PT-symmetric Quantum Mechanics Research Articles

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.

Quasi-hermiticity and the Role of a Metric in Some Boson Hamiltonians

The strategy of endowing PT-symmetric quantum mechanics with a positive definite metric, by adopting a modified inner product, has recently been explored in a simple non-hermitian quadratic boson Hamiltonian. We reconsider this analysis with emphasis on the question of a unique metric linked to the identification of an irreducible set of observables. Our results emphasise the necessity to ensure such a unique metric in order to establish a viable quantum mechanical framework.

Ambiguities in the Regularization of the Spiked Harmonic Oscillator

It has been suggested that the machinery of PT-symmetric quantum mechanics can be utilized to regularize certain singular potentials. In this contribution I point out that different regularizations lead to different results. In a particular model, that of the spiked harmonic oscillator, I use a symmetry inherent in the model to cast some light on this ambiguity.

Finite-dimensional -symmetric Hamiltonians

This paper investigates finite-dimensional representations of PT-symmetric Hamiltonians. In doing so, it clarifies some of the claims made in earlier papers on PT-symmetric quantum mechanics. In particular, 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.

Exponential Type Complex and Non-Hermitian Potentials in PT-Symmetric Quantum Mechanics

Using the NU method [A F Nikiforov and V B Uvarov 1988 “Special Functions of Mathematical Physics” (Birkhauser, Basel)], we investigated the real eigenvalues of the complex and/or PT-symmetric, non-Hermitian and the exponential type systems, such as Pöschl–Teller and Morse potentials.

Nonlinear holomorphic supersymmetry, Dolan–Grady relations and Onsager algebra

Recently, it was noticed by us that the nonlinear holomorphic supersymmetry of order n∈ N , n>1 ( n-HSUSY) has an algebraic origin. We show that the Onsager algebra underlies n-HSUSY and investigate the structure of the former in the context of the latter. A new infinite set of mutually commuting charges is found which, unlike those from the Dolan–Grady set, include the terms quadratic in the Onsager algebra generators. This allows us to find the general form of the superalgebra of n-HSUSY and fix it explicitly for the cases of n=2,3,4,5,6. The similar results are obtained for a new, contracted form of the Onsager algebra generated via the contracted Dolan–Grady relations. As an application, the algebraic structure of the known 1D and 2D systems with n-HSUSY is clarified and a generalization of the construction to the case of nonlinear pseudo-supersymmetry is proposed. Such a generalization is discussed in application to some integrable spin models and with its help we obtain a family of quasi-exactly solvable systems appearing in the PT-symmetric quantum mechanics.

Space of state vectors in 𝒫𝒯-symmetric quantum mechanics

The space of states of -symmetric quantum mechanics is examined. The requirement that eigenstates with different eigenvalues must be orthogonal leads to the conclusion that eigenfunctions belong to a space with an indefinite metric. Self-consistent expressions for the probability amplitude and the average value of operator are suggested. Further specification of the space of state vectors yields a superselection rule, redefining the notion of the superposition principle. An expression for the probability current density, satisfying the equation of continuity and vanishing for the bound state, is proposed.

GENERALIZED CONTINUITY EQUATION AND MODIFIED NORMALIZATION IN PT-SYMMETRIC QUANTUM MECHANICS

The continuity, equation relating the change in time of the position probability density to the gradient of the probability current density is generalized to PT-symmetric quantum mechanics. The normalization condition of eigenfunctions is modified in accordance with this new conservation law and illustrated with some detailed examples.

On the relation between Stokes multipliers and the T-Q systems of conformal field theory

The vacuum expectation values of the so-called Q -operators of certain integrable quantum field theories have recently been identified with spectral determinants of particular Schrödinger operators. In this paper we extend the correspondence to the T -operators, finding that their vacuum expectation values also have an interpretation as spectral determinants. As byproducts we give a simple proof of an earlier conjecture of ours, proved by another route by Suzuki, and generalise a problem in PT symmetric quantum mechanics studied by Bender and Boettcher. We also stress that the mapping between Q -operators and Schrödinger equations means that certain problems in integrable quantum field theory are related to the study of Regge poles in non-relativistic potential scattering.