M-matrix asymptotics for Sturm-Liouville problems on graphs ∗ Sonja Currie Bruce A. Watson
†
School of Mathematics University of the Witwatersrand Private Bag 3, P O WITS 2050, South Africa October 13, 2008
Abstract We consider a system formulation for Sturm-Liouville operators with formally selfadjoint boundary conditions on a graph. An M-matrix associated with the boundary value problem is defined and is related to the matrix Pr¨ ufer angle associated with the system boundary value problem and consequently with the boundary value problem on the graph. Asymptotics for the M-matrix are obtained as the eigenparameter tends to negative infinity. We show that the boundary conditions may be recovered, up to a unitary equivalence, from the M-matrix. This is the first in a series of papers devoted to the reconstruction of the Sturm-Liouville problem on a graph from its M-matrix.
1
Introduction
We consider the second order differential equation ly := −
d2 y + q(x)y = λy, dx2
(1.1)
where q is real-valued and essentially bounded, on a weighted graph G with formally self-adjoint boundary conditions at the nodes. For the characterisation of self-adjoint boundary value problems on graphs and self-adjoint boundary conditions see [3] and [10]. ∗
Keywords: m-function, Sturm-Liouville, Pr”ufer angle, differential operators on graphs. (2000)MSC Primary: , Secondary: . † Supported in part by the Centre for Applicable Analysis and Number Theory.
1
There are many applications of differential operators on graphs, for example: quantum wires, quantum chaos, scattering theory and photonic crystals, see [9, 13, 14, 18]. For a survey of physical systems giving rise to boundary value problems on graphs see [17] and the bibliography there of. Although the first graph models were used in chemistry, [20, 21], the development of the theory of differential operators on graphs is recent with most of the research in this area having been conducted in the last couple of decades. It should however be noted that both multipoint boundary value problems (less general then boundary value problems on graphs) and systems (more general then boundary value problems on graphs) were studied far earlier then this. In this paper we make use of the fact that a self-adjoint boundary value problem on a graph can be reformulated as a self-adjoint boundary value problem for a system on [0, 1] with separated boundary conditions, see [5] and Section 2. In Section 3 we give asymptotic approximations for solutions of the system boundary value problem and consequently for solutions of the boundary value problem on the graph. The WeylTitchmarsh M-matrix is defined in Section 4, where, in addition, it is shown that the M-matrix exists and is well-defined. The M-matrix is then related to the matrix Pr¨ ufer angle. The asymptotic solutions from Section 3 are then used to find an asymptotic approximation for the matrix Pr¨ ufer angle and consequently of the M-matrix as the eigenparameter tends to negative infinity. Finally, in Section 6, we show that the boundary conditions may be recovered, up to a unitary equivalence, from the M-matrix. The results found in this paper form a starting point for solving the M-matrix inverse spectral problem for boundary value problems on graphs, a subsequent work. The inverse spectral problem involving the recovery of the potential and boundary conditions for a scalar Sturm-Liouville problem from two spectra, was given by Krein, Levitan and Gel’fand in [15, 16, 8] where the boundary conditions are not dependent on the eigenparameter, and by, amongst others, Binding, Browne, Watson and Chugonova, in [1, 4], when the boundary conditions are eigenparameter dependent. Brasche, Malamud and Neidhardt, in [2], characterise the spectra of self-adjoint extensions of a symmetric operator with equal deficiency indices in terms of boundary values for their abstract M-functions. In [11], Hinton and Shaw consider the Titchmarsh-Weyl M (λ)-function for linear Hamiltonian systems. Recent contributions to the relationship of the m-function to the spectral function for singular Sturm-Liouville problems have been made by Fulton and Pearson in [7].
