Virtual eigenvalues of the high order schrödinger operator I

Jonathan Arazy; Leonid Zelenko

doi:10.1007/s00020-005-1389-x

Virtual eigenvalues of the high order schrödinger operator I

Jonathan Arazy, Leonid Zelenko

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider the Schrödinger operator H_γ = (-Δ)^ι + _γV(x)• acting in the space L₂(IR^d), where 2ι ≥ d, V(x) ≥ 0, V(x) is continuous and is not identically zero, and lim_|x|→∞ V(x) = 0. We obtain an asymptotic expansion as γ ↑ 0 of the bottom negative eigenvalue of H_γ, which is born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = (-Δ)^ι (a virtual eigenvalue). To this end we develop a supplement to the Birman-Schwinger theory on the process of the birth of eigenvalues in the gap of the spectrum of the unperturbed operator H₀. Furthermore, we extract a finite-rank portion φ(λ) from the Birman- Schwinger operator X _V(λ) = V^1/2R_λ(H ₀)V^1/2, which yields the leading terms for the desired asymptotic expansion.

Original language	English
Pages (from-to)	189-231
Number of pages	43
Journal	Integral Equations and Operator Theory
Volume	55
Issue number	2
DOIs	https://doi.org/10.1007/s00020-005-1389-x
State	Published - Jun 2006

Bibliographical note

Funding Information:
Both authors were partially supported from the Israel Science Foundation (ISF), grant number 585/00, and from the German-Israeli Foundation (GIF), grant number I-619-17.6/2001. The second author was partially supported also by the KAMEA Project for Scientific Absorption in Israel.

Keywords

Asymptotic behavior of virtual eigenvalues
Birman-Schwinger principle
Coupling constant
Schrödinger operator
Virtual eigenvalues

ASJC Scopus subject areas

Analysis
Algebra and Number Theory

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10.1007/s00020-005-1389-x

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title = "Virtual eigenvalues of the high order schr{\"o}dinger operator I",

abstract = "We consider the Schr{\"o}dinger operator Hγ = (-Δ)ι + γV(x)• acting in the space L2(IRd), where 2ι ≥ d, V(x) ≥ 0, V(x) is continuous and is not identically zero, and lim|x|→∞ V(x) = 0. We obtain an asymptotic expansion as γ ↑ 0 of the bottom negative eigenvalue of Hγ, which is born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H0) of the unperturbed operator H0 = (-Δ)ι (a virtual eigenvalue). To this end we develop a supplement to the Birman-Schwinger theory on the process of the birth of eigenvalues in the gap of the spectrum of the unperturbed operator H0. Furthermore, we extract a finite-rank portion φ(λ) from the Birman- Schwinger operator X V(λ) = V1/2Rλ(H 0)V1/2, which yields the leading terms for the desired asymptotic expansion.",

keywords = "Asymptotic behavior of virtual eigenvalues, Birman-Schwinger principle, Coupling constant, Schr{\"o}dinger operator, Virtual eigenvalues",

author = "Jonathan Arazy and Leonid Zelenko",

note = "Funding Information: Both authors were partially supported from the Israel Science Foundation (ISF), grant number 585/00, and from the German-Israeli Foundation (GIF), grant number I-619-17.6/2001. The second author was partially supported also by the KAMEA Project for Scientific Absorption in Israel.",

year = "2006",

month = jun,

doi = "10.1007/s00020-005-1389-x",

language = "English",

volume = "55",

pages = "189--231",

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AU - Arazy, Jonathan

AU - Zelenko, Leonid

N1 - Funding Information: Both authors were partially supported from the Israel Science Foundation (ISF), grant number 585/00, and from the German-Israeli Foundation (GIF), grant number I-619-17.6/2001. The second author was partially supported also by the KAMEA Project for Scientific Absorption in Israel.

PY - 2006/6

Y1 - 2006/6

N2 - We consider the Schrödinger operator Hγ = (-Δ)ι + γV(x)• acting in the space L2(IRd), where 2ι ≥ d, V(x) ≥ 0, V(x) is continuous and is not identically zero, and lim|x|→∞ V(x) = 0. We obtain an asymptotic expansion as γ ↑ 0 of the bottom negative eigenvalue of Hγ, which is born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H0) of the unperturbed operator H0 = (-Δ)ι (a virtual eigenvalue). To this end we develop a supplement to the Birman-Schwinger theory on the process of the birth of eigenvalues in the gap of the spectrum of the unperturbed operator H0. Furthermore, we extract a finite-rank portion φ(λ) from the Birman- Schwinger operator X V(λ) = V1/2Rλ(H 0)V1/2, which yields the leading terms for the desired asymptotic expansion.

AB - We consider the Schrödinger operator Hγ = (-Δ)ι + γV(x)• acting in the space L2(IRd), where 2ι ≥ d, V(x) ≥ 0, V(x) is continuous and is not identically zero, and lim|x|→∞ V(x) = 0. We obtain an asymptotic expansion as γ ↑ 0 of the bottom negative eigenvalue of Hγ, which is born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H0) of the unperturbed operator H0 = (-Δ)ι (a virtual eigenvalue). To this end we develop a supplement to the Birman-Schwinger theory on the process of the birth of eigenvalues in the gap of the spectrum of the unperturbed operator H0. Furthermore, we extract a finite-rank portion φ(λ) from the Birman- Schwinger operator X V(λ) = V1/2Rλ(H 0)V1/2, which yields the leading terms for the desired asymptotic expansion.

KW - Asymptotic behavior of virtual eigenvalues

KW - Birman-Schwinger principle

KW - Coupling constant

KW - Schrödinger operator

KW - Virtual eigenvalues

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DO - 10.1007/s00020-005-1389-x

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SN - 0378-620X

VL - 55

SP - 189

EP - 231

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

IS - 2

ER -

Virtual eigenvalues of the high order schrödinger operator I

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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