Domain perturbations, capacity and shift of eigenvalues

André Noll

Journées équations aux dérivées partielles (1999)

  • page 1-10
  • ISSN: 0752-0360

Abstract

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After introducing the notion of capacity in a general Hilbert space setting we look at the spectral bound of an arbitrary self-adjoint and semi-bounded operator H . If H is subjected to a domain perturbation the spectrum is shifted to the right. We show that the magnitude of this shift can be estimated in terms of the capacity. We improve the upper bound on the shift which was given in Capacity in abstract Hilbert spaces and applications to higher order differential operators (Comm. P. D. E., 24:759–775, 1999) and obtain a lower bound which leads to a generalization of Thirring’s inequality if the underlying Hilbert space is an L 2 -space. Moreover, a similar capacitary upper bound for the second eigenvalue is established. The results are finally applied to higher-order partial differential operators.

How to cite

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Noll, André. "Domain perturbations, capacity and shift of eigenvalues." Journées équations aux dérivées partielles (1999): 1-10. <http://eudml.org/doc/93385>.

@article{Noll1999,
abstract = {After introducing the notion of capacity in a general Hilbert space setting we look at the spectral bound of an arbitrary self-adjoint and semi-bounded operator $H$. If $H$ is subjected to a domain perturbation the spectrum is shifted to the right. We show that the magnitude of this shift can be estimated in terms of the capacity. We improve the upper bound on the shift which was given in Capacity in abstract Hilbert spaces and applications to higher order differential operators (Comm. P. D. E., 24:759–775, 1999) and obtain a lower bound which leads to a generalization of Thirring’s inequality if the underlying Hilbert space is an $L^2$-space. Moreover, a similar capacitary upper bound for the second eigenvalue is established. The results are finally applied to higher-order partial differential operators.},
author = {Noll, André},
journal = {Journées équations aux dérivées partielles},
keywords = {capacity; Hilbert space; spectral bound; domain perturbation; shift; Thirring's inequality; second eigenvalue; higher-order partial differential operators},
language = {eng},
pages = {1-10},
publisher = {Université de Nantes},
title = {Domain perturbations, capacity and shift of eigenvalues},
url = {http://eudml.org/doc/93385},
year = {1999},
}

TY - JOUR
AU - Noll, André
TI - Domain perturbations, capacity and shift of eigenvalues
JO - Journées équations aux dérivées partielles
PY - 1999
PB - Université de Nantes
SP - 1
EP - 10
AB - After introducing the notion of capacity in a general Hilbert space setting we look at the spectral bound of an arbitrary self-adjoint and semi-bounded operator $H$. If $H$ is subjected to a domain perturbation the spectrum is shifted to the right. We show that the magnitude of this shift can be estimated in terms of the capacity. We improve the upper bound on the shift which was given in Capacity in abstract Hilbert spaces and applications to higher order differential operators (Comm. P. D. E., 24:759–775, 1999) and obtain a lower bound which leads to a generalization of Thirring’s inequality if the underlying Hilbert space is an $L^2$-space. Moreover, a similar capacitary upper bound for the second eigenvalue is established. The results are finally applied to higher-order partial differential operators.
LA - eng
KW - capacity; Hilbert space; spectral bound; domain perturbation; shift; Thirring's inequality; second eigenvalue; higher-order partial differential operators
UR - http://eudml.org/doc/93385
ER -

References

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