Stability Estimates for the Lowest Eigenvalue of a Schrödinger Operator

Eric A. Carlen; Rupert L. Frank; Elliott H. Lieb

doi:10.1007/s00039-014-0253-z

Stability Estimates for the Lowest Eigenvalue of a Schrödinger Operator

Eric A. Carlen, Rupert L. Frank, Elliott H. Lieb

Mathematics

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40 Scopus citations

Abstract

There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger operator under the constraint of a given L ^p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for Hölder's inequality, which we believe to be new.

Original language	English (US)
Pages (from-to)	63-84
Number of pages	22
Journal	Geometric and Functional Analysis
Volume	24
Issue number	1
DOIs	https://doi.org/10.1007/s00039-014-0253-z
State	Published - Feb 2014

All Science Journal Classification (ASJC) codes

Analysis
Geometry and Topology

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title = "Stability Estimates for the Lowest Eigenvalue of a Schr{\"o}dinger Operator",

abstract = "There is a family of potentials that minimize the lowest eigenvalue of a Schr{\"o}dinger operator under the constraint of a given L p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for H{\"o}lder's inequality, which we believe to be new.",

author = "Carlen, {Eric A.} and Frank, {Rupert L.} and Lieb, {Elliott H.}",

note = "Funding Information: Work partially supported by NSF grants DMS 0901632, DMS 1201354 (E.A.C.), PHY 1068285, PHY 1347399 (R.L.F.), PHY 0965859, PHY 1265118 (E.H.L.) and the Simons Foundation grant #230207 (E.H.L.).",

year = "2014",

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doi = "10.1007/s00039-014-0253-z",

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T1 - Stability Estimates for the Lowest Eigenvalue of a Schrödinger Operator

AU - Carlen, Eric A.

AU - Frank, Rupert L.

AU - Lieb, Elliott H.

N1 - Funding Information: Work partially supported by NSF grants DMS 0901632, DMS 1201354 (E.A.C.), PHY 1068285, PHY 1347399 (R.L.F.), PHY 0965859, PHY 1265118 (E.H.L.) and the Simons Foundation grant #230207 (E.H.L.).

PY - 2014/2

Y1 - 2014/2

N2 - There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger operator under the constraint of a given L p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for Hölder's inequality, which we believe to be new.

AB - There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger operator under the constraint of a given L p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for Hölder's inequality, which we believe to be new.

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DO - 10.1007/s00039-014-0253-z

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EP - 84

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 1

ER -

Stability Estimates for the Lowest Eigenvalue of a Schrödinger Operator

Abstract

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