On Dirichlet-Schrödinger operators with strong potentials

Gabriele Grillo

Studia Mathematica (1995)

  • Volume: 116, Issue: 3, page 239-254
  • ISSN: 0039-3223

Abstract

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We consider Schrödinger operators H = -Δ/2 + V (V≥0 and locally bounded) with Dirichlet boundary conditions, on any open and connected subdomain D ⊂ ℝ n which either is bounded or satisfies the condition d ( x , D c ) → 0 as |x| → ∞. We prove exponential decay at the boundary of all the eigenfunctions of H whenever V diverges sufficiently fast at the boundary ∂D, in the sense that d ( x , D C ) 2 V ( x ) → ∞ as d ( x , D C ) → 0 . We also prove bounds from above and below for Tr(exp[-tH]), and in particular we give criterions for the finiteness of such trace. Applications to pointwise bounds for the integral kernel of exp[-tH] and to the computation of expected values of the Feynman-Kac functional with respect to Doob h-conditioned measures are given as well.

How to cite

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Grillo, Gabriele. "On Dirichlet-Schrödinger operators with strong potentials." Studia Mathematica 116.3 (1995): 239-254. <http://eudml.org/doc/216231>.

@article{Grillo1995,
abstract = {We consider Schrödinger operators H = -Δ/2 + V (V≥0 and locally bounded) with Dirichlet boundary conditions, on any open and connected subdomain $D ⊂ ℝ^n$ which either is bounded or satisfies the condition $d(x,D^\{c\}) → 0$ as |x| → ∞. We prove exponential decay at the boundary of all the eigenfunctions of H whenever V diverges sufficiently fast at the boundary ∂D, in the sense that $d(x,D^C)^\{2\}V(x) → ∞$ as $d(x,D^C) → 0$. We also prove bounds from above and below for Tr(exp[-tH]), and in particular we give criterions for the finiteness of such trace. Applications to pointwise bounds for the integral kernel of exp[-tH] and to the computation of expected values of the Feynman-Kac functional with respect to Doob h-conditioned measures are given as well.},
author = {Grillo, Gabriele},
journal = {Studia Mathematica},
keywords = {probabilistic methods; exponential decay at the boundary; Feynman-Kac functional},
language = {eng},
number = {3},
pages = {239-254},
title = {On Dirichlet-Schrödinger operators with strong potentials},
url = {http://eudml.org/doc/216231},
volume = {116},
year = {1995},
}

TY - JOUR
AU - Grillo, Gabriele
TI - On Dirichlet-Schrödinger operators with strong potentials
JO - Studia Mathematica
PY - 1995
VL - 116
IS - 3
SP - 239
EP - 254
AB - We consider Schrödinger operators H = -Δ/2 + V (V≥0 and locally bounded) with Dirichlet boundary conditions, on any open and connected subdomain $D ⊂ ℝ^n$ which either is bounded or satisfies the condition $d(x,D^{c}) → 0$ as |x| → ∞. We prove exponential decay at the boundary of all the eigenfunctions of H whenever V diverges sufficiently fast at the boundary ∂D, in the sense that $d(x,D^C)^{2}V(x) → ∞$ as $d(x,D^C) → 0$. We also prove bounds from above and below for Tr(exp[-tH]), and in particular we give criterions for the finiteness of such trace. Applications to pointwise bounds for the integral kernel of exp[-tH] and to the computation of expected values of the Feynman-Kac functional with respect to Doob h-conditioned measures are given as well.
LA - eng
KW - probabilistic methods; exponential decay at the boundary; Feynman-Kac functional
UR - http://eudml.org/doc/216231
ER -

References

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