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A Brunn-Minkowski inequality for Schrödinger operators with Kato class potentials

Alessandro Carbotti

Abstract

In this paper we prove a Brunn-Minkowski inequality for the first Dirichlet eigenvalue of a Schrödinger type operator $\mathcal{H}_V:=-\operatorname{div}(A\nabla)+V$, where $V$ is convex and Kato decomposable, using the trace class property of the generated semigroup. As a consequence, using the ultracontractivity of the semigroup we obtain the log-concavity of the ground state.

A Brunn-Minkowski inequality for Schrödinger operators with Kato class potentials

Abstract

In this paper we prove a Brunn-Minkowski inequality for the first Dirichlet eigenvalue of a Schrödinger type operator , where is convex and Kato decomposable, using the trace class property of the generated semigroup. As a consequence, using the ultracontractivity of the semigroup we obtain the log-concavity of the ground state.

Paper Structure

This paper contains 6 sections, 13 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Trace class semigroups
  4. The Schrödinger semigroup in $L^2(\Omega)$ with Kato class potential
  5. Log-concavity of the heat kernel
  6. Log-concavity of the ground state

Key Result

Theorem 1.1

Let $\Omega\subseteq\mathbb{R}^N$ be a measurable and convex set and $f,g,h:\Omega\rightarrow (0,+\infty)$ measurable functions such that for every $x,y\in \Omega$ and every $r\in[0,1]$. Then

Theorems & Definitions (24)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3: Trace Class Operator
  • Theorem 2.4: Spectral Mapping Theorem for Point Spectrum
  • Proposition 2.5: Trace of a Semigroup
  • Remark 2.6
  • Definition 2.7
  • Theorem 2.8
  • Corollary 2.9
  • ...and 14 more