introduced the abbreviation e0 = inf σ(Hθ0 ,θR ) ∪ σ(Hθ00 ,θR0 ) = min(eθ0 ,θR , eθ00 ,θR0 ).
(4.8)
Moreover, applying Theorem 3.11 one concludes that actually, (Hθ00 ,θR0 − zI)1/2 (Hθ0 ,θR − zI)−1 (Hθ00 ,θR0 − zI)1/2 − I = −(Hθ00 ,θR0 − zI)1/2 (Hθ00 ,θR0 − zI)−1 − (Hθ0 ,θR − zI)−1 (Hθ00 ,θR0 − zI)1/2 is a finite-rank (and hence a trace class) operator on L2 ((0, R); dx), (4.9) 0 θ00 , θR ∈ [0, 2π)\{0, π}, θ0 , θR ∈ [0, 2π), z ∈ C\[e0 , ∞).
To see the finite-rank property one can argue asfollows: By (3.70)–(3.72), the C2 -vector γθ00 ,θR0 (Hθ0 ,θR − zI)−1 f , f ∈ L2 (0, R); dx , is of the type C1 u+,θR (z, ·), f L2 ((0,R);dx) , (4.10) γθ00 ,θR0 (Hθ0 ,θR − zI)−1 f = C2 u−,θ0 (z, ·), f L2 ((0,R);dx) 0 , θ0 , θR ), j = 1, 2, and hence, since obviously u+,θR (z, ·) for some Cj = Cj (z, θ00 , θR and u−,θ0 (z, ·) belong to H 1 ((0, R)),
γθ00 ,θR0 (Hθ0 ,θR − zI)−1 (Hθ00 ,θR0 − zI)1/2 g C1 u+,θR (z, ·), (Hθ00 ,θR0 − zI)1/2 g L2 ((0,R);dx) = C2 u−,θ0 (z, ·), (Hθ00 ,θR0 − zI)1/2 g L2 ((0,R);dx) C1 [(Hθ00 ,θR0 − zI)1/2 ]∗ u+,θR (z, ·), g L2 ((0,R);dx) , g ∈ H 1 ((0, R)), (4.11) = 1/2 ∗ 0 0 C2 [(Hθ0 ,θR − zI) ] u−,θ0 (z, ·), g L2 ((0,R);dx) extends by continuity to all g ∈ L2 (0, R); dx . Similarly, using [16, eq. (3.54)], one infers for any [a0 aR ]> ∈ C2 that ∗ γθ00 ,θR0 ((Hθ0 ,θR )∗ − zI)−1 [a0 aR ]> (4.12) = D1 a0 u+,θR (z, ·) + D2 aR u−,θ0 (z, ·) ∈ H 1 ((0, R)), 0 for some Dj = Dj (z, θ00 , θR , θ0 , θR ), j = 1, 2. Consequently, ∗ 1/2 (Hθ00 ,θR0 − zI) γθ00 ,θR0 ((Hθ0 ,θR )∗ − zI)−1 [a0 aR ]> ∈ L2 (0, R); dx
(4.13)
> 2 is well-defined for (4.11) (for arbitrary all [a0 aR ] ∈ C . Thus, combining 2 g ∈ L (0, R); dx ) and (4.13) (for arbitrary [a0 aR ]> ∈ C2 ) with the finite-rank property of the second terms on the right-hand sides in (3.76)–(3.78) yields the asserted finite-rank property in (4.9).
