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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

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

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.

Related papers

PT-Symmetric Quantum Mechanics

This paper proposes to broaden the canonical formulation of quantum mechanics.Ordinarily, one imposes the condition H † = H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition.This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum.However, replacing this mathematical condition by the weaker and more physical requirement H ‡ = H, where ‡ represents combined parity reflection and time reversal PT , one obtains new classes of complex Hamiltonians whose spectra are still real and positive.This generalization of Hermiticity is investigated using a complex deformation H = p 2 + x 2 (ix) ǫ of the harmonic oscillator Hamiltonian, where ǫ is a real parameter.The system exhibits two phases: When ǫ ≥ 0, the energy spectrum of H is real and positive as a consequence of PT symmetry.However, when -1 < ǫ < 0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because PT symmetry is spontaneously broken.The phase transition that occurs at ǫ = 0 manifests itself in both the quantum-mechanical system and the underlying classical system.Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians H = p 2 + x 2N (ix) ǫ with N integer and ǫ > -N ; each of these complex Hamiltonians exhibits a phase transition at ǫ = 0.These PT -symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.

WORLD SCIENTIFIC (EUROPE) eBooks•Nov 21, 2018•1 citation

Admissible perturbations and false instabilities in PT -symmetric quantum systems

Miloslav Znojil

One of the most characteristic mathematical features of the $\mathcal{PT}$-symmetric quantum mechanics is the explicit Hamiltonian dependence of its physical Hilbert space of states $\mathcal{H}=\mathcal{H}(H)$. Some of the most important physical consequences are discussed, with emphasis on the dynamical regime in which the system is close to phase transition. Consistent perturbation treatment of such a regime is proposed. An illustrative application of the innovated perturbation theory to a non-Hermitian but $\mathcal{PT}$-symmetric user-friendly family of $J$-parametric ``discrete anharmonic'' quantum Hamiltonians $H=H(\stackrel{P\vec}{\ensuremath{\lambda}})$ is provided. The models are shown to admit the standard probabilistic interpretation if and only if the parameters remain compatible with the reality of the spectrum, $\stackrel{P\vec}{\ensuremath{\lambda}}\ensuremath{\in}{\mathcal{D}}^{(\mathrm{physical})}$. In contradiction to conventional wisdom, the systems are then shown to be stable with respect to admissible perturbations, inside the domain ${\mathcal{D}}^{(\mathrm{physical})}$, even in the immediate vicinity of the phase-transition boundaries $\ensuremath{\partial}{\mathcal{D}}^{(\mathrm{physical})}$.

arXiv:1803.01949•Physical Review A•Mar 16, 2018•23 citations

Introduction

Jan Nesemann

In the theory of quantum mechanics the Hamiltonian H is typically selfadjoint, i.e., H = H *. The self-adjointness ensures that the spectrum of the Hamiltonian, representing the energy spectrum of H , is real but it is not a necessary condition. In the literature on so-called PT -symmetric quantum mechanics (see, e.g., [BB98], [BBM99], [BBJ03], [Ben04b] and [Ben07]), it is believed that self-adjointness is rather a mathematical requirement than a physically established fact. Therefore, it was considered a surprise that operators exist which are not self-adjoint in the given quantum mechanical Hilbert space, but have real spectrum and that – if e.g. complex eigenvalues were present – they occurred only in complex conjugate pairs.

Vieweg+Teubner eBooks•Jan 1, 2011•0 citations

Spectra of PT -symmetric operators and perturbation theory

Emanuela Caliceti, S. Graffi, Johannes Sjöstrand

A criterion is formulated for existence and another for the non-existence of complex eigenvalues for a class of non-self-adjoint operators in Hilbert space invariant under a particular discrete symmetry. Applications to the PT -symmetric Schrödinger operators are discussed.

arXiv:math-ph/0407052•Journal of Physics A Mathematical and General•Dec 9, 2004•62 citations

Non-Hermitian PT-Symmetric Theories

Alireza Beygi

This dissertation concerns the concept of PT symmetry as a replacement of the conventional symmetry of the Dirac Hermiticity of the operators, in particular, for the Hamiltonian. After solving a controversial issue surrounding the claimed violation of the special relativity by PT-symmetric quantum mechanics, we move further to discuss PT symmetry for the systems whose associated time reversal symmetry is odd, i.e., T^2 = −1. Our results, as published papers, are incorporated in the subsequent chapters. Then we conclude by suggestions for future work.