2
Preliminaries
Let G denote a connected directed graph with a finite number of nodes and edges, with each edge parametrized by path-length and having finite length. Each edge, ei , of length say li , can thus be considered as the interval [0, li ]. Having made this identification, it is possible to consider the differential equation (1.1) on the graph G as the collection of differential equations −
d2 yi + qi (x)yi = λyi , dx2
x ∈ [0, li ], i = 1, ..., K,
2
(2.1)
where qi and yi denote the restriction of q and y to ei , respectively. At each node, ν, the boundary conditions can be specified in terms of the values of y and y ′ at ν on each of the incident edges. In particular, if the edges which originate at node ν are ei , i ∈ Λs (ν), and the edges which terminate at node ν are ei , i ∈ Λe (ν), then the boundary conditions at ν are of the form X X γij yj (lj ) + δij y ′ j (lj ) = 0, i = 1, ..., N (ν), (2.2) αij yj (0) + βij y ′ j (0) + j∈Λe (ν)
j∈Λs (ν)
where N (ν) is the number of linearly independent boundary conditions at node ν. Remark It should be noted that by setting αij = 0 = βij for i = 1, ..., N (ν) with j 6∈ Λs (ν) and γij = 0 = δij for i = 1, ..., N (ν) with j 6∈ Λe (ν), after relabelling the conditions (2.2) may be written more conveniently as K X j=1
K X γij yj (lj ) + δij y ′ j (lj ) = 0, αij yj (0) + βij y ′ j (0) +
i = 1, ..., N,
(2.3)
j=1
where N is the total number of linearly independent boundary conditions. If the boundary conditions (2.3) are formally self-adjoint with respect to (2.1), then N = 2K. Define L2 (G) to be the set of all f : G → C with fi ∈ L2 (0, li ) and inner product K Z li X f |ei g¯|ei dt, (f, g) = i=1
0
making L2 (G) a Hilbert space. The above boundary value problem on G can be reformulated as an operator eigenvalue problem, see [3], by setting Lf = −f ′′ + qf
with domain D(L) = {f | f, f ′ ∈ AC, l(f ) ∈ L2 (G), f obeying (2.2)}. In this setting, the formal self-adjointness of (2.1)-(2.2) ensures that the operator L on L2 (G) is a (closed densely defined) self-adjoint operator. In [5] it was shown that by letting t = lxi and y˜i (t) = yi (li t) the formally self-adjoint boundary value problem (2.1)-(2.2) can reformulated, on [0, 1], as the formally selfadjoint system boundary value problem −W Y˜ ′′ + QY˜ = λY˜ , (2.4) ′ ′ ˜ Y˜ (0) + C˜ Y˜ (1) + D ˜ Y˜ (1) = 0, A˜Y˜ (0) + B (2.5) y˜1 h i h i i h 1 1 ˜ = δij ˜ = βij , C˜ = [γij ], D where W = diag l2 , . . . , l2 , Y˜ = ... , A˜ = [αij ], B lj lj 1 K y˜K and Q = diag [Q1 , . . . , QK ] := diag [q1 (l1 t), . . . , qK (lK t)]. It was shown in [5, Theorem 5.1] that (2.4)-(2.5), formally self-adjoint with respect to the inner product Z 1 Z 1 K X 1 ¯ Fi Gi dt = li F T W − 2 G dt, < F, G >W = (2.6) i=1
0
0
3
is equivalent to the system −M T ′′ + P T = λT,
(2.7)
with boundary conditions A∗ T (0) − B ∗ T ′ (0) = 0, ∗
∗
(2.8)
′
Γ T (1) − ∆ T (1) = 0, (2.9) 0 Q t+1 0 I −I ∗ 2 , P = , A = , −B ∗ = W 0 Q 1−t 0 0 2
W where M = 4 0 0 0 ˜ and −∆∗ = 2[D ˜ − B], ˜ which is formally self-adjoint, with respect , Γ∗ = [C˜ A] I I to the inner product Z 2K X li < F, G >M = i=1
1
¯ i dt = Fi G
Z
1
1
F T M − 2 G dt.
(2.10)
0
0
Here by equivalent we mean that the spectrum is preserved, and that there is a linear bijection between the spaces L2 (G) and (L2 [0, 1])2K which is inner product preserving and maps eigenfunctions to eigenfunctions. This map can be given explicitly by Y˜ t+1 2 . In particular the boundary value problem (2.1)-(2.2) on the graph T (t) = ˜ 1−t Y 2 G with K edges, is equivalent to the system boundary value problem (2.7)-(2.9) of dimension 2K, having separated boundary conditions.
