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WORLD SCIENTIFIC eBooks

CONJECTURE ON THE REALITY OF SPECTRA OF NON-HERMITIAN HAMILTONIANS

Carl M. Bender, Stefan Boettcher, Peter N. Meisinger

Published Jun 1, 2001• 0 citations
Publisher
DOI
10.1142/9789812811240_0010

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We introduce and study a class of non-Hermitian Hamiltonians which have velocity dependent potentials. Since stability cannot be advocated directly from the classical potential, we show that the energy spectra are real and bounded from below which proves the stability of the spectra of all members in the class. We find that the introduced class of non-Hermitian Hamiltonians do have a corresponding superpartner class of non-Hermitian Hamiltonians. We were able to introduce supercharges which in conjunction with the corresponding super Hamiltonians constitute a closed super algebra. Among the introduced Hamiltonians, we show that non-[Formula: see text]-symmetric Hamiltonians can be transformed into their corresponding superpartner Hamiltonians via a specific canonical transformation while the [Formula: see text]-symmetric ones failed to be mapped to their corresponding superpartner Hamiltonians via the same canonical transformation. Since canonical transformations preserve the spectrum, we conclude that non-[Formula: see text]-symmetric Hamiltonians out of the introduced class of Hamiltonians have the same spectrum as the corresponding superpartner Hamiltonians and thus supersymmetry (Susy) is broken for such Hamiltonians. This kind of intertwining of [Formula: see text]-symmetry and Susy is new as all the so far discussed cases concentrate on Hamiltonians of broken [Formula: see text]-symmetry that have broken Susy too while we showed that Susy can be also broken for non-[Formula: see text]-symmetric and non-Hermitian Hamiltonians.