Resolutions of Identity for Some Non-Hermitian Hamiltonians. II. Proofs
نویسندگان
چکیده
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 eigenand 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.
منابع مشابه
Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties
We study complex potentials and related non-diagonalizable Hamiltonians with special emphasis on formal definitions of associated functions and Jordan cells. The non-linear SUSY for complex potentials is considered and the theorems characterizing its structure are presented. We define the class of complex potentials invariant under SUSY transformations for (non-)diagonalizable Hamiltonians and ...
متن کاملNon-Hermitian von Roos Hamiltonian’s η-weak-pseudo-Hermiticity and exact solvability
A complexified von Roos Hamiltonian is considered and a Hermitian first-order intertwining differential operator is used to obtain the related position dependent mass η-weak-pseudo-Hermitian Hamiltonians. Two ”user -friendly” reference-target maps are introduced to serve for exactsolvability of some non-Hermitian η-weak-pseudo-Hermitian position dependent mass Hamiltonians. A non-Hermitian PT -...
متن کاملNon-Hermitian Quantum Mechanics of Non-diagonalizable Hamiltonians: puzzles with self-orthogonal states
We consider QM with non-Hermitian quasi-diagonalizable Hamiltonians, i.e. the Hamiltonians having a number of Jordan cells in particular biorthogonal bases. The ”self-orthogonality” phenomenon is clarified in terms of a correct spectral decomposition and it is shown that ”self-orthogonal” states never jeopardize resolution of identity and thereby quantum averages of observables. The example of ...
متن کاملPseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
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-Hermit...
متن کاملPseudo-Hermiticity versus PT -Symmetry II: A complete characterization of non-Hermitian Hamiltonians with a real spectrum
We give a necessary and sufficient condition for the reality of the spectrum of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal eigenvectors. Recently, we have explored in [1] the basic mathematical structure underlying the spectral properties of PT -symmetric Hamiltonians [2]. In particular, we have shown that these properties are associated with a class of more general (...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011