Schrödinger Operators with δ -potentials Supported on Unbounded Lipschitz Hypersurfaces
Result description
In this note we consider the self-adjoint Schrödinger operator Aα in L2(ℝd), d≥ 2, with a δ -potential supported on a Lipschitz hypersurface Σ ⊆ ℝd of strength α∈ Lp(Σ ) + L∞(Σ ). We show the uniqueness of the ground state and, under some additional conditions on the coefficient α and the hypersurface Σ, we determine the essential spectrum of Aα. In the special case that Σ is a hyperplane we obtain a Birman-Schwinger principle with a relativistic Schrödinger operator as Birman-Schwinger operator. As an application we prove an optimization result for the bottom of the spectrum of Aα.
Keywords
Birman-Schwinger operatorEigenvalue optimizationEssential spectrumGround stateSchrödinger operatorSingular potential
The result's identifiers
Result code in IS VaVaI
Result on the web
DOI - Digital Object Identifier
Alternative languages
Result language
angličtina
Original language name
Schrödinger Operators with δ -potentials Supported on Unbounded Lipschitz Hypersurfaces
Original language description
In this note we consider the self-adjoint Schrödinger operator Aα in L2(ℝd), d≥ 2, with a δ -potential supported on a Lipschitz hypersurface Σ ⊆ ℝd of strength α∈ Lp(Σ ) + L∞(Σ ). We show the uniqueness of the ground state and, under some additional conditions on the coefficient α and the hypersurface Σ, we determine the essential spectrum of Aα. In the special case that Σ is a hyperplane we obtain a Birman-Schwinger principle with a relativistic Schrödinger operator as Birman-Schwinger operator. As an application we prove an optimization result for the bottom of the spectrum of Aα.
Czech name
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Czech description
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Classification
Type
C - Chapter in a specialist book
CEP classification
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OECD FORD branch
10102 - Applied mathematics
Result continuities
Project
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Book/collection name
Operator Theory: Advances and Applications
ISBN
978-3-031-31138-3
Number of pages of the result
28
Pages from-to
123-150
Number of pages of the book
698
Publisher name
Birkhäuser
Place of publication
Cham
UT code for WoS chapter
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Basic information
Result type
C - Chapter in a specialist book
OECD FORD
Applied mathematics
Year of implementation
2023