3
Asymptotic Solutions
2 Theorem 3.1 Let ρ = λ. The solution matrix Y of (2.7) obeying the initial condition Y (0) I 0 = is entire in ρ and can be represented as Y ′ (0) 0 I
Y = [U V](ρ, t) = [C(ρ, t) , S(ρ, t)] + [O(ρ, 1) , O(ρ, 2)], with derivative, Y ′ = [U′ V′ ](ρ, t) = [−ρ2 M −1 S(ρ, t) , C(ρ, t)] + [O(ρ, 0) , O(ρ, 1)], asymptotically for |ρ| → ∞. Here C and S are the diagonal matrices C(ρ, t) = diag (cos1 (ρ, t), . . . , cos2K (ρ, t)) , S(ρ, t) = diag (sin1 (ρ, t), . . . , sin2K (ρ, t)) , where cosi (ρ, t) := cos li2ρt and sini (ρ, t) := li2ρ sin li2ρt . Here, for ρ ∈ C and k ∈ Z, O(ρ, k) = diag O
etl1 |ℑρ|/2 ρk
4
!
,...,O
etl2K |ℑρ|/2 ρk
!!
.
Proof: Let Y (t) be the solution of (2.7) with initial conditions as stated above. If we fix i ∈ {1, . . . , 2K} and denote u(t) = Yij (t), where j 6∈ {i, i + 2K}, then u is the solution of a second order linear differential equation with initial conditions u(0) = 0 = u′ (0) and is thus zero on [0, 1]. Hence all entries of Y other than Yii and Yi i+2K , i = 1, . . . , 2K, are identically zero. Now consider u(t) = Yii (t). Here u is the solution of u′′ +
li2 2 (ρ − Pii )u = 0 4
(3.1)
obeying the initial conditions u(0) = 1 and u′ (0) = 0. Thus, from [12, Appendix], ! e|ℑ(tli ρ|/2 u(t) = cosi (ρ, t) + O , ρ !! 2 ρ2 |ℑ(tli ρ|/2 l e u′ (t) = − i sini (ρ, t) + O . 4 ρ2 Finally, for u = Yi,i+2K , u is the solution of (3.1) with Pii replaced by Pi,i+2K obeying the initial conditions u(0) = 0 and u′ (0) = 1. Thus, from [12, Appendix], ! e|ℑ(tli ρ|/2 , u(t) = sini (ρ, t) + O ρ2 ! |ℑ(tli ρ|/2 e u′ (t) = cosi (ρ, t) + O . ρ Remark In the case of P ≡ 0, the O(·) terms in the above theorem are identically zero.
4
The M-matrix and the matrix Pr¨ ufer angle
The boundary value problem (2.7)-(2.9) can be rewritten as the first order system Y′ =Z
Z ′ = −G(x)Y,
and
(4.1)
with boundary conditions A∗ Y (0) − B ∗ Z(0) = 0, ∗
∗
Γ Y (1) − ∆ Z(1) = 0.
(4.2) (4.3)
Here G(x) = M −1 (λ − P ). Without loss of generality it may be assumed that the following three properties hold: (1) G(x) is continuous and symmetric. (2) A∗ B = B ∗ A and Γ∗ ∆ = ∆∗ Γ. (3) A∗ A + B ∗ B = I and Γ∗ Γ + ∆∗ ∆ = I. Property (1) follows directly from the assumptions made throughout this paper. For formally self-adjoint boundary conditions it was shown in [5, Lemma 7.1] that (2) and (3) do not pose additional constraints.
