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Spectral Shift Functions of Lamé Operators

Ubong Sam Idiong, Unanaowo Nyong Bassey

TL;DR

This work computes Kreĭn's spectral shift functions for the Lamé operator in Weierstrass form and the Brioschi-Halphen operator by constructing Green functions through Fourier transforms of distributions and connecting them to the spectral data via the Kreĭn trace formula. The Lamé case leverages its known spectral polynomial structure and elliptic potential, while the Brioschi-Halphen operator is analyzed through a two-step Lamé transformation and a Fourier-symbol analysis, yielding explicit expressions for the associated SSFs in terms of distributional kernels and Heaviside factors. The results establish a concrete methodology to obtain SSFs for complex elliptic differential operators, providing a framework that can be extended to other Fuchsian equations. The findings contribute to the spectral theory of elliptic and hyperelliptic operators and have potential applications in quantum mechanics and mathematical physics where spectral flow and perturbation determinants are relevant.

Abstract

The search for spectral shift functions of operators remains an open area of research. In this paper, the Kreĭn's spectral shift functions are computed for the Lamé operator in the Weierstrass form and the Brioschi-Halphen operator through Green functions obtained by applying the technique of Fourier transform of distributions.

Spectral Shift Functions of Lamé Operators

TL;DR

This work computes Kreĭn's spectral shift functions for the Lamé operator in Weierstrass form and the Brioschi-Halphen operator by constructing Green functions through Fourier transforms of distributions and connecting them to the spectral data via the Kreĭn trace formula. The Lamé case leverages its known spectral polynomial structure and elliptic potential, while the Brioschi-Halphen operator is analyzed through a two-step Lamé transformation and a Fourier-symbol analysis, yielding explicit expressions for the associated SSFs in terms of distributional kernels and Heaviside factors. The results establish a concrete methodology to obtain SSFs for complex elliptic differential operators, providing a framework that can be extended to other Fuchsian equations. The findings contribute to the spectral theory of elliptic and hyperelliptic operators and have potential applications in quantum mechanics and mathematical physics where spectral flow and perturbation determinants are relevant.

Abstract

The search for spectral shift functions of operators remains an open area of research. In this paper, the Kreĭn's spectral shift functions are computed for the Lamé operator in the Weierstrass form and the Brioschi-Halphen operator through Green functions obtained by applying the technique of Fourier transform of distributions.

Paper Structure

This paper contains 4 sections, 7 theorems, 125 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Conclusion

Key Result

Theorem 1

Let $f:X\times Y\longrightarrow [0,\infty]$ be measurable with respect to the $\sigma$-algebra $\sigma(\mathcal{K})$ and let $\mu\times \nu$ the product measure so that where $\mu$ and $\nu$ are the finite measure on the the measurable spaces $(X,\mathcal{F})$ and $(Y,\mathcal{S})$ respectively. Then where $\mathcal{K}$ is the family of compact sets.

Theorems & Definitions (15)

  • Theorem 1: KUT, § 9.10, p. 264
  • proof
  • Lemma 2
  • proof
  • Corollary 3
  • proof
  • Remark 4
  • Theorem 5: RS, Theorem VI.23, p.210
  • proof
  • Theorem 6: MGK, p.134
  • ...and 5 more