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Sensitizing Non-Hermitian Hamiltonian for Study of Real Spectra

Biswanath Rath

Published Feb 1, 2015• 0 citations
OpenAlex

Abstract

.We notice many non-Hermitian Hamiltonians yield real spectra .Interestingly some of them belong to PT symmetry and others non-PT symmetry in nature.In the case of simple quantum systems, one can calculate the eigen- spectra analytically, however in other cases one has seek numerical results .We give specific atten tion on stable real spectra .

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.We notice many non-Hermitian Hamiltonians yield real spectra .Interestingly some of them belong to PT symmetry and others non-PT symmetry in nature.In the case of simple quantum systems, one can calculate the eigen- spectra analytically, however in other cases one has seek numerical results .We give specific atten tion on stable real spectra .
We notice that non-linear oscillator Hamiltonian having the character of non-Hermitian (H≠H+)(H \ne H^{+})(H=H+), non-PT Symmetry ([H,PT]≠0)([H,PT] \ne 0)([H,PT]=0) and non-Pseudohermiticity (ηHη−1≠H+)(\eta H \eta^{-1}\ne H^{+})(ηHη−1=H+) can yield real eigenvalues.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudosupersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler–DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.