5
Let V be the solution of (2.7) satisfying the initial conditions V(0) = B,
(4.4)
′
V (0) = A,
(4.5)
making A∗ V(0) − B ∗ V ′ (0) = 0. Thus V = Y and V ′ = Z with Y and Z as described above. The following result of Etgen, see [6, Theorem C], is fundamental to both the definition of the matrix Pr¨ ufer angle and our study thereof. Theorem 4.1 Let {Y (x), Z(x)} be the solution pair of (4.1) with initial conditions Y (0) = B, Z(0) = A. Then there exists a continuous, symmetric matrix H(x) and a nonsingular, continuously differentiable (in x) matrix J(x) such that Y (x) = S∗ (x)J(x),
Z(x) = C∗ (x)J(x)
for each λ, where {S(x), C(x)} is the solution of S′ = H(x)C,
C′ = −H(x)S, ∗
∗
S(0) = B ,
C(0) = A .
(4.6) (4.7)
Moreover, J(x) is the solution of J ′ = [SC∗ − CGS∗ ]J,
J(0) = I
with H(x) = CC∗ + SGS∗ .
(4.8)
From [5, 6] we have that the matrix Pr¨ ufer angle corresponding to (2.7)-(2.9) is given by F = (V − iU )−1 (V + iU ) where U (x) = S(x)Γ − C(x)∆,
V (x) = C(x)Γ + S(x)∆,
and {S(x), C(x)} is as given by (4.6)-(4.7). The following lemma provides an important link between V(1), V ′ (1) and F (1), needed in the proof of Theorem 4.4. Lemma 4.2 Let F, V, Γ and ∆ be as defined above, then F ∗ = [(Γ∗ + i∆∗ )V ′ + (∆∗ − iΓ∗ )V][(Γ∗ − i∆∗ )V ′ + (iΓ∗ + ∆∗ )V]−1 . Proof: Since V = Y and V ′ = Z, it follows that U ∗J
= Γ∗ S∗ J − ∆∗ C∗ J,
V ∗J
= Γ∗ C∗ J + ∆∗ S∗ J.
6
(4.9)
Thus U ∗J
= Γ∗ Y − ∆ ∗ Z = Γ∗ V − ∆ ∗ V ′ ,
V ∗J
= Γ∗ Z + ∆∗ Y = Γ∗ V ′ + ∆∗ V.
Hence, since J is non-singular, F ∗ = (V ∗ J − iU ∗ J)(V ∗ J + iU ∗ J)−1 = [Γ∗ V ′ + ∆∗ V − i(Γ∗ V − ∆∗ V ′ )] ×[Γ∗ V ′ + ∆∗ V + i(Γ∗ V − ∆∗ V ′ )]−1 . Let χ(x) = V(x)[∆∗ V ′ (1) − Γ∗ V(1)]−1 .
(4.10)
The determinant (as a function of λ), det(∆∗ V ′ (1) − Γ∗ V(1)), has zeros at the eigenvalues of the system (2.7)-(2.9) with the order of the zero equal to the multiplicity of the eigenvalue, see [19]. Thus, for each x, χ is analytic in λ except at the eigenvalues of (2.7)-(2.9). In order to define the Weyl M-matrix we need to define two solutions, W1 and W2 , of −Γ ∆ (2.7). Let R = and ∆ Γ W1 (x) W2 (x) W (x) = , (4.11) W1′ (x) W2′ (x) where W1 and W2 are the solutions of (2.7) such that W (x) obeys the terminal condition W (1) = R.
