Semi-classical limit of the bottom of spectrum of a Schrödinger operator on a path space over a compact Riemannian manifold

Shigeki Aida

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8 Citations (Scopus)

Abstract

We determine the limit of the bottom of spectrum of Schrödinger operators with variable coefficients on Wiener spaces and path spaces over finite-dimensional compact Riemannian manifolds in the semi-classical limit. These are extensions of the results in [S. Aida, Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space, J. Funct. Anal. 203 (2) (2003) 401-424]. The problem on path spaces over Riemannian manifolds is considered as a problem on Wiener spaces by using Ito's map. However the coefficient operator is not a bounded linear operator and the dependence on the path is not continuous in the uniform convergence topology if the Riemannian curvature tensor on the underling manifold is not equal to 0. The difficulties are solved by using unitary transformations of the Schrödinger operators by approximate ground state functions and estimates in the rough path analysis.

Original languageEnglish
Pages (from-to)59-121
Number of pages63
JournalJournal of Functional Analysis
Volume251
Issue number1
DOIs
Publication statusPublished - 2007 Oct 1

Keywords

  • Logarithmic Sobolev inequality
  • Path integral
  • Path space
  • Rough path analysis
  • Schrödinger operator
  • Semi-classical limit
  • Witten Laplacian

ASJC Scopus subject areas

  • Analysis

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