26
F. GESZTESY AND M. ZINCHENKO
Thus, the Fredholm determinant, more precisely, the symmetrized perturbation determinant, detL2 ((0,R);dx) (Hθ00 ,θR0 − zI)1/2 (Hθ0 ,θR − zI)−1 (Hθ00 ,θR0 − zI)1/2 , (4.14) 0 θ0 , θR ∈ [0, 2π), θ00 , θR ∈ (0, 2π)\{π}, z ∈ C\[e0 , ∞), is well-defined, and an application of Theorem 2.8 to Hθ00 ,θR0 and Hθ0 ,θR yields trL2 ((0,R);dx) (Hθ00 ,θR0 − zI)−1 − (Hθ0 ,θR − zI)−1 d = − ln detL2 ((0,R);dx) (Hθ00 ,θR0 − zI)1/2 (Hθ0 ,θR − zI)−1 (Hθ00 ,θR0 − zI)1/2 , dz 0 θ0 , θR ∈ [0, 2π), θ00 , θR ∈ (0, 2π)\{π}, z ∈ C\[e0 , ∞), (4.15) whenever detL2 ((0,R);dx) (Hθ00 ,θR0 − zI)1/2 (Hθ0 ,θR − zI)−1 (Hθ00 ,θR0 − zI)1/2 6= 0. Next, we show that the symmetrized (Fredholm) perturbation determinant (4.14) associated with the pair (Hθ00 ,θR0 , Hθ0 ,θR ) can essentially be reduced to the 2 × 2 θ 0 ,θ 0
determinant of the boundary data map Λθ00 ,θR (z): R 0 ∈ (0, 2π)\{π}, and suppose Theorem 4.1. Assume that θ0 , θR ∈ [0, 2π), θ00 , θR that V satisfies (4.1). Let Hθ0 ,θR and Hθ00 ,θR0 be defined as in (3.5). Then, detL2 ((0,R);dx) (Hθ00 ,θR0 − zI)1/2 (Hθ0 ,θR − zI)−1 (Hθ00 ,θR0 − zI)1/2 0 0 (4.16) sin(θ0 ) sin(θR ) θ0 ,θR 2 Λ det = (z) , z ∈ C\[e , ∞). 0 C θ ,θ 0 ) 0 R sin(θ00 ) sin(θR
Proof. Let z ∈ C\[e0 , ∞). By (3.8) and (3.9) it suffices to consider θ0 , θR ∈ [0, π), 0 ∈ (0, π). Moreover, we will assume that θ0 6= 0 and θR 6= 0. The cases where θ00 , θR θ0 = 0 and/or θR = 0 follow along the same lines. In addition, simplifying the proof a bit, we will choose z < 0, |z| sufficiently large, and introduce the following convenient abbreviations: H 0 = Hθ00 ,θR0 ,
H = Hθ0 ,θR ,
R Λ(z) = Λθθ00 ,θ ,θ 0 (z), 0
R
γ = γθ0 ,θR , γ 0 = γθ00 ,θR0 , sin(θ0 − θ00 ) 0 , S = Sθ0 −θ00 ,θR −θR0 = 0 0 sin(θR − θR )
u ˇ+ (z, ·) = u+,θR0 (z, ·), u ˇ− (z, ·) = u−,θ00 (z, ·), (4.17) ˇ (z) = W (ˇ W u+ (z, ·), u ˇ− (z, ·)) = W (u+,θR0 (z, ·), u−,θ00 (z, ·)), ∗ B(z) = (H 0 − zI)1/2 γ(H 0 − zI)−1 ∈ B C2 , L2 ((0, R); dx) . That B(z) ∈ B C2 , L2 ((0, R); dx) follows as in (4.12), (4.13). In addition, we recall that ∗ sin(θ0 ) 0 γ(H 0 − zI)−1 (a0 aR )> = [ˇ u (z, 0) + cot(θ0 )ˇ u− (z, 0)]a0 u ˇ+ (z, ·) ˇ (z) − W sin(θR ) 0 − [ˇ u (z, R) − cot(θR )ˇ u+ (z, R)]aR u ˇ− (z, ·), (a0 aR )> ∈ C2 (4.18) ˇ (z) + W (cf. [16, eq. (3.54)]). Thus, B(z)(a0 aR )> =
sin(θ0 ) 0 [ˇ u (z, 0) + cot(θ0 )ˇ u− (z, 0)]a0 (H 0 − zI)1/2 u ˇ+ (z, ·) ˇ (z) − W
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
−
27
sin(θR ) 0 [ˇ u (z, R) − cot(θR )ˇ u+ (z, R)]aR (H 0 − zI)1/2 u ˇ− (z, ·), ˇ (z) + W (a0 aR )> ∈ C2 ,
and hence B(z)∗ ∈ B L2 ((0, R); dx), C
2
(4.19)
is given by
∗
B(z) f =
sin(θ0 ) 0 [ˇ u− (z, 0) + cot(θ0 )ˇ u− (z, 0)]((H 0 − zI)1/2 u ˇ+ (z, ·), f )L2 ((0,R);dx) ˇ , W (z) R) 0 0 1/2 2 − sin(θ [ˇ u (z, R) − cot(θ )ˇ u (z, R)]((H − zI) u ˇ (z, ·), f ) R + − L ((0,R);dx) + ˇ (z) W
f ∈ L2 ((0, R); dx).