(4.12)
We define the Titchmarsh-Weyl M-matrix, M = M(λ), of (2.7)-(2.9) to be the matrix M given by Ψ = W1 + W2 M, (4.13) with the constraint that Ψ obey (2.8). Proposition 4.3 The M-matrix as defined in (4.13) exists and is well-defined for λ not an eigenvalue of (2.7)-(2.9). ∗ ∗ ∈ Proof: Let S = A v − B u = 0 . Let W1 and W2 be solutions of (2.7) W2 (1) such that the terminal condition (4.12) is satisfied. Then W2 obeys (2.9) and W2′ (1) W2 (x) has rank 2K. Consequently has rank 2K at each x, since R has rank 4K. W2′ (x) As λ is not an eigenvalue of (2.7)-(2.9),
v u
C4K
< W 2 > ∩ S = {0},
7
(4.14)
W2 (0) where W 2 := and < W 2 > is the linear subspace of C4K spanned by the W2′ (0) columns of W 2 (i.e. the column space of W 2 ). Consider the matrices W 1 and W 2 in column form, i.e. W 1 = {c11 , c12 , . . . , c12K } and W 2 = {c21 , c22 , . . . , c22K } where cij are 4K × 1 columns for i = 1, 2, and j = 1, . . . , 2K, then < c21 , c22 , . . . , c22K , c1j > ∩ S =< W 2 , c1j > ∩ S has dimension 1 by (4.14), since c11 , . . . , c12K , c21 , . . . , c22K are linearly independent and S has dimension 2K. Thus there is v j 6= 0 such that < W 2 , c1j > ∩ S =< v j >,
(4.15)
and each such v j has a unique representation as v j = αj c1j + W 2 kj where αj 6= 0 and kj ∈ C2k , (since c1j , c21 , . . . , c22K are linearly independent). In particular there exists a unique v j ∈< W 2 , c1j > and kj ∈ C2K such that v j = c1j + W 2 k j , and in this case [v 1 , . . . , v 2K ] = W 1 + W 2 [ k 1 , . . . , k2K ] W1 W2 = (0) + (0)M, W1′ W2′ where M(λ) = [ k1 , . . . , k 2K ]. Observe that < v 1 , . . . , v 2K > = S, i.e. {v 1 , . . . , v 2K } is a basis for S and hence the solution W1 + W2 M is a solution of (2.7)-(2.8) of maximal rank, giving the existence of M. The uniqueness of M follows from the uniqueness of kj for j = 1, . . . , 2K. Note that there exists a constant invertible matrix C = C(λ) such that χ = ΨC. Defining Φ(x) =
Ψ(x) W2 (x) Ψ′ (x) W2′ (x)
, it follows that Φ(x) = W (x)
I 0 M I
.
We are now ready to relate the Weyl M-matrix to the matrix Pr¨ ufer angle F (x). Theorem 4.4 The M-matrix satisfies M∗ = i(F ∗ (1) − I)−1 (F ∗ (1) + I). The poles of the determinant of M∗ are prescisely the eigenvalues of (2.7)-(2.9). Proof: Since A∗ B = B ∗ A, from (4.4)-(4.10), A∗ χ(0) − B ∗ χ′ (0) = A∗ B[∆∗ V ′ − Γ∗ V]−1 (1) − B ∗ A[∆∗ V ′ − Γ∗ V]−1 (1) = 0. Now, by (4.13) and since χ = ΨC, C constant and invertible, we get that [A∗ (W1 (0) + W2 (0)M) − B ∗ (W1′ (0) + W2′ (0)M)]C = 0.
8
Multiplying the above equation on the right by C −1 gives A∗ (W1 (0) + W2 (0)M) − B ∗ (W1′ (0) + W2′ (0)M) = 0. Taking adjoints and rearranging the terms above gives W1∗ (0)A − W1′∗ (0)B = M∗ [W2′∗ (0)B − W2∗ (0)A],
(4.16)
where [W2′∗ (0)B − W2∗ (0)A] is invertible everywhere except at the eigenvalues of (2.7)(2.9). We now show that the matrix-Wronskians W2′∗ V − W2∗ V ′ and W1∗ V ′ − W1′∗ V are constant. Observe that (4.17) [W2′∗ V − W2∗ V ′ ]′ = W2′′∗ V − W2∗ V ′′ , and, since W2 and V are solutions of (2.7), −W2′′∗ M + W2∗ P = λW2∗
and
− V ′′∗ M + V ∗ P = λV ∗ .
