Leander Geisinger
Refined Semiclassical Asymptotics for Fractional Powers of the Laplace Operator.
Fri, 05/27/2011 - 15:01 — SomNambulAuthors: Leander Geisinger, Rupert L. Frank
Subjects: Spectral Theory
AbstractWe consider the fractional Laplacian on a domain and investigate the
asymptotic behavior of its eigenvalues. Extending methods from semi-classical
analysis we are able to prove a two-term formula for the sum of eigenvalues
with the leading (Weyl) term given by the volume and the subleading term by the
surface area. Our result is valid under very weak assumptions on the regularity
of the boundary.Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain.
Fri, 05/27/2011 - 15:01 — SomNambulAuthors: Leander Geisinger, Rupert L. Frank
Subjects: Spectral Theory
AbstractLet -\Delta denote the Dirichlet Laplace operator on a bounded open set in
\mathbb{R}^d. We study the sum of the negative eigenvalues of the operator -h^2
\Delta - 1 in the semiclassical limit h \to 0+. We give a new proof that yields
not only the first term of the asymptotic formula but also the second term
involving the surface area of the boundary of the set. The proof is valid under
weak smoothness assumptions on the boundary.Universal Bounds for Traces of the Dirichlet Laplace Operator.
Tue, 09/08/2009 - 12:16 — SomNambulAuthors: Leander Geisinger, Timo Weidl
Subjects: Mathematical Physics
AbstractWe derive upper bounds for the trace of the heat kernel $Z(t) = \sum_{k \in
\mathbb{N}} e^{-\lambda_k t}$, where $(\lambda_k)_{k \in \mathbb{N}}$ denote
the eigenvalues of the Dirichlet Laplace operator in an open set $\Omega
\subset \R^d$, $d \geq 2$. The result improves an inequality of Kac and is
applicable to any open set with finite volume. The bound decays exponentially
as $t$ tends to infinity and it contains the sharp first term and a correction
term reflecting the properties of the short time asymptotics of $Z(t)$.Universal Bounds for Traces of the Dirichlet Laplace Operator.
Tue, 09/08/2009 - 12:16 — SomNambulAuthors: Leander Geisinger, Timo Weidl
Subjects: Mathematical Physics
AbstractWe derive upper bounds for the trace of the heat kernel $Z(t) = \sum_{k \in
\mathbb{N}} e^{-\lambda_k t}$, where $(\lambda_k)_{k \in \mathbb{N}}$ denote
the eigenvalues of the Dirichlet Laplace operator in an open set $\Omega
\subset \R^d$, $d \geq 2$. The result improves an inequality of Kac and is
applicable to any open set with finite volume. The bound decays exponentially
as $t$ tends to infinity and it contains the sharp first term and a correction
term reflecting the properties of the short time asymptotics of $Z(t)$.
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