(4.20)
Using the following version of the Krein-type resolvent formula (3.76) ∗ θ 0 ,θ 0 (H − zI)−1 = (H 0 − zI)−1 − γ(H 0 − zI)−1 S −1 Λθ00 ,θR γ(H 0 − zI)−1 , (4.21) R one obtains (H 0 − zI)1/2 (H − zI)−1 (H 0 − zI)1/2 = I − (H 0 − zI)1/2 (H 0 − zI)−1 − (H − zI)−1 (H 0 − zI)1/2 θ 0 ,θ 0
= I − B(z)S −1 Λθ00 ,θR B(z)∗ , R
(4.22)
and thus, detL2 ((0,R);dx) (H 0 − zI)1/2 (H − zI)−1 (H 0 − zI)1/2 θ 0 ,θ 0 ∗ = detL2 ((0,R);dx) I − B(z)S −1 Λθ00 ,θR B(z) R 0 0 −1 θ0 ,θR ∗ = detC2 I2 − S Λθ0 ,θR B(z) B(z) ,
(4.23)
using cyclicity for determinants . Next we turn to the computation of the 2 × 2 matrix B(z)∗ B(z): By equations (4.19) and (4.20) one infers B(z)∗ B(z) = Cj,k (z) j,k=1,2 , (4.24) C1,1 (z) =
sin2 (θ0 ) 0 [ˇ u (z, 0) + cot(θ0 )ˇ u− (z, 0)]2 ˇ (z)2 − W × ((H 0 − zI)1/2 u ˇ+ (z, ·), (H 0 − zI)1/2 u ˇ+ (z, ·))L2 ((0,R);dx) ,
C1,2 (z) = −
sin(θ0 ) sin(θR ) ˇ (z)2 W × [ˇ u0− (z, 0) + cot(θ0 )ˇ u− (z, 0)][ˇ u0+ (z, R) − cot(θR )ˇ u+ (z, R)] × ((H 0 − zI)1/2 u ˇ+ (z, ·), (H 0 − zI)1/2 u ˇ− (z, ·))L2 ((0,R);dx) ,
C2,1 (z) = −
(4.26)
sin(θ0 ) sin(θR ) ˇ (z)2 W 0 × [ˇ u− (z, 0) + cot(θ0 )ˇ u− (z, 0)][ˇ u0+ (z, R) − cot(θR )ˇ u+ (z, R)] × ((H 0 − zI)1/2 u ˇ− (z, ·), (H 0 − zI)1/2 u ˇ+ (z, ·))L2 ((0,R);dx) , 2
C2,2 (z) =
(4.25)
sin (θR ) 0 [ˇ u (z, R) − cot(θR )ˇ u+ (z, R)]2 ˇ (z)2 + W
(4.27)
28
F. GESZTESY AND M. ZINCHENKO
× ((H 0 − zI)1/2 u ˇ− (z, ·), (H 0 − zI)1/2 u ˇ− (z, ·))L2 ((0,R);dx) .
(4.28)
A straightforward, although rather tedious computation then yields the following simplification of Cj,k (z), j, k = 1, 2, and hence of B(z)∗ B(z): sin2 (θ0 − θ00 ) 1 , ˇ0+ (z, 0) + cot(θ00 ) sin2 (θ00 ) u 0 sin(θ0 − θ00 ) sin(θR − θR ) u ˇ− (z, 0) C1,2 (z) = 0 0 ) 0 sin(θ0 ) sin(θR ) u ˇ0− (z, R) − cot(θR 0 sin(θ0 − θ00 ) sin(θR − θR ) u ˇ+ (z, R) =− 0 0 sin(θ0 ) sin(θR ) u ˇ0+ (z, 0) + cot(θ00 ) = C2,1 (z), C1,1 (z) = −
2
C2,2 (z) =
0 θR )
sin (θR − 0 ) sin2 (θR
u ˇ0− (z, R)
1 0 ). − cot(θR
(4.29) (4.30) (4.31) (4.32) (4.33)
To arrrive at (4.29)–(4.33) one repeatedly uses the identity cot(x) − cot(y) =
sin(y − x) , sin(y) sin(x)
ˇ, the following expressions for the Wronskian W 0 ˇ (z) = u W ˇ+ (z, R)[ˇ u0− (z, R) − cot(θR )] = −ˇ u− (z, 0)[ˇ u0+ (z, 0) + cot(θ00 )],
(4.34)
(4.35)
and ((H 0 − zI)1/2 u ˇ+ (z, ·), (H 0 − zI)1/2 u ˇ+ (z, ·))L2 ((0,R);dx) = −[ˇ u0+ (z, 0) + cot(θ00 )], 0
1/2
((H − zI) =
u ˇ+ (z, ·), (H − zI)
−ˇ u− (z, 0)[ˇ u0+ (z, 0) 0
(4.36)
0
1/2
= ((H − zI)
+
1/2
cot(θ00 )] 0
u ˇ− (z, ·))L2 ((0,R);dx) 0 =u ˇ+ (z, R)[ˇ u0− (z, R) − cot(θR )],
u ˇ− (z, ·), (H − zI)
1/2
(4.37)
u ˇ+ (z, ·))L2 ((0,R);dx) ,
((H 0 − zI)1/2 u ˇ− (z, ·), (H 0 − zI)1/2 u ˇ− (z, ·))L2 ((0,R);dx) 0 = [ˇ u0− (z, R) − cot(θR )].