(4.18)
Substituting (4.18) into (4.17) we obtain [W2′∗ V − W2∗ V ′ ]′ = W2∗ (P − λ)M −1 V − W2∗ M −1 (P − λ)V, and since M −1 and P − λ are diagonal matrices (P − λ)M −1 = M −1 (P − λ). Therefore [W2′∗ V − W2∗ V ′ ]′ = 0, and W2′∗ V − W2∗ V ′ is constant. The proof that W1∗ V ′ − W1′∗ V is constant, is similar. Consequently [W2′∗ V − W2∗ V ′ ](0) = [W2′∗ V − W2∗ V ′ ](1) and thus W2′∗ (0)B − W2∗ (0)A = Γ∗ V(1) − ∆∗ V ′ (1). In the case of W1 we have that W1∗ (0)A − W1′∗ (0)B = −Γ∗ V ′ (1) − ∆∗ V(1), since W1 (1) = −Γ = −W2′ (1) and W2 (1) = ∆ = W1′ (1). Equation (4.16) consequently can be written as M∗ (∆∗ V ′ − Γ∗ V)(1) = (Γ∗ V ′ + ∆∗ V)(1). (4.19) Lemma 4.2 with (4.19) give F ∗ (1) = [M∗ (∆∗ V ′ − Γ∗ V)(1) + i(∆∗ V ′ − Γ∗ V)(1)] ×[M∗ (∆∗ V ′ − Γ∗ V)(1) − i(∆∗ V ′ − Γ∗ V)(1)]−1 = [(M∗ + iI)(∆∗ V ′ − Γ∗ V)(1)][(M∗ − iI)(∆∗ V ′ − Γ∗ V)(1)]−1 . Since λ is not an eigenvalue of (2.7)-(2.9), det(∆∗ V ′ (1) − Γ∗ V(1)) 6= 0, and thus F ∗ (1) = (M∗ + iI)(M∗ − iI)−1 . Solving for M∗ we get M∗ = i(F ∗ (1) − I)−1 (F ∗ (1) + I). Now, from (4.19), λ is a pole of det(M∗ ) of order n if and only if λ is a zero of det[∆∗ V ′ (1) − Γ∗ V(1)] of order n, which is equivalent, see [19], to λ being an eigenvalue of (2.7)-(2.9) of multiplicity n. Corollary 4.5 The matrix Pr¨ ufer angle, F (1), determines the M-function, M, and vice-versa.
9
5
M-matrix asymptotics
Let ρ = iσ, where we recall that λ = ρ2 . Asymptotics for the matrix Pr¨ ufer angle, F (1), as σ → ∞ will be found, and, by Corollary 4.5, these will provide us with asymptotics for the M-function. Theorem 5.1 Asymptotically as σ → +∞, the matrix Pr¨ ufer angle, F , takes the form 1 F (1) = (Γ + i∆)−1 I + O (Γ − i∆) . (5.1) σ Proof: Let V be as in (4.4)-(4.5), then V(t) = C(t)B + S(t)A, where C and S are as given in Theorem 3.1. Consequently ! 1 M − 2 ρt V(t) = cos B + O(ρ, 1, t), 2 ! ! Z t 1 1 1 M−2 1 M − 2 ρt M − 2 ρt ′ V (t) = − B+ B P (x) dx cos ρ sin 2 2 2 2 0 ! 1 M − 2 ρt + cos A + O(ρ, 1, t), 2 tl |ℑρ|/2 tl |ℑ(ρ)|/2 where O(ρ, k, t) = diag O e 1ρk , . . . , O e 2Kρk . Hence
! ! 1 1 1 M−2 ρ M−2 ρ M−2 ρ sin B + cos (A + iB) V (1) + iV(1) = − 2 2 2 ! Z 1 1 1 M−2 ρ + P (x) dx cos B + O(ρ, 1, 1). 2 2 0 ′
Let D be the block matrix ′
D = V (1) + iV(1) = Since A∗ =
I −I 0 0
1, . . . , 4, where d1j
=
d2j
=
d3j
=
d4j
=
and −B ∗ =
0 0 I I
D1 D2 D3 D4
.
we have Di = diag[di1 , . . . , diK ], i =
! lj ρ elj |ℑρ|/2 , cos +O 2 ρ ! ρlj ρlj lj ρ pj elj |ℑρ|/2 , sin + i cos − +O 2 2 2 2 ρ ! elj |ℑρ|/2 lj ρ , +O − cos 2 ρ ! ρlj ρlj lj ρ pj+K elj |ℑρ|/2 , sin + i cos − +O 2 2 2 2 ρ
10
where
R1 0
P (x) dx = diag[p1 , . . . p2K ].
