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$L_1$-Estimates for Eigenfunctions of the Dirichlet Laplacian

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Periodic and Ergodic Spectral Problems

Co-authors: Michiel van den Berg (U Bristol), J”urgen Voigt (TU Dresden)

For $d in {f N}$ and $Omega e mptyset$ an open set in ${f R}^d$, we consider the eigenfunctions $Phi$ of the Dirichlet Laplacian $-Delta_Omega$ of $Omega$. We do {it not} require $Omega$ to be of finite volume. % If $Phi$ is associated with an eigenvalue below the essential spectrum of $-Delta_Omega$, we provide estimates for the $L_1$-norm of $Phi$ in terms of the $L_2$-norm of $Phi$ and suitable spectral data of $-Delta_Omega$. The main idea in obtaining such estimates consists in finding a—-sufficiently small—-subset $Omega’ ubset Omega$ where $Phi$ is localized in the sense that $Phi$ decays exponentially as one moves away from $Omega’$.

These $L_1$-estimates are then used in the comparison of the
heat content of $Omega$ at time $t>0$ and
the heat trace at times $t' > 0$, where a two-sided estimate is established.


This is joint work with Michiel van den Berg (Bristol) and J”urgen Voigt (Dresden), with improvements by Hendrik Vogt (Dresden).

This talk is part of the Isaac Newton Institute Seminar Series series.

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