Self-adjoint, globally defined Hamiltonian operators for  systems with boundaries

Nuno Costa Dias; Andrea Posilicano; João Nuno Prata

doi:10.3934/cpaa.2011.10.1687

Article Contents

2011, Volume 10, Issue 6: 1687-1706. Doi: 10.3934/cpaa.2011.10.1687

This issue Previous Article Unbounded solutions of the nonlocal heat equation Next Article Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations

Self-adjoint, globally defined Hamiltonian operators for systems with boundaries

1.
Universidade Lusófona de Humanidades e Tecnologias, Av. Campo Grande 376, 1749-024 Lisboa
2.
Dipartimento di Scienze Fisiche e Matematiche, Università dell'Insubria, via valleggio 11, I-22100 Como

Received: February 28, 2010

Revised: January 31, 2011

Published: October 2011

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Abstract

For a general self-adjoint Hamiltonian operator $H_0$ on the Hilbert space $L^2(R^d)$, we determine the set of all self-adjoint Hamiltonians $H$ on $L^2(R^d)$ that dynamically confine the system to an open set $\Omega \subset \RE^d$ while reproducing the action of $ H_0$ on an appropriate operator domain. In the case $H_0=-\Delta +V$ we construct these Hamiltonians explicitly showing that they can be written in the form $H=H_0+ B$, where $B$ is a singular boundary potential and $H$ is self-adjoint on its maximal domain. An application to the deformation quantization of one-dimensional systems with boundaries is also presented.

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Mathematics Subject Classification: Primary: 81Q10, 47B25; Secondary: 81S30.

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References

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