Let the block matrix H be defined by ′
−1
H = (V (1) + iV(1))
=
H1 H2 H3 H4
,
then Hi = diag[hi1 , . . . , hiK ], i = 1, . . . , 4, where d1j h1j + d2j h3j
= 1 = d3j h2j + d4j h4j
d1j h2j + d2j h4j
= 0 = d3j h1j + d4j h3j ,
for j = 1, . . . , K, and, more concisely, 1 h1j h2j d4j −d2j = . h3j h4j d4j d1j − d2j d3j −d3j d1j So, as σ → +∞, d4j d1j − d2j d3j
and d4j d1j
ρlj ρlj = ρlj sin cos + O elj |ℑρ| 2 2 −σl j + O(1) , = eσlj 4
−4e−σlj 1 = − d2j d3j σlj
1 1+O , σ
giving h4j
=
h2j
=
h3j
=
h1j
=
−2e− σlj
σlj 2
1 1+O , σ σl 1 − 2j −e 1+O , σ σlj 1 −2e− 2 1+O , σlj σ σlj 1 . e− 2 1 + O σ
Therefore h σl1 σl i − K − 1 1 j+1 , j = 1, 2, (−1) diag e 2 1 + O σ , . . . , e 2 1 + O σ σlK σl1 Hj = − − diag −2eσl 2 1 + O σ1 , . . . , −2eσl 2 1 + O σ1 , j = 3, 4. 1 K
Multiplying H on the left by V ′ (1) − iV(1) gives E1 E2 ′ (V (1) − iV(1))H = E3 E4 where Ej =
(
diag 1 + O σ1 , . . . , 1 + O diag O σ1 , . . . , O σ1 ,
11
1 σ
, j = 1, 4, j = 2, 3.
(5.2)
Hence, as σ → ∞, ′
′
−1
(V (1) − iV(1))(V (1) + iV(1))
1 =I +O , σ
and thus from equation (4.9) we have F ∗ (1) = (Γ∗ + i∆∗ )[V ′ (1) − iV(1)][V ′ (1) + iV(1)]−1 (Γ∗ − i∆∗ )−1 1 ∗ ∗ = (Γ + i∆ ) I + O (Γ∗ − i∆∗ )−1 . σ
6
Recovery of the boundary conditions
˜ ∗, ∆ ˜ ∗ , P˜ ) Theorem 6.1 Let (Γ∗ , ∆∗ , P ) denote the boundary value problem (2.7)-(2.9) and (Γ ˜ ˜ ˜ the boundary value problem (2.7)-(2.9) but with Γ replaced by Γ, ∆ by ∆ and P by P . If ˜ and Γ = U Γ ˜ where U is the unitary their respective M-matrices are equal, then ∆ = U ∆ matrix ˜ ∗ + ∆∆ ˜ ∗. U = ΓΓ ˜ ∗, ∆ ˜ ∗ , P˜ ) have the same M-matrix, M(λ), by Corollary 4.5 Proof: Since (Γ∗ , ∆∗ , P ) and (Γ they have the same matrix Pr¨ ufer angle F ( 1). So, from (5.1), we have ˜ ∗ + i∆ ˜ ∗ )(Γ ˜ ∗ − i∆ ˜ ∗ )−1 . (Γ∗ + i∆∗ )(Γ∗ − i∆∗ )−1 = (Γ ˜ ∗ − i∆ ˜ ∗ are unitary, Since Γ∗ − i∆∗ and Γ ˜ ∗ + i∆ ˜ ∗ )(Γ ˜ + i∆), ˜ (Γ∗ + i∆∗ )(Γ + i∆) = (Γ ˜ ∗ + i∆ ˜ ∗ = (Γ ˜ − i∆) ˜ −1 and Γ∗ − i∆∗ = (Γ + i∆)−1 , we get and, since Γ ∗ ∗ ˜ − i∆)(Γ ˜ ˜ + i∆)(Γ ˜ (Γ + i∆∗ ) = (Γ − i∆∗ ).
