Weyl–Titchmarsh $M$-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators

Abstract

Abstract. We explicitly determine the high-energy asymptotics for WeylTitchmarsh matrices associated with general Dirac-type operators on half-lines and on R. We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small differences of Weyl-Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac operators and use it to prove a Borg-type theorem.

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

“…dx 2 + Q Σ is in the limit point case at ±∞ can be proved in analogy to Wienholtz's proof [109] of a result originally due to Povzner [84], reproduced as Theorem 35 in [41, p. 58]. The corresponding details will be presented in [22]. Equation At this point we cannot yet infer continuity of Q Σ at the boundary point x 0 .…”

Initial Value Problems for Integrable Systems on a Semi-Strip

“…We note that the theory of Weyl functions (Weyl-Titchmarsh theory) is actively developing in recent years (see, e.g., [14,15,21,23,24,35,48,52,54,56] and references therein) and its applications to initial-boundary value problems are of growing interest.…”

“…Important results on the asymptotics of Weyl functions are given, for instance, in [14,26]. However, it would be fruitful to know, under which conditions the asymptotic series (for Weyl functions) from [14,26] converge or at least uniquely define the corresponding Weyl function.…”

Real-valued algebro-geometric solutions of the Camassa–Holm hierarchy

“…This material is standard and can be found, for instance, in Kogan & Rofe-Beketov (1974), Hinton & Shaw (1981, 1983, 1984, Clark & Gesztesy (2002) and Lesch & Malamud (2003), and references therein.…”