Semi-classical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space: II. <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>P</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:math>-model on a finite volume

In the classical limit, we construct the well-known formal WKB expansions for the eigenfunctions of a Schrödinger operator in the vicinity of a single nondegenerate C∞-potential well. Our elementary approach covers the case of eigenvalues degenerate in the harmonic approximation.

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Stability Estimates for the Lowest Eigenvalue of a Schrödinger Operator

Geometric and Functional Analysis ◽

10.1007/s00039-014-0253-z ◽

2014 ◽

Vol 24 (1) ◽

pp. 63-84 ◽

Cited By ~ 14

Author(s):

Eric A. Carlen ◽

Rupert L. Frank ◽

Elliott H. Lieb

Keyword(s):

Schrödinger Operator ◽

Stability Estimates ◽

Schrodinger Operator ◽

Lowest Eigenvalue

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Stationary Scattering in the Semi-classical Limit for a Schrödinger Operator in a Gauge Field

Journal of Functional Analysis ◽

10.1006/jfan.1999.3416 ◽

1999 ◽

Vol 165 (2) ◽

pp. 347-389

Author(s):

R. Brummelhuis ◽

P. Levy-Bruhl ◽

J. Nourrigat

Keyword(s):

Gauge Field ◽

Schrödinger Operator ◽

Classical Limit ◽

Schrodinger Operator ◽

Stationary Scattering

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

Journal of Functional Analysis ◽

10.1016/j.jfa.2007.06.009 ◽

2007 ◽

Vol 251 (1) ◽

pp. 59-121 ◽

Cited By ~ 8

Author(s):

Shigeki Aida

Keyword(s):

Riemannian Manifold ◽

Schrödinger Operator ◽

Classical Limit ◽

Compact Riemannian Manifold ◽

Path Space ◽

Schrodinger Operator ◽

Bottom Of Spectrum

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Random media and quasi-classical limit of Schrödinger operator

Proceedings of the Japan Academy Series A Mathematical Sciences ◽

10.3792/pjaa.60.43 ◽

1984 ◽

Vol 60 (2) ◽

pp. 43-45

Author(s):

Shin Ozawa

Keyword(s):

Schrödinger Operator ◽

Classical Limit ◽

Random Media ◽

Schrodinger Operator

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Geometric and spectral estimates based on spectral Ricci curvature assumptions

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2020-0026 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Gilles Carron ◽

Christian Rose

Keyword(s):

Fundamental Group ◽

Ricci Curvature ◽

Schrödinger Operator ◽

Ricci Tensor ◽

Isoperimetric Inequalities ◽

Spectral Condition ◽

Schrodinger Operator ◽

Spectral Estimates ◽

Finite Fundamental Group ◽

Lowest Eigenvalue

AbstractWe obtain a Bonnet–Myers theorem under a spectral condition: a closed Riemannian {(M^{n},g)} manifold for which the lowest eigenvalue of the Ricci tensor ρ is such that the Schrödinger operator {\Delta+(n-2)\rho} is positive has finite fundamental group. Further, as a continuation of our earlier results, we obtain isoperimetric inequalities from Kato-type conditions on the Ricci curvature. We also obtain the Kato condition for the Ricci curvature under purely geometric assumptions.

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