Dresden 2011 – scientific programme
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A: Fachverband Atomphysik
A 7: Poster I
A 7.6: Poster
Monday, March 14, 2011, 16:00–18:30, P1
Numerical signatures of non-selfadjointness in quantum Hamiltonians — •Matthias Ruf1, Carsten Müller1, and Rainer Grobe2 — 1Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg — 2Intense Laser Physics Theory Unit and Department of Physics, Illinois State University, Normal, IL 61790-4560 USA
There are quantum mechanical Hamiltonians, which may lose their self-adjointness if certain parameters exceed certain values. For example, it is well known that the Dirac Hamiltonian for the Coulomb potential V(r)=−Z/r loses its essential self-adjointness if the nuclear charge Z exceeds the critical value of Zcr=118 [1].
While non-selfadjoint quantum mechanical operators do not necessarily possess eigenvalues, finite N× N matrix representations of these, however, may be hermitian and therefore have a finite set of N real eigenvalues. Using the momentum operator, the kinetic energy operator, and the relativistic Hamiltonian of the Coulomb problem for the Klein-Gordon equation as examples, we examine analytically and numerically the properties of the spectrum and eigenvectors in finite dimensional Hilbert spaces. We study the limit of N→∞ for which some eigenvalues cease to exist as the corresponding operators are not selfadjoint.
[1] B. Thaller, The Dirac Equation, Springer (1992)