(4.38)
Relations (4.36)–(4.38) are a consequence of one integration by parts in (3.11). Finally, we compute Λ(z)S, starting with (3.29) and (4.17): Λ(z)S = Kj,k (z) j,k=1,2 , (4.39) ˇ0+ (z, 0) + cot(θ0 ) sin(θ0 − θ00 ) sin(θ0 ) u 0 sin(θ0 ) u ˇ0+ (z, 0) + cot(θ00 ) sin(θ0 − θ00 ) sin(θ0 ) , = C1,1 (z) + sin(θ00 ) 0 sin(θR − θR ) sin(θ0 ) u ˇ− (z, 0) + cot(θ0 )ˇ u− (z, 0) K1,2 (z) = − 0 ) 0 ) sin(θR u ˇ0− (z, R) − cot(θR 0 sin(θ0 − θ00 ) sin(θR − θR ) u ˇ− (z, 0) = 0 0 0 ) sin(θ0 ) sin(θR ) u ˇ0− (z, R) − cot(θR = C1,2 (z) = C2,1 (z) K1,1 (z) =
(4.40)
(4.41)
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS 0 sin(θ0 − θ00 ) sin(θR − θR ) u ˇ+ (z, R) 0 0 0 sin(θ0 ) sin(θR ) u ˇ+ (z, 0) + cot(θ00 ) = K2,1 (z),
29
=−
0 ˇ0− (z, R) − sin(θR − θR ) sin(θR ) u 0 ) sin(θR u ˇ0− (z, R) + 0 sin(θR − θR ) sin(θR ) C2,2 (z) + . 0 ) sin(θR
K2,2 (z) = =
(4.42)
cot(θR ) 0 ) cot(θR (4.43)
In particular, Λ(z)S = B(z)∗ B(z) +
sin(θ0 −θ00 ) sin(θ0 ) sin(θ00 )
0
!
0
An insertion of (4.44) into (4.23) finally yields detL2 ((0,R);dx) (H 0 − zI)1/2 (H − zI)−1 (H 0 − zI)1/2 θ 0 ,θ 0 ∗ = detC2 I2 − S −1 Λθ00 ,θR B(z) B(z) , R " sin(θ0 −θ 0 ) sin(θ0 ) = detC2
= detC2
−1
I2 − [Λ(z)S] −1
[Λ(z)S]
sin(θ0 −θ00 ) sin(θ0 ) sin(θ00 )
0
0
0
0 sin(θR −θR ) sin(θR ) 0 ) sin(θR
sin(θ00 )
0
(4.44)
!#!
!!
0 sin(θR −θR ) sin(θR ) 0 ) sin(θR 0 0 ) sin(θ0 ) sin(θR ) sin(θ0 − θ0 ) sin(θR − θR S −1 0 0 ) sin(θ0 ) sin(θR
0
0 0 θ ,θ = detC2 Λθ00 ,θR (z) detC2 R =
Λ(z)S −
.
0 sin(θR −θR ) sin(θR ) 0 ) sin(θR
0 0 sin(θ0 ) sin(θR ) θ ,θ detC2 Λθ00 ,θR (z) . 0 0 R sin(θ0 ) sin(θR )
(4.45)
Since the Fredholm determinant on the left-hand side of (4.16) vanishes for θ0 = 0 and/or θR = 0, we now briefly consider the nullspace of the operators involved:. 0 Lemma 4.2. Assume that θ0 , θR ∈ [0, 2π), θ00 , θR ∈ (0, 2π)\{π}, z ∈ C\[e0 , ∞), and suppose that V satisfies (4.1). Let Hθ0 ,θR and Hθ00 ,θR0 be defined as in (3.5). Then recalling the factorization,
(Hθ00 ,θR0 − zI)1/2 (Hθ0 ,θR − zI)−1 (Hθ00 ,θR0 − zI)1/2 (4.46) ∗ 1/2 −1/2 1/2 −1/2 = (Hθ00 ,θR0 − zI) (Hθ0 ,θR − zI) (Hθ00 ,θR0 − zI) (Hθ0 ,θR − zI) one obtains ∗ (Hθ00 ,θR0 − zI)1/2 (Hθ0 ,θR − zI)−1/2 = f ∈ L2 ((0, R); dx) f = (Hθ00 ,θR0 − zI)1/2 ψ(z, ·);
ker
ψ(z, ·), ψ 0 (z, ·) ∈ AC([0, R]); −ψ 00 (z, ·) + (V (·) − z)ψ(z, ·) = 0; 0 ψ 0 (z, R) − cot(θR )ψ(z, R) = 0 if θ0 = 0, θR 6= 0; 0
ψ (z, 0) +
cot(θ00 )ψ(z, 0)
= 0 if θ0 6= 0, θR = 0;
no boundary conditions on ψ(z, ·) if θ0 = θR = 0 ,
(4.47)
30
F. GESZTESY AND M. ZINCHENKO
in particular, ∗ dim ker (Hθ00 ,θR0 − zI)1/2 (Hθ0 ,θR − zI)−1/2 ( 2, θ0 = θR = 0, = 1, θ0 = 0, θR 6= 0 or θ0 6= 0, θR = 0.
