Trace formulae for Schrödinger operators with singular interactions

Jussi Behrndt; Matthias Langer; Vladimir Lotoreichik

doi:10.4171/175-1/6

Trace formulae for Schrödinger operators with singular interactions

Jussi Behrndt, Matthias Langer, Vladimir Lotoreichik

Mathematics And Statistics

Research output: Chapter in Book/Report/Conference proceeding › Chapter (peer-reviewed) › peer-review

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Abstract

Let Σ⊂ℝ^d be a C^∞-smooth closed compact hypersurface, which splits the Euclidean space ℝ^d into two domains Ω_±. In this note self-adjoint Schrödinger operators with δ and δ'-interactions supported on Σ are studied. For large enough m∈ℕ the difference of mth powers of resolvents of such a Schrödinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in L²(ℝ^d) is written in terms of Neumann-to-Dirichlet maps on the boundary space L²(Σ).

Original language	English
Title of host publication	Functional Analysis and Operator Theory for Quantum Physics
Subtitle of host publication	The Pavel Exner Anniversary Volume
Editors	Jaroslav Dittrich, Hynek Kovarik, Ari Laptev
Place of Publication	Switzerland
Pages	129-152
Number of pages	24
ISBN (Electronic)	9783037196755
DOIs	https://doi.org/10.4171/175-1/6
Publication status	Published - 31 May 2017

Publication series

Name	EMS Series of Congress Reports
Publisher	European Mathematical Society
Volume	12

Keywords

trace formula
delta interaction
Schroedinger operator
singular potential

Access to Document

10.4171/175-1/6

Behrndt-etal-EMS-2016-Trace-formulae-for-Schrodinger-operators-with-singularAccepted author manuscript, 505 KB

Cite this

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abstract = "Let Σ⊂ℝd be a C∞-smooth closed compact hypersurface, which splits the Euclidean space ℝd into two domains Ω±. In this note self-adjoint Schr{\"o}dinger operators with δ and δ'-interactions supported on Σ are studied. For large enough m∈ℕ the difference of mth powers of resolvents of such a Schr{\"o}dinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in L2(ℝd) is written in terms of Neumann-to-Dirichlet maps on the boundary space L2(Σ). ",

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Trace formulae for Schrödinger operators with singular interactions. / Behrndt, Jussi; Langer, Matthias; Lotoreichik, Vladimir.
Functional Analysis and Operator Theory for Quantum Physics: The Pavel Exner Anniversary Volume. ed. / Jaroslav Dittrich; Hynek Kovarik; Ari Laptev. Switzerland, 2017. p. 129-152 (EMS Series of Congress Reports; Vol. 12).

Research output: Chapter in Book/Report/Conference proceeding › Chapter (peer-reviewed) › peer-review

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AU - Langer, Matthias

AU - Lotoreichik, Vladimir

PY - 2017/5/31

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N2 - Let Σ⊂ℝd be a C∞-smooth closed compact hypersurface, which splits the Euclidean space ℝd into two domains Ω±. In this note self-adjoint Schrödinger operators with δ and δ'-interactions supported on Σ are studied. For large enough m∈ℕ the difference of mth powers of resolvents of such a Schrödinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in L2(ℝd) is written in terms of Neumann-to-Dirichlet maps on the boundary space L2(Σ).

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KW - trace formula

KW - delta interaction

KW - Schroedinger operator

KW - singular potential

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M3 - Chapter (peer-reviewed)

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A2 - Dittrich, Jaroslav

A2 - Kovarik, Hynek

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CY - Switzerland

ER -

Trace formulae for Schrödinger operators with singular interactions

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