Miloslav Znojil

Miloslav Znojil

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

Miloslav Znojil

Schroedinger equation with imaginary PT-symmetric potential $V^{}(x) = i\,x^3$ is studied using the numerical discretization methods in both the coordinate and momentum representations. In the former case our results confirm that the model generates an infinite number of bound states with real energies. In the latter case the differential equation is of the third order and a square-well, solvable approximation of kinetic energy is recommended and discussed. One finds that in the strong-coupling limit, the exact PT-symmetric solutions converge to their Hermitian predecessors.

Miloslav Znojil

Sextic oscillator in D dimensions is considered as a typical quasi-exactly solvable (QES) model. Usually, its QES N-plets of bound states have to be computed using the coupled Magyari's nonlinear algebraic equations. We propose and describe an alternative linear method which is N-independent and works with power series in 1/\sqrt(D). Main merit: simultaneous exact solvability (for all the QES states) in the first two leading orders (the degeneracy is completely removed, the unperturbed spectrum is equidistant). An additional merit: All the perturbation corrections are given by explicit matrix formulae in integer arithmetics (there are no rounding errors).