Hendrik B. Geyer

Quasi-Hermitian operators in quantum mechanics and the variational principle

We establish a general criterion for a set of non-Hermitian operators to constitute a consistent quantum mechanical system, which allows for the normal quantum-mechanical interpretation. This involves the construction of a metric (if it exists) for the given set of non-Hermitian observables. We discuss uniqueness of this metric. We also show that it is not always necessary to construct the metric for the whole set of observables under consideration, but that it is sufficient for some calculational purposes to construct it for a subset only, even though this metric is, in general, not unique. The restricted metric turns out to be particularly useful in the implementation of a variational principle, which we also formulate.

Choice of a metric for the non-Hermitian oscillator

The harmonic oscillator Hamiltonian, when augmented by a non-Hermitian PT -symmetric part, can be transformed into a Hermitian Hamiltonian. This is achieved by introducing a metric which, in general, renders other observables such as the usual momentum or position as non-Hermitian operators. The metric depends on one real parameter, the full range of which is investigated. The explicit functional dependence of the metric and each associated Hamiltonian is given. A specific choice of this parameter determines a specific combination of position and momentum as being an observable; this can be in particular either standard position or momentum, but not both simultaneously. Singularities of the metric are explored and their removability is investigated. The physical significance of these findings is discussed.

Construction of a unique metric in quasi-Hermitian quantum mechanics: Nonexistence of the charge operator in a 2 × 2 matrix model

Abstract For a specific exactly solvable 2 × 2 matrix model with a PT -symmetric Hamiltonian possessing a real spectrum, we construct all the eligible physical metrics Θ > 0 and show that none of them admits a factorization Θ = CP in terms of an involutive charge operator C . Alternative ways of restricting the physical metric to a unique form are briefly discussed.

Moyal products—a new perspective on quasi-Hermitian quantum mechanics

The rationale for introducing non-Hermitian Hamiltonians and other observables is reviewed and open issues identified. We present a new approach based on Moyal products to compute the metric for quasi-Hermitian systems. This approach is not only an efficient method of computation, but also suggests a new perspective on quasi-Hermitian quantum mechanics which invites further exploration. In particular, we present some first results which link the Berry connection and curvature to non-perturbative properties and the metric.

Operator equations and Moyal products–metrics in quasi-Hermitian quantum mechanics

Abstract The Moyal product is used to cast the equation for the metric of a non-Hermitian Hamiltonian in the form of a differential equation. For Hamiltonians of the form p 2 + V ( i x ) with V polynomial this is an exact equation. Solving this equation in perturbation theory recovers known results. Explicit criteria for the hermiticity and positive definiteness of the metric are formulated on the functional level.

The large N behaviour of the Lipkin model and exceptional points

The ubiquitous Lipkin model is investigated for an interaction parameter beyond the traditional critical point. It is argued that a phase transition occurs higher up in the spectrum for such larger interaction, where, using appropriate scaling of the energies, the position of the phase transition becomes independent of the particle number. The phase transition is related to near singularities in the complex interaction plane, the exceptional points. Consideration of finite temperature yields the well-known physical features associated with phase transitions.

Loosely bound hyperons in the SU(3) Skyrme model

Hyperon pairs bound in deuteronlike states are obtained with the SU(3) Skyrme model in agreement with general expectations from boson exchange models. The central binding from the flavor-symmetry-breaking terms increases with the strangeness contents of the interacting baryons whereas the kinetic nonlinear [sigma]-model term fixes the spin and isospin of the bound pair. We give a complete account of the interactions of octet baryons within the product approximation to baryon number [ital B]=2 configurations.

The Casimir energy of strongly bound B = 2 configurations in the Skyrme model

Abstract The status of the H-dibaryon in the SU(3) Skyrme model is reconsidered by taking the Casimir energy of the relevant B = 2 soliton into account. Since the negative Casimir-energy corrections for the B = 2 configuration are much smaller than the corresponding ones for two isolated B = 1 systems a strongly bound H-dibaryon is excluded and a slightly bound one unlikely.

Non-abelian bosonization from factored coset models in path integrals

We present a derivation of abelian and non-abelian bosonization in a path integral setting by expressing the generating functional for current-current correlation functions as a product of a G/G-coset model, which is dynamically trivial, and a bosonic part which contains the dynamics. A BRST symmetry can be identified which leads to smooth bosonization in both the abelian and non-abelian cases.We present a derivation of abelian and non-abelian bosonization in a path integral setting by expressing the generating functional for current-current correlation functions as a product of a

Sturm-Schrödinger equations: Formula for metric

G/G