Combining and simplifying the above two equations yields
which has hermitian adjoint
˜ ∗ = ∆Γ ˜ ∗, Γ∆
(6.1)
˜ ∗ = Γ∆ ˜ ∗. ∆Γ
(6.2)
˜ ∗ − i∆ ˜ ∗ )(Γ ˜ + i∆) ˜ = I and Γ∗ − i∆∗ = (Γ + i∆)−1 giving Now (Γ ∗ ˜ ∗ − i∆ ˜ ∗ )(Γ ˜ + i∆)(Γ ˜ (Γ + i∆)(Γ − i∆∗ ) = I.
After some manipulation this yields ˜ ∗ + i∆Γ ˜ ∗ − iΓ∆ ˜ ∗ + ∆∆˜∗ )(ΓΓ ˜ ∗ + i∆Γ ˜ ∗ − iΓ∆ ˜ ∗ + ∆∆ ˜ ∗ ) = I. (ΓΓ Subsituting from (6.1) and (6.2) into the above equation we obtain ˜ ∗ + iΓ∆ ˜ ∗ − iΓ∆ ˜ ∗ + ∆∆˜∗ )(ΓΓ ˜ ∗ + iΓ∆ ˜ ∗ − iΓ∆ ˜ ∗ + ∆∆ ˜ ∗ ) = I, (ΓΓ giving
˜ ∗ + ∆∆˜∗ )(ΓΓ ˜ ∗ + ∆∆ ˜ ∗ ) = I. (ΓΓ
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(6.3)
˜ ∗ + ∆∆˜∗ is unitary. Pre-multiplying (6.2) by Γ∗ gives Therefore U := ΓΓ ˜ ∗ = Γ∗ Γ∆ ˜ ∗. Γ∗ ∆ Γ Since ∆∗ Γ = Γ∗ ∆ and Γ∗ Γ + ∆∗ ∆ = I the above equation becomes ˜ ∗ = (I − ∆∗ ∆)∆ ˜ ∗. ∆ ∗ ΓΓ Hence
˜ ∗ + ∆∆˜∗ ) = ∆ ˜ ∗. ∆∗ U = ∆∗ (ΓΓ
˜ Similarly Γ = U Γ. ˜ Taking adjoints gives ∆ = U ∆. The above theorem shows that we can recover the boundary conditions from the Mmatrix up to a unitary equivalence. Let U be a unitary matrix such that M = U ∗ M U . Let A1 = U ∗ A, B1 = U ∗ B, Γ1 = U ∗ Γ, ∆1 = U ∗ ∆, P1 = U ∗ P U . Let Y1 = U ∗ Y . If A∗1 Y1 (0) − B1∗ Y1′ (0) = 0,
(6.4)
then A ∗ U U ∗ Y (0) − B ∗ U U ∗ Y ′ (0) = 0, giving that Y obeys (2.8). Similarly, if Γ∗1 Y1 (1) − ∆∗1 Y1′ (1) = 0,
(6.5)
−M Y1′′ + P1 Y1 = λY1 ,
(6.6)
then Y obeys (2.9). Also if
then −M U ∗ Y ′′ + U ∗ P U U ∗ Y = λU ∗ Y , which gives −U ∗ M Y ′′ + U ∗ P Y = U ∗ λY , from which it follows directly that Y obeys (2.7). Conversely, it can be shown in a similar manner that if Y obeys (2.7)-(2.9) then Y1 obeys (6.4)-(6.6). Hence the boundary value problems are equivalent and spectrally indistinguishable. Thus a unitarily related class of boundary value problems is the strongest form of uniqueness that can be hoped for in the inverse problem discussed here.
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