Virtual eigenvalues of the high order schrödinger operator II

Jonathan Arazy; Leonid Zelenko

doi:10.1007/s00020-005-1390-4

Virtual eigenvalues of the high order schrödinger operator II

Jonathan Arazy, Leonid Zelenko

Department of Mathematics

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

Abstract

We consider the Schrödinger operator H _γ =(-Δ) ^l +γ V(x)• acting in the space L₂ (double scprit R sign^d ), where 2l ≥d, V (x)≥ 0,V (x) is continuous and is not identically zero, and lim |x| →∞ V(x) = 0. We study the asymptotic behavior as γ,↑,0 of the non-bottom negative eigenvalues of H _γ, which are born at the moment γ=0 from the lower bound λ=0 of the spectrum σ(H ₀) of the unperturbed operator H ₀=(-Δ) ^l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.

Original language	English
Pages (from-to)	305-345
Number of pages	41
Journal	Integral Equations and Operator Theory
Volume	55
Issue number	3
DOIs	https://doi.org/10.1007/s00020-005-1390-4
State	Published - Jul 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
Coupling constant
Lieb-Thirring estimates
Puiseux-Newton diagram
Schrödinger operator
Virtual eigenvalues

ASJC Scopus subject areas

Analysis
Algebra and Number Theory

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10.1007/s00020-005-1390-4

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

abstract = "We consider the Schr{\"o}dinger operator H γ =(-Δ) l +γ V(x)• acting in the space L2 (double scprit R signd ), where 2l ≥d, V (x)≥ 0,V (x) is continuous and is not identically zero, and lim |x| →∞ V(x) = 0. We study the asymptotic behavior as γ,↑,0 of the non-bottom negative eigenvalues of H γ, which are born at the moment γ=0 from the lower bound λ=0 of the spectrum σ(H 0) of the unperturbed operator H 0=(-Δ) l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.",

keywords = "Asymptotic behavior of virtual eigenvalues, Coupling constant, Lieb-Thirring estimates, Puiseux-Newton diagram, 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",

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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 - 2006/7

Y1 - 2006/7

N2 - We consider the Schrödinger operator H γ =(-Δ) l +γ V(x)• acting in the space L2 (double scprit R signd ), where 2l ≥d, V (x)≥ 0,V (x) is continuous and is not identically zero, and lim |x| →∞ V(x) = 0. We study the asymptotic behavior as γ,↑,0 of the non-bottom negative eigenvalues of H γ, which are born at the moment γ=0 from the lower bound λ=0 of the spectrum σ(H 0) of the unperturbed operator H 0=(-Δ) l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.

AB - We consider the Schrödinger operator H γ =(-Δ) l +γ V(x)• acting in the space L2 (double scprit R signd ), where 2l ≥d, V (x)≥ 0,V (x) is continuous and is not identically zero, and lim |x| →∞ V(x) = 0. We study the asymptotic behavior as γ,↑,0 of the non-bottom negative eigenvalues of H γ, which are born at the moment γ=0 from the lower bound λ=0 of the spectrum σ(H 0) of the unperturbed operator H 0=(-Δ) l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.

KW - Asymptotic behavior of virtual eigenvalues

KW - Coupling constant

KW - Lieb-Thirring estimates

KW - Puiseux-Newton diagram

KW - Schrödinger operator

KW - Virtual eigenvalues

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DO - 10.1007/s00020-005-1390-4

M3 - Article

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SN - 0378-620X

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EP - 345

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

IS - 3

ER -

Virtual eigenvalues of the high order schrödinger operator II

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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