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Session 25. Nonlocal Phenomena: Levy processes and related operators
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Decay of eigenfunctions for nonlocal Schrödinger operators |
Kamil Kaleta, Wrocław University of Technology and University of Warsaw, Poland
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The talk is based on the joint work with J. Lörinczi (Loughborough University)
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The spatial decay of eigenfunctions at infinity for Schrödinger operators \(H = -\Delta + V\), where \(V\) is a suitably chosen potential, has been widely studied for many years. A basic interest in this property is that it tells of how well a quantum particle
described by \(H\) is localized in physical space. When \(V\) is a pinning potential, i.e., \(V(x) \to \infty\) as \(|x| \to \infty\), then the decay is known to be typically faster than exponential. Questions about the decay of eigenfunctions, motivated by the problems in a relativistic quantum mechanics, also appear in the case of the so-called nonlocal Schrödinger operators
\[
H=-L+V,
\]
where \(L\) is a nonlocal operator which is the generator of the jump Lévy process.
I will present the recent results on the pointwise bounds at infinity of the eigenfunctions of \(H\) for a wide class of operators \(L\) and signed potentials \(V\). These estimates explicitly depend on the density of the Lévy measure of the process generated by \(L\) and the growth of \(V\) at infinity. For the ground state eigenfunction (which is known to be strictly positive) they are even two-sided and sharp. Our methods are mainly probabilistic (stochastic Feynman-Kac-type representation of the semigroup \(e^{-tH}\)) and are based on a precise analysis of jumps of the process and some specific self-improving estimates iterated infinitely many times. I will also discuss some interesting consequences and applications of these results: the properties of domination (of the semigroup \(e^{-tH}\) and other eigenfunctions) by the ground state eigenfunction, the asymptotic behaviour of the semigroup \(e^{-tH}\) for large \(t\) (intrinsic ultracontractivity-type properties), the asymptotic behaviour of paths of the related ground state-transformed jump processes (integral tests of the Kolmogorov-type, LILs, etc.).
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References- K. Kaleta, J. L\H orinczi, Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman-Kac semigroups for a class of Lévy processes , Ann. Probab., to appear, 2014.
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