Weyl-Titchmarsh <i>M</i>
-Function Asymptotics for Matrix-valued Schrödinger Operators

Clark, Stephen L.; Gesztesy, Fritz

doi:10.1112/plms/82.3.701

Cited by 46 publications

(79 citation statements)

References 59 publications

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“…Here we recall just a special case of the asymptotic expansion proved in [21] which is most suited for our discussion in Section 4. We denote by C ε ⊂ C + the open sector with vertex at zero, symmetry axis along the positive imaginary axis, and opening angle ε, with 0 < ε < π/2.…”

Section: ])mentioning

confidence: 99%

“…Inverse monodromy problems have recently been discussed in [4], [5], [6], [15], [68], [69], [96], [101], and the literature cited therein. More specific inverse spectral problems, such as compactness of the isospectral set of periodic Schrödinger operators [13], special isospectral matrix-valued Schrödinger operators, and Borg-type uniqueness theorems (for periodic coefficients as well as eigenvalue problems on compact intervals) were recently studied in [17], [18], [19], [21], [23], [26], [51], [52], [68], [69], [101], [102], [103], [104]. Moreover, direct spectral theory in the particular case of periodic Schrödinger operators (i.e., Floquet theory and the like) has been studied in [12], [14], [23], [25], [26], [34], [53], [59], [60], [86], [102], [103], [110]- [112], with many more pertinent references to be found therein.…”

Section: Introductionmentioning

confidence: 99%

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A Class of Matrix-valued Schrödinger Operators with Prescribed Finite-band Spectra

Gesztesy

Sakhnovich²

2003

Reproducing Kernel Spaces and Applications

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Abstract. We construct a class of matrix-valued Schrödinger operators with prescribed finite-band spectra of maximum spectral multiplicity. The corresponding matrix potentials are shown to be stationary solutions of the KdV hierarchy. The methods employed in this paper rely on matrix-valued Herglotz functions, Weyl-Titchmarsh theory, pencils of matrices, and basic inverse spectral theory for matrix-valued Schrödinger operators.

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Section: ])mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Class of Matrix-valued Schrödinger Operators with Prescribed Finite-band Spectra

Gesztesy

Sakhnovich²

2003

Reproducing Kernel Spaces and Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…The basic results for spectral theory for the matrix case were obtained by Lyapunov and Poincaré [YS]. We also mention the new papers [BBK], [C1], [C2], [CK], [CHGL], [CG], [CG1], [GKM], [GS], [Sh] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Conformal spectral theory for the monodromy matrix

Korotyaev

2010

Trans. Amer. Math. Soc.

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Abstract. For any N × N monodromy matrix we define the Lyapunov function which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the Hill operator. The Lyapunov function has (real or complex) branch points, which we call resonances. We determine the asymptotics of the periodic, anti-periodic spectrum and of the resonances at high energy. We show that the endpoints of each gap are periodic (anti-periodic) eigenvalues or resonances (real branch points). Moreover, the following results are obtained: 1) We define the quasimomentum as an analytic function on the Riemann surface of the Lyapunov function; various properties and estimates of the quasimomentum are obtained. 2) We construct the conformal mapping with imaginary part given by the Lyapunov exponent, and we obtain various properties of this conformal mapping, which are similar to the case of the Hill operator. 3) We determine various new trace formulae for potentials and the Lyapunov exponent. 4) We obtain a priori estimates of gap lengths in terms of the Dirichlet integral. We apply these results to the Schrödinger operators and to first order periodic systems on the real line with a matrix-valued complex self-adjoint periodic potential.

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“…recent papers [2], [16] and literature therein). Due to space limitation we discuss these important issues separately in [15].…”

Section: Theorem 31 Let Q Be Real-valued and Qmentioning

confidence: 99%

“…Our research in this area is, in part, inspired by a recent new interest in asymptotic representations in connection with the inverse spectral theory for scalar and matrix valued Schrödinger operators [2], [3], [8], [9], [16].…”

Section: Introductionmentioning

confidence: 99%

On a complete analysis of high-energy scattering matrix asymptotics for one dimensional Schrödinger operators with integrable potentials

Rybkin

2001

Proc. Amer. Math. Soc.

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Abstract. For the general one dimensional Schrödinger operatorwith real q ∈ L 1 (R) we present a complete streamlined treatment of large spectral parameter power asymptotics of Jost solutions and the scattering matrix. We find simple necessary and sufficient conditions relating the number of exact terms in the asymptotics with the smoothness of q. These conditions are expressed in terms of the Fourier transform of some functions related to q. In particular, under the usual conditions q (N) ∈ L 1 (R) , N ∈ N 0 , we derive up to two extra terms in the asymptotic expansion of the Jost solution and for the transmission coefficient we derive twice as many terms. Our main results are complete.

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Weyl-Titchmarsh M -Function Asymptotics for Matrix-valued Schrödinger Operators

Abstract: Abstract. We explicitly determine the high-energy asymptotics for WeylTitchmarsh matrices associated with general matrix-valued Schrödinger operators on a half-line.

Cited by 46 publications

References 59 publications

A Class of Matrix-valued Schrödinger Operators with Prescribed Finite-band Spectra

A Class of Matrix-valued Schrödinger Operators with Prescribed Finite-band Spectra

Conformal spectral theory for the monodromy matrix

On a complete analysis of high-energy scattering matrix asymptotics for one dimensional Schrödinger operators with integrable potentials

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