Spectral analysis of the complex cubic oscillator

Consistent Treatment of Quantum Systems with a Time-Dependent Hilbert Space

We consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a time-independent Hilbert space. We show that in general the Hamiltonian operator does not represent an observable of the system even if it is a self-adjoint operator. This is related to a hidden geometric aspect of quantum mechanics arising from the presence of an operator-valued gauge potential. We also offer a careful treatment of quantum systems whose Hilbert space is obtained by endowing a time-independent vector space with a time-dependent inner product.

Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics

A non-Hermitian operator H defined in a Hilbert space with inner product 〈·|·〉 may serve as the Hamiltonian for a unitary quantum system if it is η-pseudo-Hermitian for a metric operator (positive-definite automorphism) η. The latter defines the inner product 〈·|η·〉 of the physical Hilbert space Hη of the system. For situations where some of the eigenstates of H depend on time, η becomes time-dependent. Therefore, the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian. Their proper treatment requires a geometric framework which clarifies the notion of the energy observable and leads to a geometric extension of quantum mechanics (GEQM). We provide a general introduction to the subject, review some of the recent developments, offer a straightforward description of the Heisenberg-picture formulation of the dynamics for quantum systems having a time-dependent Hilbert space, and outline the Heisenberg-picture formulation of dynamics in GEQM.

Visualization of branch points in PT-symmetric waveguides

The visualization of an exceptional point in a PT-symmetric directional coupler (DC) is demonstrated. In such a system the exceptional point can be probed by varying only a single parameter. Using the Rayleigh-Schrödinger perturbation theory we prove that the spectrum of a PT-symmetric Hamiltonian is real as long as the radius of convergence has not been reached. We also show how one can use a PT-symmetric directional coupler to measure the radius of convergence for non-PT-symmetric structures. For such systems the physical meaning of the rather mathematical term radius of convergence is exemplified.

Wave packet pseudomodes of variable coefficient differential operators

The pseudospectra of non-selfadjoint linear ordinary differential operators with variable coefficients are considered. It is shown that when a certain winding number or twist condition is satisfied, closely related to Hörmander's commutator condition for partial differential equations, ϵ -pseudoeigenfunctions of such operators for exponentially small values of ϵ exist in the form of localized wave packets. In contrast to related results of Davies and of Dencker, Sjöstrand & Zworski, the symbol need not be smooth. Applications in fluid mechanics, non-hermitian quantum mechanics and other areas are presented with the aid of high-accuracy numerical computations.