(4.48)
Proof. Let z ∈ C\[e0 , ∞). To determine the precise characterization of the nullspace in (4.47) one can argue as follows: Suppose first that θ0 = θR = 0 and that f ⊥ ran (Hθ00 ,θR0 − zI)1/2 (H0,0 − zI)−1/2 , (4.49) g ∈ ran (Hθ00 ,θR0 − zI)1/2 (H0,0 − zI)−1/2 , implying g = (Hθ00 ,θR0 − zI)1/2 h for some h ∈ H01 ((0, R)),
(4.50)
and hence h(0) = h(R) = 0. Thus, introducing ψ(z, ·) = (Hθ00 ,θR0 − zI)−1/2 f , one obtains using (3.11) again, 0 = (g, f )L2 ((0,R);dx) = (Hθ00 ,θR0 − zI)1/2 h, (Hθ00 ,θR0 − zI)1/2 (Hθ00 ,θR0 − zI)−1/2 f = (Hθ00 ,θR0 − zI)1/2 h, (Hθ00 ,θR0 − zI)1/2 ψ(z) L2 ((0,R);dx) Z R = dx [h0 (x)ψ 0 (z, x) + (V (x) − z)h(x)ψ(z, x)]
L2 ((0,R);dx)
0 0 − cot(θ00 )h(0)ψ(z, 0) − cot(θR )h(R)ψ(z, R) Z R R = h(x)ψ 0 (z, x) 0 + dx h(x)[ψ 00 (z, x) + (V (x) − z)ψ(z, x)] 0
Z =
R
dx h(x)[ψ 00 (z, x) + (V (x) − z)ψ(z, x)],
h ∈ H01 ((0, R)).
(4.51)
0
Hence one concludes that ψ(z, ·), ψ 0 (z, ·) ∈ AC([0, R]),
(4.52)
and that ψ 00 (z, ·) + (V (·) − z)ψ(z, ·) = 0 in the sense of distributions. (4.53) 1/2 −1/2 As g ∈ ran (Hθ00 ,θR0 − zI) (H0,0 − zI) was chosen arbitrarily, one concludes ∗ 1/2 is of the form that any element f in ker (Hθ00 ,θR0 − zI) (H0,0 − zI)−1/2 f = (Hθ00 ,θR0 − zI)1/2 ψ(z, ·).
(4.54)
The fact that ψ(z, ·) satisfies no boundary conditions then shows that the dimension of the nullspace in (4.47) is precisely two if θ0 = θR = 0. Next we consider the case θ0 = 0, θR 6= 0 (the case θ0 6= 0, θR = 0 being completely analogous): Again we assume f ⊥ ran (Hθ00 ,θR0 − zI)1/2 (H0,θR − zI)−1/2 , (4.55) g ∈ ran (Hθ00 ,θR0 − zI)1/2 (H0,θR − zI)−1/2 , implying g = (Hθ00 ,θR0 − zI)1/2 h for some h ∈ H 1 ((0, R)) with h(0) = 0.
(4.56)
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
31
Introducing once more ψ(z, ·) = (Hθ00 ,θR0 − zI)−1/2 f , one obtains again via (3.11) that 0 = (g, f )L2 ((0,R);dx) = (Hθ00 ,θR0 − zI)1/2 h, (Hθ00 ,θR0 − zI)1/2 (Hθ00 ,θR0 − zI)−1/2 f = (Hθ00 ,θR0 − zI)1/2 h, (Hθ00 ,θR0 − zI)1/2 ψ(z) L2 ((0,R);dx) Z R = dx [h0 (x)ψ 0 (z, x) + (V (x) − z)h(x)ψ(z, x)]
L2 ((0,R);dx)
0 0 − cot(θ00 )h(0)ψ(z, 0) − cot(θR )h(R)ψ(z, R) Z R R = h(x)ψ 0 (z, x) 0 + dx h(x)[ψ 00 (z, x) + (V (x) − z)ψ(z, x)] 0
0 − cot(θR )h(R)ψ(z, R) 0 )ψ(z, R)] = h(R)[ψ 0 (z, R) − cot(θR Z R + dx h(x)[ψ 00 (z, x) + (V (x) − z)ψ(z, x)],
(4.57) h ∈ H 1 ((0, R)), h(0) = 0.