•Jan 1, 2019•0 citations

Surprising spectra of PT-symmetric point interactions

Petr Siegl

Spectra of the second derivative operators corresponding to the special PT-symmetric point interactions are studied. The results are partly the completion of those obtained in [1]. The particular PT-symmetric point interactions causing unusual spectral effects are investigated for the systems defined on a finite interval as well. The spectrum of this type of interactions is very far from the self-adjoint case despite of PT-symmetry, P-pseudo-Hermiticity and T-self-adjointness.

arXiv:0906.0226•Jun 1, 2009•2 citations
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 PTPTPT-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.
This paper proposes to broaden the canonical formulation of quantum mechanics.Ordinarily, one imposes the condition H † = H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition.This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum.However, replacing this mathematical condition by the weaker and more physical requirement H ‡ = H, where ‡ represents combined parity reflection and time reversal PT , one obtains new classes of complex Hamiltonians whose spectra are still real and positive.This generalization of Hermiticity is investigated using a complex deformation H = p 2 + x 2 (ix) ǫ of the harmonic oscillator Hamiltonian, where ǫ is a real parameter.The system exhibits two phases: When ǫ ≥ 0, the energy spectrum of H is real and positive as a consequence of PT symmetry.However, when -1 < ǫ < 0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because PT symmetry is spontaneously broken.The phase transition that occurs at ǫ = 0 manifests itself in both the quantum-mechanical system and the underlying classical system.Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians H = p 2 + x 2N (ix) ǫ with N integer and ǫ > -N ; each of these complex Hamiltonians exhibits a phase transition at ǫ = 0.These PT -symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.
One of the most characteristic mathematical features of the PT\mathcal{PT}PT-symmetric quantum mechanics is the explicit Hamiltonian dependence of its physical Hilbert space of states H=H(H)\mathcal{H}=\mathcal{H}(H)H=H(H). Some of the most important physical consequences are discussed, with emphasis on the dynamical regime in which the system is close to phase transition. Consistent perturbation treatment of such a regime is proposed. An illustrative application of the innovated perturbation theory to a non-Hermitian but PT\mathcal{PT}PT-symmetric user-friendly family of JJJ-parametric ``discrete anharmonic'' quantum Hamiltonians $H=H(\stackrel{P\vec}{\ensuremath{\lambda}})$ is provided. The models are shown to admit the standard probabilistic interpretation if and only if the parameters remain compatible with the reality of the spectrum, $\stackrel{P\vec}{\ensuremath{\lambda}}\ensuremath{\in}{\mathcal{D}}^{(\mathrm{physical})}$. In contradiction to conventional wisdom, the systems are then shown to be stable with respect to admissible perturbations, inside the domain D(physical){\mathcal{D}}^{(\mathrm{physical})}D(physical), even in the immediate vicinity of the phase-transition boundaries $\ensuremath{\partial}{\mathcal{D}}^{(\mathrm{physical})}$.
In the theory of quantum mechanics the Hamiltonian H is typically selfadjoint, i.e., H = H *. The self-adjointness ensures that the spectrum of the Hamiltonian, representing the energy spectrum of H , is real but it is not a necessary condition. In the literature on so-called PT -symmetric quantum mechanics (see, e.g., [BB98], [BBM99], [BBJ03], [Ben04b] and [Ben07]), it is believed that self-adjointness is rather a mathematical requirement than a physically established fact. Therefore, it was considered a surprise that operators exist which are not self-adjoint in the given quantum mechanical Hilbert space, but have real spectrum and that – if e.g. complex eigenvalues were present – they occurred only in complex conjugate pairs.
This dissertation concerns the concept of PT symmetry as a replacement of the conventional symmetry of the Dirac Hermiticity of the operators, in particular, for the Hamiltonian. After solving a controversial issue surrounding the claimed violation of the special relativity by PT-symmetric quantum mechanics, we move further to discuss PT symmetry for the systems whose associated time reversal symmetry is odd, i.e., T^2 = −1. Our results, as published papers, are incorporated in the subsequent chapters. Then we conclude by suggestions for future work.