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Trace formula and Levinson's theorem as an index pairing in the presence of resonances

Angus Alexander

Abstract

We realise the number of bound states of a Schrödinger operator on $\mathbb{R}^n$ as an index pairing in all dimensions. Expanding on ideas of Guillopé and others, we use high-energy corrections to find representatives of the $K$-theory class of the scattering operator. These representatives allow us to compute the number of bound states using an integral formula involving heat kernel coefficients.

Trace formula and Levinson's theorem as an index pairing in the presence of resonances

Abstract

We realise the number of bound states of a Schrödinger operator on as an index pairing in all dimensions. Expanding on ideas of Guillopé and others, we use high-energy corrections to find representatives of the -theory class of the scattering operator. These representatives allow us to compute the number of bound states using an integral formula involving heat kernel coefficients.
Paper Structure (11 sections, 37 theorems, 173 equations)

This paper contains 11 sections, 37 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on scattering theory
  3. The scattering matrix and the spectral shift function
  4. The form of the wave operator
  5. Generalised wave operators and index pairings
  6. The index pairing
  7. Trace class properties of the scattering matrix
  8. Low and high energy behaviour of the scattering matrix and Levinson's theorem
  9. Low energy corrections in dimension $n = 1$ and $n = 3$
  10. High energy corrections in dimension $n \geq 2$
  11. Levinson's theorem and the spectral shift function at zero

Key Result

Proposition 2.1

Suppose that $V$ satisfies eq:ass11 for some $\rho > 1$. Let $N$ be the total number of eigenvalues of $H = H_0+V$, counted with multiplicity. Then

Theorems & Definitions (75)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3: jensen81
  • Definition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 65 more