How to test for diagonalizability: the discretized PT-invariant square-well potential

Given a non-Hermitian matrix M, the structure of its minimal polynomial encodes whether M is diagonalizable or not. This note explains how to determine the minimal polynomial of a matrix without going through its characteristic polynomial. The approach is applied to a quantum mechanical particle moving in a square well under the influence of a piece-wise constant PT-symmetric potential. Upon discretizing the configuration space, the system is described by a matrix of dimension three which turns out not to be diagonalizable for a critical strength of the interaction. The systems develops a three-fold degenerate eigenvalue, and two of the three eigenfunctions disappear at this exceptional point, giving a difference between the algebraic and geometric multiplicity of the eigenvalue equal to two. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Finite-Dimensional PT-Symmetric Hamiltonians

This paper investigates finite-dimensional PT-symmetric Hamiltonians. It is shown here that there are two ways to extend real symmetric Hamiltonians into the complex domain: (i) The usual approach is to generalize such Hamiltonians to include complex Hermitian Hamiltonians. (ii) Alternatively, one can generalize real symmetric Hamiltonians to include complex PT-symmetric Hamiltonians. In the first approach the spectrum remains real, while in the second approach the spectrum remains real if the PT symmetry is not broken. Both generalizations give a consistent theory of quantum mechanics, but if D>2, a D-dimensional Hermitian matrix Hamiltonian has more arbitrary parameters than a D-dimensional PT-symmetric matrix Hamiltonian.     

Application of asymptotic iteration method to the Khare-Mandal and its PT-symmetric QES partner potential

We study the application of the asymptotic iteration method to the Khare-Mandal potential and its PT-symmetric partner. The eigenvalues and eigenfunctions for both potentials are obtained analytically. We have shown that although the quasi-exactly solvable energy eigenvalues of the Khare-Mandal potential are found to be in complex conjugate pairs for certain values of potential parameters, its PT-symmetric partner exhibits real energy eigenvalues in all cases.      

PT-symmetric Quantum Mechanics: A Precise and Consistent Formulation

The physical condition that the expectation values of physical observables are real quantities is used to give a precise formulation of PT-symmetric quantum mechanics. A mathematically rigorous proof is given to establish the physical equivalence of PT-symmetric and conventional quantum mechanics. The results reported in this paper apply to arbitrary PT-symmetric Hamiltonians with a real and discrete spectrum. They hold regardless of whether the boundary conditions defining the spectrum of the Hamiltonian are given on the real line or a complex contour.     

PT Symmetry in Quantum Field Theory

The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian. The Hermiticity of H guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). In this talk we investigate an alternative formulation of quantum mechanics in which the mathematical requirement of Hermiticity is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. We show that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are H=p ²+ix ³ and H=p ²-x ⁴. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry that we call C. Using C, we show how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables exhibit CPT symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.     

PT-Symmetric Quantum Field Theories and the Langevin Equation

Many non-Hermitian but PT-symmetric theories are known to have a real, positive spectrum, and for quantum-mechanical versions of these theories, there exists a consistent probabilistic interpretation. Since the action is complex for these theories, Monte Carlo methods do not apply. In this paper a field-theoretic method for numerical simulations of PT-symmetric Hamiltonians is presented. The method is the complex Langevin equation, which has been used previously to study complex Hamiltonians in statistical physics and in Minkowski space. We compute the equal-time one-point and two-point Green's functions in zero and one dimension, where comparisons to known results can be made. The method should also be applicable in four-dimensional space-time. This approach may grant insight into the formulation of a probabilistic interpretation for path integrals in PT-symmetric quantum field theories.     

$\mathcal{PT}$ Symmetry in Statistical Mechanics and the Sign Problem

Generalized PT\mathcal{PT} symmetry provides crucial insight into the sign problem for two classes of models. In the case of quantum statistical models at non-zero chemical potential, the free energy density is directly related to the ground state energy of a non-Hermitian, but generalized PT\mathcal{PT}-symmetric Hamiltonian. There is a corresponding class of PT\mathcal{PT}-symmetric classical statistical mechanics models with non-Hermitian transfer matrices. We discuss a class of Z(N) spin models with explicit PT\mathcal{PT} symmetry and also the ANNNI model, which has a hidden PT\mathcal{PT} symmetry. For both quantum and classical models, the class of models with generalized PT\mathcal{PT} symmetry is precisely the class where the complex weight problem can be reduced to real weights, i.e., a sign problem. The spatial two-point functions of such models can exhibit three different behaviors: exponential decay, oscillatory decay, and periodic behavior. The latter two regions are associated with PT\mathcal{PT} symmetry breaking, where a Hamiltonian or transfer matrix has complex conjugate pairs of eigenvalues. The transition to a spatially modulated phase is associated with PT\mathcal{PT} symmetry breaking of the ground state, and is generically a first-order transition. In the region where PT\mathcal{PT} symmetry is unbroken, the sign problem can always be solved in principle using the equivalence to a Hermitian theory in this region. The ANNNI model provides an example of a model with PT\mathcal{PT} symmetry which can be simulated for all parameter values, including cases where PT\mathcal{PT} symmetry is broken.     

1/L Expansions for a Class of PT-symmetric Potentials

The 1/L perturbative series for a special class of PT-symmetric potentials with centrifugal repulsive core are computed. Reality of obtained spectra is shown.     