0
Choosing temporarily h ∈ H01 ((0, R)), one obtains Z R 0= dx h(x)[ψ 00 (z, x) + (V (x) − z)ψ(z, x)],
h ∈ H01 ((0, R)),
(4.58)
0
and hence again concludes that ψ(z, ·), ψ 0 (z, ·) ∈ AC([0, R]),
(4.59)
and that ψ 00 (z, ·) + (V (·) − z)ψ(z, ·) = 0 in the sense of distributions.
(4.60)
Taking into account (4.60) in (4.57) then yields 0 0 = h(R)[ψ 0 (z, R) − cot(θR )ψ(z, R)],
h ∈ H 1 ((0, R)), h(0) = 0,
(4.61)
implying 0 [ψ 0 (z, R) − cot(θR )ψ(z, R)] = 0. (4.62) As before, this proves (4.47) in the case θ0 = 0, θR 6= 0. The boundary condition (4.62) then yields that the nullspace (4.47) is one-dimensional in this case.
Remark 4.3. We emphasize the interesting fact that relation (4.16) represents yet another reduction of an infinite-dimensional Fredholm determinant (more precisely, a symmetrized perturbation determinant) to a finite-dimensional determinant. This is analogous to the following well-known situations: (i) The Jost–Pais formula [39] (see also [14]) in the context of half-line Schr¨odinger operators (relating the perturbation determinant of the corresponding Birman– Schwinger kernel with the Jost function and hence a Wronski determinant). (ii) Schr¨ odinger operators on the real line [65] (relating the perturbation determinant of the corresponding Birman–Schwinger kernel with the transmission coefficient and hence again a Wronski determinant). (iii) One-dimensional periodic Schr¨odinger operators [56] (relating the Floquet discriminant with an appropriate Fredholm determinant). These cases, and much more general situations in connection with semi-separable
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F. GESZTESY AND M. ZINCHENKO
integral kernels (which typically apply to one-dimensional differential and difference operators with matrix-valued coefficients) were studied in great deal in [26] (see also [32] and the multi-dimensional discussion in [31]). We conclude this section by pointing out that determinants (especially, ζ-function regularized determinants) for various elliptic boundary value problems on compact intervals (including cases with regular singular coefficients) have received considerable attention and we refer, for instance, to Burghelea, Friedlander, and Kappeler [13], Buslaev and Faddeev [14], Carron [15], Dreyfus and Dym [21], Forman [24], Kirsten, Loya, and Park [42], Kirsten and McKane [43], Lesch [50], Lesch and Tolksdorf [51], Lesch and Vertman [52], and Levit and Smilansky [53] in this context. 5. Trace Formulas and the Spectral Shift Function In this section we derive the trace formula for the resolvent difference of Hθ0 ,θR and Hθ00 ,θR0 in terms of the spectral shift function ξ(·; Hθ00 ,θR0 , Hθ0 ,θR ) and establish θ 0 ,θ 0
the connection between Λθ00 ,θR (·) and ξ(·; Hθ00 ,θR0 , Hθ0 ,θR ). R To prepare the ground for the basic trace formula we now state the following fact (which does not require Hθ0 ,θR and Hθ00 ,θR0 to be self-adjoint): 0 ∈ S2π , and let Hθ0 ,θR and Hθ00 ,θR0 be defined Lemma 5.1. Assume that θ0 , θR , θ00 , θR θ 0 ,θ 0
as in (3.5). Then, with Λθ00 ,θR (·)Sθ00 −θ0 ,θR0 −θR given by (3.61), R ∗ d θ00 ,θR0 Λθ0 ,θR (z)Sθ00 −θ0 ,θR0 −θR , γθ00 ,θR0 (Hθ0 ,θR − zI)−1 γθ0 ,θ0 (Hθ∗0 ,θR − zI)−1 = 0 R dz z ∈ ρ(Hθ0 ,θR ). (5.1) Proof. Employing the resolvent equation for Hθ∗0 ,θR , one verifies that ∗ ∗ d γθ00 ,θR0 γθ0 ,θ0 (Hθ∗0 ,θR − zI)−1 = γθ00 ,θR0 γθ0 ,θ0 (Hθ∗0 ,θR − zI)−2 0 0 R R dz ∗ = γθ00 ,θR0 (Hθ0 ,θR − zI)−1 γθ0 ,θ0 (Hθ∗0 ,θR − zI)−1 . 0
Together with (3.61) this proves (5.1).
