TY - JOUR

T1 - Semi-classical limit of the bottom of spectrum of a Schrödinger operator on a path space over a compact Riemannian manifold

AU - Aida, Shigeki

PY - 2007/10/1

Y1 - 2007/10/1

N2 - 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.

AB - 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.

KW - Logarithmic Sobolev inequality

KW - Path integral

KW - Path space

KW - Rough path analysis

KW - Schrödinger operator

KW - Semi-classical limit

KW - Witten Laplacian

UR - http://www.scopus.com/inward/record.url?scp=34548035808&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34548035808&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2007.06.009

DO - 10.1016/j.jfa.2007.06.009

M3 - Article

AN - SCOPUS:34548035808

VL - 251

SP - 59

EP - 121

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -