arXiv:quant-ph/0206112

Point Interactions: PT-Hermiticity and Reality of the Spectrum

Sergio A Albeverio, S.-M. Fei, Pavel Kurasov

Published Jun 18, 2002• 4 citations

Abstract

General point interactions for the second derivative operator in one dimension are studied. In particular, ${\mathcal P \mathcal T}$-self-adjoint point interactions with the support at the origin and at points $\pm l$ are considered. The spectrum of such non-Hermitian operators is investigated and conditions when the spectrum is pure real are presented. The results are compared with those for standard self-adjoint point interactions.

DOI: 10.48550/arxiv.quant-ph/0206112

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arXiv:quant-ph/0206112

Point Interactions: PT-Hermiticity and Reality of the Spectrum

Sergio A Albeverio, S.-M. Fei, Pavel Kurasov

Published Jun 18, 2002• 4 citations

arXiv HTML PDF

Abstract

General point interactions for the second derivative operator in one dimension are studied. In particular,

{\mathcal P \mathcal T}

-self-adjoint point interactions with the support at the origin and at points

\pm l

are considered. The spectrum of such non-Hermitian operators is investigated and conditions when the spectrum is pure real are presented. The results are compared with those for standard self-adjoint point interactions.

DOI: 10.48550/arxiv.quant-ph/0206112

Related papers

Surprising spectra of PT-symmetric point interactions

Petr Siegl

arXiv:0906.0226Jun 1, 20092 citations

Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian

Ali Mostafazadeh

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 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.

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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.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.

Jul 1, 20014 citations

On domains of {\cal P}{\cal T} symmetric operators related to − y ″( x ) + (− 1) n x 2 n y ( x )

T. Ya. Azizov, Carsten Trunk

In recent years a generalization of Hermiticity has been investigated using a complex deformation H = p 2 + x 2 (i x ) of the harmonic oscillator Hamiltonian, where is a real parameter. These complex Hamiltonians, possessing symmetry (the product of parity and time reversal), can have a real spectrum. We will consider the most simple case: even. In this paper we describe all self-adjoint (Hermitian) and at the same time symmetric operators associated with H = p 2 + x 2 (i x ) . Surprisingly, it turns out that there is a large class of self-adjoint operators associated with H = p 2 + x 2 (i x ) which are not symmetric.In recent years a generalization of Hermiticity has been investigated using a complex deformation H = p 2 + x 2 (i x ) of the harmonic oscillator Hamiltonian, where is a real parameter. These complex Hamiltonians, possessing symmetry (the product of parity and time reversal), can have a real spectrum. We will consider the most simple case: even. In this paper we describe all self-adjoint (Hermitian) and at the same time symmetric operators associated with H = p 2 + x 2 (i x ) . Surprisingly, it turns out that there is a large class of self-adjoint operators associated with H = p 2 + x 2 (i x ) which are not symmetric.

arXiv:0911.1284Journal of Physics A Mathematical and TheoreticalApr 14, 201012 citations

Criteria for the reality of the spectrum of PT-symmetric Schroedinger operators and for the existence of PT-symmetric phase transitions

Emanuela Caliceti, S. Graffi

A major mathematical problem in PT-symmetric quantum mechanics is to determine whether or not the spectrum of any given non self-adjoint but PT-symmetric Schroedinger operator is real. In this case the PT-symmetry is called "proper". Clearly in this connection an equally important issue is the spontaneous violation of the PT-symmetry which might occur in a PT-symmetric operator family. The spontaneous violation of the PT-symmetry is defined as the transition from real values of the spectrum to complex ones at the variation of the parameter labeling the family. Its occurrence is referred to also as the $PT$ -symmetric phase transition. \nThis chapter intends to present a review of recent results concerning these two mathematical points, within the standard notions of spectral theory for Hilbert space operators. The main technical instrument is represented by perturbation theory.A major mathematical problem in PT-symmetric quantum mechanics is to determine whether or not the spectrum of any given non self-adjoint but PT-symmetric Schroedinger operator is real. In this case the PT-symmetry is called "proper". Clearly in this connection an equally important issue is the spontaneous violation of the PT-symmetry which might occur in a PT-symmetric operator family. The spontaneous violation of the PT-symmetry is defined as the transition from real values of the spectrum to complex ones at the variation of the parameter labeling the family. Its occurrence is referred to also as the $PT$-symmetric phase transition. \nThis chapter intends to present a review of recent results concerning these two mathematical points, within the standard notions of spectral theory for Hilbert space operators. The main technical instrument is represented by perturbation theory.

Wiley eBooksJan 1, 20150 citations

General point interactions for the second derivative operator in one dimension are studied. In particular,

{\mathcal P \mathcal T}

-self-adjoint point interactions with the support at the origin and at points

\pm l

PT

-symmetric phase transition. \nThis chapter intends to present a review of recent results concerning these two mathematical points, within the standard notions of spectral theory for Hilbert space operators. The main technical instrument is represented by perturbation theory.