(5.2)
R
Combining Theorems 3.11 and 2.8 with Lemma 5.1 then yields the following result: 0 Theorem 5.2. Assume that θ0 , θR , θ00 , θR ∈ [0, 2π), and suppose that V satisfies (4.1). Let Hθ0 ,θR and Hθ00 ,θR0 be defined as in (3.5). Then, trL2 ((0,R);dx) (Hθ00 ,θR0 − zI)−1 − (Hθ0 ,θR − zI)−1 h i−1 d h 0 0 i 0 θ00 ,θR θ0 ,θR = −trC2 Λθ0 ,θR (z) Λ (z) dz θ0 ,θR 0 0 d θ ,θ = − ln detC2 Λθ00 ,θR (z) , z ∈ C\[e0 , ∞). (5.3) R dz 0 If, in addition, θ00 , θR ∈ (0, 2π)\{π}, then trL2 ((0,R);dx) (Hθ00 ,θR0 − zI)−1 − (Hθ0 ,θR − zI)−1 d = − ln detL2 ((0,R);dx) (Hθ00 ,θR0 − zI)1/2 (Hθ0 ,θR − zI)−1 (Hθ00 ,θR0 − zI)1/2 , dz z ∈ C\[e0 , ∞). (5.4)
SYMMETRIZED PERTURBATION DETERMINANTS AND BOUNDARY DATA MAPS
33
Proof. The second equality in (5.3) is obvious. Next, we temporarily suppose that 0 θ0 6= θ00 and θR 6= θR . Then the first equality in (5.3) follows upon taking the trace in (3.76), using cyclicity of the trace, and applying (5.1) (keeping in mind that Sθ00 −θ0 ,θR0 −θR is invertible and z-independent). The remaining cases where θ0 = θ00 0 0 or θR = θR follow similarly (the case where θ0 = θ00 and θR = θR instantly follows from (3.25)). Relation (5.4) follows from (2.57), (4.16), and (5.3). Next, we note that the rank-two behavior of the difference of the resolvents of Hθ0 ,θR and Hθ00 ,θR0 displayed in Theorem 3.11 permits one to define the spectral shift function ξ( · ; Hθ00 ,θR0 , Hθ0 ,θR ) associated with the pair (Hθ00 ,θR0 , Hθ0 ,θR ). Using the standard normalization in the context of self-adjoint operators bounded from below, ξ( · ; Hθ00 ,θR0 , Hθ0 ,θR ) = 0, λ < e0 = inf σ(Hθ0 ,θR ) ∪ σ(Hθ00 ,θR0 ) , (5.5) Krein’s trace formula (see, e.g., [76, Ch. 8], [77]) reads trL2 ((0,R);dx) (Hθ00 ,θR0 − zI)−1 − (Hθ0 ,θR − zI)−1 Z ξ(λ; Hθ00 ,θR0 , Hθ0 ,θR ) dλ , z ∈ ρ(Hθ0 ,θR ) ∩ ρ(Hθ00 ,θR0 ), =− (λ − z)2 [e0 ,∞)
(5.6)
where ξ(· ; Hθ00 ,θR0 , Hθ0 ,θR ) satisfies ξ(· ; Hθ00 ,θR0 , Hθ0 ,θR ) ∈ L1 R; (λ2 + 1)−1 dλ .
(5.7)
Since the spectra of Hθ0 ,θR and Hθ00 ,θR0 are purely discrete, ξ( · ; Hθ00 ,θR0 , Hθ0 ,θR ) is a piecewise constant function on R with jumps of size 1 at the (necessarily simple) eigenvalues of Hθ0 ,θR and Hθ00 ,θR0 . In particular, ξ( · ; Hθ00 ,θR0 , Hθ0 ,θR ) represents the difference of the eigenvalue counting functions of Hθ00 ,θR0 and Hθ0 ,θR . Moreover, ξ(· ; Hθ00 ,θR0 , Hθ0 ,θR ) permits a representation in terms of nontangential 0 0 θ ,θ boundary values to the real axis of detC2 Λθ00 ,θR (·) (resp., of the symmetrized R perturbation determinant (4.14)), to be described in Theorem 5.3. Since by (3.8) and (3.9) it suffices to consider θ0 , θR ∈ [0, π) when considering the operator Hθ0 ,θR , we will restrict the boundary condition parameters accordingly next: 0 Theorem 5.3. Assume that θ0 , θR ∈ [0, π), θ00 , θR ∈ (0, π), and suppose that V satisfies (4.1). Let Hθ0 ,θR and Hθ00 ,θR0 be defined as in (3.5). Then, 0 0 θ ,θ ξ(λ; Hθ00 ,θR0 , Hθ0 ,θR ) = π −1 lim Im ln η(θ0 , θR ) detC2 Λθ00 ,θR (λ + iε) R ε↓0 (5.8) for a.e. λ ∈ R,
where (
θ0 , θR ∈ (0, π) or θ0 = θR = 0, θ0 = 0, θR ∈ (0, π) or θ0 ∈ (0, π), θR = 0. Proof. We recall the definition of e0 = inf σ(Hθ0 ,θR ) ∪ σ(Hθ00 ,θR0 ) in (5.5). η(θ0 , θR ) =
