Uniqueness Results for Matrix-Valued Schr�dinger, Jacobi, and Dirac-Type Operators (original) (raw)
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Uniqueness Results for Matrix-Valued Schrödinger, Jacobi, and Dirac-Type Operators
Mathematische Nachrichten, 2002
Let g(z, x) denote the diagonal Green's matrix of a self-adjoint m × m matrix-valued Schrödinger operator H = − d 2 dx 2 Im + Q(x) in L 2 (R) m , m ∈ N. One of the principal results proven in this paper states that for a fixed x 0 ∈ R and all z ∈ C + , g(z, x 0 ) and g ′ (z, x 0 ) uniquely determine the matrixvalued m × m potential Q(x) for a.e. x ∈ R. We also prove the following local version of this result. Let g j (z, x), j = 1, 2 be the diagonal Green's matrices of the self-adjoint Schrödinger operators
2004
Abstract We announce three results in the theory of Jacobi matrices and Schr��dinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schr��dinger operator��� d 2/dx 2+ V (x) on L 2 (0,���) with V��� L 2 (0,���) and the boundary condition u (0)= 0. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szeg�� asymptotics.
Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators
Journal of Differential Equations, 2005
Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A, B in the self-adjoint Jacobi operator H = AS + + A − S − + B (with S ± the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E − , E + ], E − < E + , we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric self-adjoint Dirac difference operators D with spectrum given by − E
A priori estimates for the Hill and Dirac operators
Russian Journal of Mathematical Physics, 2008
Consider the Hill operator T y = −y ′′ + q ′ (t)y in L 2 (R), where q ∈ L 2 (0, 1) is a 1periodic real potential. The spectrum of T is is absolutely continuous and consists of bands separated by gaps γ n , n 1 with length |γ n | 0. We obtain a priori estimates of the gap lengths, effective masses, action variables for the KDV. For example, if µ ± n are the effective masses associated with the gap γ n = (λ − n , λ + n), then |µ − n +µ + n | C|γ n | 2 n −4 for some constant C = C(q) and any n 1. In order prove these results we use the analysis of a conformal mapping corresponding to quasimomentum of the Hill operator. That makes possible to reformulate the problems for the differential operator as the problems of the conformal mapping theory. Then the proof is based on the analysis of the conformal mapping and the identities. Moreover, we obtain the similar estimates for the Dirac operator.
Weyl solutions and j-selfadjointness for Dirac operators
Journal of Mathematical Analysis and Applications, 2019
We consider a non-selfadjoint Dirac-type differential expression (0.1) D(Q)y := J n dy dx + Q(x)y, with a non-selfadjoint potential matrix Q ∈ L 1 loc (I, C n×n) and a signature matrix J n = −J −1 n = −J * n ∈ C n×n. Here I denotes either the line R or the half-line R +. With this differential expression one associates in L 2 (I, C n) the (closed) maximal and minimal operators D max (Q) and D min (Q), respectively. One of our main results for the whole line case states that D max (Q) = D min (Q) in L 2 (R, C n). Moreover, we show that if the minimal operator D min (Q) in L 2 (R, C n) is j-symmetric with respect to an appropriate involution j, then it is j-selfadjoint. Similar results are valid in the case of the semiaxis R +. In particular, we show that if n = 2p and the minimal operator D + min (Q) in L 2 (R + , C 2p) is j-symmetric, then there exists a 2p × p-Weyl-type matrix solution Ψ(z, •) ∈ L 2 (R + , C 2p×p) of the equation D + max (Q)Ψ(z, •) = zΨ(z, •). A similar result is valid for the expression (0.1) with a potential matrix having a bounded imaginary part. This leads to the existence of a unique Weyl function for the expression (0.1). The main results are proven by means of a reduction to the self-adjoint case by using the technique of dual pairs of operators. The differential expression (0.1) is of significance as it appears in the Lax formulation of the vector-valued nonlinear Schrödinger equation.
On spectral theory for Schr�dinger operators with strongly singular potentials
Math Nachr, 2006
We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schr\"odinger operators on [a,\infty), a\in\bbR, with a regular finite end point a and the case of Schr\"odinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2\times 2 matrix-valued Herglotz functions representing the associated Weyl-Titchmarsh coefficients. Second, we contrast this with the case of self-adjoint half-line Schr\"odinger operators on (a,\infty) with a potential strongly singular at the end point a. We focus on situations where the potential is so singular that the associated maximally defined Schr\"odinger operator is self-adjoint (equivalently, the associated minimally defined Schr\"odinger operator is essentially self-adjoint) and hence no boundary condition is required at the finite end point a. For this case we show that the Weyl-Titchmarsh coefficient in this strongly singular context still determines the associated spectral function, but ceases to posses the Herglotz property. However, as will be shown, Herglotz function techniques continue to play a decisive role in the spectral theory for strongly singular Schr\"odinger operators.
On the basis property of root function systems of Dirac operators with regular boundary conditions
2019
where B is a nonsingular diagonal n× n matrix, B = diag(b−1 1 In1, . . . , b −1 r Inr) ∈ Cn×n, n = n1 + . . . nr, with complex entries bj 6= bk, and Q(x) is a potential matrix takes its origin in the paper by Birkho and Langer [4]. Afterwards their investigations were developed in many directions. Malamud and Oridoroga in [20] established rst general results on completeness of root function systems of boundary value problems for di erential systems (1). A little bit later Lunyov and Malamud in [17] obtained rst general results on Riesz basis property (Riesz basis property with parentheses) for mentioned boundary value problems with a potential matrix Q(x) ∈ L∞. There is an enormous literature related to the spectral theory outlined above, and we refer to [6, 7, 16, 22, 25] and their extensive reference lists for this activity. In the present paper, we study the Dirac system