Supersymmetry Without Hermiticity

The PT symmetry requirement of a potential defines an inhomogeneous system of first-order differential equations for the real/imaginary and even/odd components of the relevant superpotential. By identifying the general solutions of this system we search for non-trivial supersymmetric partner potentials and analyze whether they both possess PT symmetry. As an illustrative example we present the case of the Rosen-Morse I potential.     

Periodic Square-Well Potential and Spontaneous Breakdown of PT-symmetry

A particle moving on a circle in a purely imaginary one-step potential is studied in both the exact and broken PT-symmetric regime.     

Calculating the C Operator in PT-symmetric Quantum Mechanics

It has recently been shown that a non-Hermitian Hamiltonian H possessing an unbroken PT symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator C, which was originally defined as a sum over the eigenfunctions of H. However, using this definition it is cumbersome to calculate C in quantum mechanics and impossible in quantum field theory. An alternative method is devised here for calculating C directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method can be used to calculate the C operator in quantum field theory. The C operator is a new time-independent observable in PT-symmetric quantum field theory.     

Charge Conjugation from Space-Time Inversion

We show that the CPT group of the Dirac field emerges naturally from the PT and P (or T) subgroups of the Lorentz group.     

The C Operator in iϕ 3 Field Theory

The C operator defines a dynamically-determined positive-definite metric in PT-symmetric theories. We show how the operator formalism for the perturbative calculation of C can be extended from quantum mechanics to quantum field theory with a cubic self interaction.     

Real Spectra of PT-symmetric Operators and Perturbation Theory

A criterion is provided for the reality of the spectrum for a class of non self-adjoint operators in a Hilbert space invarariant under a particular discrete symmetry. Applications to the PT-symmetric Schrödinger operators are discussed.     

A Krein Space Approach to PT-symmetry

In this note we apply Krein space methods to PT-symmetric problems to obtain conditions for the spectrum to be real and estimates of the number of non-real spectral points. An erratum to this article is available at.     

Bound and Scattering States of Extended Calogero Model with an Additional PT Invariant Interaction

Here we discuss two many-particle quantum systems, which are obtained by adding some nonhermitian but PT (i.e. combined parity and time reversal) invariant interaction to the Calogero model with and without confining potential. It is shown that the energy eigenvalues are real for both of these quantum systems. For the case of extended Calogero model with confining potential, we obtain discrete bound states satisfying generalised exclusion statistics. On the other hand, the extended Calogero model without confining term gives rise to scattering states with continuous spectrum. The scattering phase shift for this case is determined through the exchange statistics parameter. We find that, unlike the case of usual Calogero model, the exclusion and exchange statistics parameters differ from each other in the presence of PT invariant interaction.     

The Scattering Problem in PT‐Symmetric Periodic Structures of 1D Two‐Material Waveguide Networks

A PT‐symmetric periodic structure with two‐material waveguide networks is constructed. In this study, how changing the number of cells affects the transmission properties is investigated. The results show that the PT‐unbroken (broken) region of the system is only determined by the cell structure, regardless of the number of unit cells. This means that any system has the same exceptional points (EPs), regardless of the number of cells and as long as the cell structure is consistent. In addition, it is confirmed that the coherent perfect absorbers and lasers (CPA lasers) occur in our model. The transfer matrix method is used to derive a sufficient condition for achieving the CPA laser point. A simple, effective formula for predicting the CPA laser state in an N unit cell system is derived.     

A Reality Proof in PT-Symmetric Quantum Mechanics

We review the proof of a conjecture concerning the reality of the spectra of certain PT-symmetric quantum mechanical systems, obtained via a connection between the theories of ordinary differential equations and integrable models. Spectral equivalences inspired by the correspondence are also discussed.     

$\mathcal{PT}$-Symmetric Periodic Optical Potentials

In quantum theory, any Hamiltonian describing a physical system is mathematically represented by a self-adjoint linear operator to ensure the reality of the associated observables. In an attempt to extend quantum mechanics into the complex domain, it was realized few years ago that certain non-Hermitian parity-time (PT\mathcal{PT}) symmetric Hamiltonians can exhibit an entirely real spectrum. Much of the reported progress has been remained theoretical, and therefore hasn’t led to a viable experimental proposal for which non Hermitian quantum effects could be observed in laboratory experiments. Quite recently however, it was suggested that the concept of PT\mathcal{PT}-symmetry could be physically realized within the framework of classical optics. This proposal has, in turn, stimulated extensive investigations and research studies related to PT\mathcal{PT}-symmetric Optics and paved the way for the first experimental observation of PT\mathcal{PT}-symmetry breaking in any physical system. In this paper, we present recent results regarding PT\mathcal{PT}-symmetric Optics.     

On nonlinear coherent states properties for electron-phonon dynamics

This work addresses a construction of a dual pair of nonlinear coherent states (NCS) in the context of changes of bases in the underlying Hilbert space for a model pertaining to an electron-phonon model in the condensed matter physics, obeying a f-deformed Heisenberg algebra. The existence and properties of reproducing kernel in the NCS Hilbert space are studied and discussed; the probability density and its dynamics in the basis of constructed coherent states are provided. A Glauber-Sudarshan P-representation of the density matrix and relevant issues related to the reproducing kernel properties are presented. Moreover, a NCS quantization of classical phase space observables is performed and illustrated in a concrete example of q-deformed coherent states. Finally, an exposition of quantum optical properties is given.