1, −1,
(5.9)
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Combining (5.3) and (5.6) one obtains 0 0 Z ξ(λ; Hθ00 ,θR0 , Hθ0 ,θR ) dλ d θ ,θ , ln η(θ0 , θR ) detC2 Λθ00 ,θR (z) = R dz (λ − z)2 [e0 ,∞)
(5.10)
z ∈ ρ(Hθ0 ,θR ) ∩ ρ(Hθ00 ,θR0 ), since η(θ0 , θR ) is z-independent. Next, combining (3.39) and (3.49), and using the fact that φ(z, x) and θ(z, x) are both for z, x ∈ R, one concludes that ∆(z, R, θ0 , θR ), and hence real-valued θ 0 ,θ 0
0 detC2 Λθ00 ,θR (z) are real-valued for z ∈ R and θ0 , θR , θ00 , θR ∈ [0, π). Moreover, R using the fact that 0 0 θ ,θ detC2 Λθ00 ,θR (z) 6= 0, z < e0 , (5.11) R
and invoking the asymptotic behavior (3.53) as z ↓ −∞, one actually concludes that 0 0 θ ,θ η(θ0 , θR ) detC2 Λθ00 ,θR (z) > 0, z < e0 . (5.12) R Integrating (5.10) with respect to the z-variable along the real axis from z0 to z, assuming z < z0 < e0 , one obtains 0 0 0 0 θ ,θ θ0 ,θR 2 Λ ln η(θ0 , θR ) detC2 Λθ00 ,θR (z) − ln η(θ , θ ) det (z ) 0 R 0 C θ ,θ 0 R R Z z Z ξ(λ; Hθ00 ,θR0 , Hθ0 ,θR ) dλ = dζ (λ − ζ)2 z0 [e0 ,∞) Z z Z [ξ+ (λ; Hθ00 ,θR0 , Hθ0 ,θR ) − ξ− (λ; Hθ00 ,θR0 , Hθ0 ,θR )] dλ = dζ (λ − ζ)2 z0 [e0 ,∞) Z Z z dζ = [ξ+ (λ; Hθ00 ,θR0 , Hθ0 ,θR ) − ξ− (λ; Hθ00 ,θR0 , Hθ0 ,θR )] dλ 2 [e0 ,∞) z0 (λ − ζ) Z 1 1 − = ξ(λ; Hθ00 ,θR0 , Hθ0 ,θR ) dλ , z < z0 < e0 . (5.13) λ−z λ − z0 [e0 ,∞) Here we split ξ into its positive and negative parts, ξ± = [|ξ| ± ξ]/2, and applied the Fubini–Tonelli theorem to interchange the integrations with respect to λ and ζ. Moreover, we chose the branch of ln(·) such that ln(x) ∈ R for x > 0, compatible with the normalization of ξ( · ; Hθ00 ,θR0 , Hθ0 ,θR ) in (5.5). An analytic continuation of the first and last line of (5.13) with respect to z then yields 0 0 0 0 θ ,θ θ0 ,θR 2 Λ ln η(θ0 , θR ) detC2 Λθ00 ,θR (z) − ln η(θ , θ ) det (z ) 0 R 0 C θ0 ,θR R Z 1 1 − , z ∈ C\[e0 , ∞). (5.14) = ξ(λ; Hθ00 ,θR0 , Hθ0 ,θR ) dλ λ−z λ − z0 [e0 ,∞) Since by (5.12), 0 0 θ ,θ ln η(θ0 , θR ) detC2 Λθ00 ,θR (z ) ∈ R, 0 R
z0 < e0 ,
(5.15)
the Stieltjes inversion formula separately applied to the absolutely continuous measures ξ± (λ; Hθ00 ,θR0 , Hθ0 ,θR ) dλ (cf., e.g., [5, p. 328], [75, App. B]), then yields (5.8).
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35
For interesting connections between the Dirichlet-to-Neumann map and the spectral function we also refer the reader to [15]. Acknowledgments. We are indebted to Steve Clark, Steve Hofmann, Alan McIntosh, Marius Mitrea, and Yuri Tomilov for helpful discussions. We also thank the anonymous referee for his comments. Fritz Gesztesy gratefully acknowledges the kind invitation and hospitality of the Department of Mathematics of the Western Michigan University, Kalamazoo, during a week in April of 2010, where the early parts of this paper were developed.
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[email protected] URL: http://www.math.missouri.edu/personnel/faculty/gesztesyf.html Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA New address: Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA E-mail address:
[email protected] URL: http://www.math.ucf.edu/~maxim/
