194 research outputs found
Resolutions of Identity for Some Non-Hermitian Hamiltonians. II. Proofs
This part is a continuation of the Part I where we built resolutions of
identity for certain non-Hermitian Hamiltonians constructed of biorthogonal
sets of their eigen- and associated functions for the spectral problem defined
on entire axis. Non-Hermitian Hamiltonians under consideration are taken with
continuous spectrum and the following cases are examined: an exceptional point
of arbitrary multiplicity situated on a boundary of continuous spectrum and an
exceptional point situated inside of continuous spectrum. In the present work
the rigorous proofs are given for the resolutions of identity in both cases
Hidden Symmetry from Supersymmetry in One-Dimensional Quantum Mechanics
When several inequivalent supercharges form a closed superalgebra in Quantum
Mechanics it entails the appearance of hidden symmetries of a
Super-Hamiltonian. We examine this problem in one-dimensional QM for the case
of periodic potentials and potentials with finite number of bound states. After
the survey of the results existing in the subject the algebraic and analytic
properties of hidden-symmetry differential operators are rigorously elaborated
in the Theorems and illuminated by several examples
Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
Resolutions of identity for certain non-Hermitian Hamiltonians constructed
from biorthogonal sets of their eigen- and associated functions are given for
the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under
consideration possess the continuous spectrum and the following peculiarities
are investigated: (1) the case when there is an exceptional point of arbitrary
multiplicity situated on a boundary of continuous spectrum; (2) the case when
there is an exceptional point situated inside of continuous spectrum. The
reductions of the derived resolutions of identity under narrowing of the
classes of employed test functions are revealed. It is shown that in the case
(1) some of associated functions included into the resolution of identity are
normalizable and some of them may be not and in the case (2) the bounded
associated function corresponding to the exceptional point does not belong to
the physical state space. Spectral properties of a SUSY partner Hamiltonian for
the Hamiltonian with an exceptional point are examined
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