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Positivity of spectral shift functions and infinite-dimensional BMV conjecture

Chandan Pradhan, Anna Skripka

TL;DR

The paper addresses the infinite-dimensional BMV conjecture and the positivity of higher-order spectral shift functions for Schatten-class perturbations of self-adjoint operators. It develops a unified approach based on multilinear operator integration to extend finite-dimensional positivity results to the infinite-dimensional setting, deriving trace formulas and sign properties for spectral shift functions eta_{n,H,V}. The authors establish Lévy-Khintchine and Bernstein-function representations for trace differences Tr(f(H+tV)-f(H)) under suitable conditions, proving that even-order shifts are nonnegative and odd-order shifts have sign consistent with V. These results yield an infinite-dimensional BMV-type representation and extend related perturbation-theory results to non-trace-class perturbations and unbounded operators, enriching the toolkit for spectral analysis in quantum and operator theory contexts.

Abstract

We obtain a solution to the Bessis-Moussa-Villani conjecture for a trace-class perturbation of a semi-bounded operator and answer affirmatively the question on positivity of higher order spectral shift functions in the setting of Schatten--von Neumann perturbations of (possibly unbounded) self-adjoint operators.

Positivity of spectral shift functions and infinite-dimensional BMV conjecture

TL;DR

The paper addresses the infinite-dimensional BMV conjecture and the positivity of higher-order spectral shift functions for Schatten-class perturbations of self-adjoint operators. It develops a unified approach based on multilinear operator integration to extend finite-dimensional positivity results to the infinite-dimensional setting, deriving trace formulas and sign properties for spectral shift functions eta_{n,H,V}. The authors establish Lévy-Khintchine and Bernstein-function representations for trace differences Tr(f(H+tV)-f(H)) under suitable conditions, proving that even-order shifts are nonnegative and odd-order shifts have sign consistent with V. These results yield an infinite-dimensional BMV-type representation and extend related perturbation-theory results to non-trace-class perturbations and unbounded operators, enriching the toolkit for spectral analysis in quantum and operator theory contexts.

Abstract

We obtain a solution to the Bessis-Moussa-Villani conjecture for a trace-class perturbation of a semi-bounded operator and answer affirmatively the question on positivity of higher order spectral shift functions in the setting of Schatten--von Neumann perturbations of (possibly unbounded) self-adjoint operators.

Paper Structure

This paper contains 5 sections, 16 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Multilinear Operator Integration
  3. Positivity preserving property of operator derivatives
  4. Positivity preserving property of spectral shift functions
  5. Infinite-dimensional BMV conjecture

Key Result

Theorem 1.1

Let $n \in \mathbb{N}$, and let $H, V$ be self-adjoint operators in $\mathcal{H}$ such that $V \in \mathcal{S}^n$. Then, there exists a unique $\eta_{n,H,V} \in L^1(\mathbb{R})$ such that for every $f \in C_c^{n+1}(\mathbb{R})$. The function $\eta_{n,H,V}$ is supported in the set $G_{[0,1]}$ defined in def:m-M.

Theorems & Definitions (27)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • Lemma 2.7
  • ...and 17 more