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Miloslav Znojil

Publications and source records attributed to Miloslav Znojil.

At least 235 records · Page 13Linked to original sources
arXiv2002

Non-Hermitian supersymmetry and singular PT symmetrized oscillators

SUSY partnership between singular potentials often breaks down. Via regularization it can be restored on certain ad hoc subspaces of Hilbert space [Das and Pernice, Nucl. Phys. B 561 (1999) 357]. Within the naturally complexified (so called PT symmetric) quantum mechanics we show how SUSY between strongly singular harmonic oscillators can completely be re-established. Our recipe leads to a new form of the bosonic creation and annihilation operators and proves continuous near the usual regular (i.e., linear harmonic) limit.

hep-th↗
arXiv2001

Spontaneous breakdown of PT symmetry in the solvable square-well model

In many PT symmetric models with real spectra, apparently, energy levels "merge and disappear" at a point of the spontaneous PT-symmetry breaking. We argue that such an oversimplified and discontinuous physical interpretation of this mechanism as proposed, e.g., by one of us in Phys. Lett. A 285 (2001), p. 7 would be inappropriate. Using the elementary square-well model of the above reference in the strongly non-Hermitian regime we exemplify how the doublets of states with broken PT symmetry continue to exist at complex conjugate energies. In contrast to many other exactly solvable examples of such a mechanism (we listed some of them in quant-ph/0110064), our present model of symmetry breaking does not complexify all the spectrum at once. "Realistically", it rather proceeds step by step, starting from the low-lying part of the spectrum and involving more and more excited states with an increasing degree of non-Hermiticity of the underlying interaction.

hep-th↗
arXiv2001

Generalized Rayleigh-Schroedinger perturbation theory as a method of linearization of the so called quasi-exactly solvable models

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).

math-ph↗
arXiv2001

Anomalous doublets of states in a PT symmetric quantum model

We complexify one of the Natanzon's exactly solvable potentials in PT symmetric manner and discover that it supports the pairs of bound states with the same number of nodal zeros. This could indicate that the Sturm Liouville oscillation theorem does not admit an immediate generalization.

quant-ph↗
arXiv2001

A generalization of the concept of PT symmetry

We show that the PT symmetric Hamiltonians (and their generalizations defined in the text) may be all assigned the projected (so called Feshbach or effective) nonlinear Hamiltonians which are "locally" Hermitian. This implies that many (if not all) of the bound-state energies may be real in a broad domain of Hermiticity-violating interactions. A complexification of a Winternitz's super-integrable example in two dimensions is conjectured as an illustration.

math-ph↗
arXiv2001

Annihilation and creation operators in non-Hermitian supersymmetric quantum mechanics

Harmonic oscillators with a centrifugal spike are analysed, via a non-Hermitian regularization, within a complexified SUSY quantum mechanics. The formalism enables us to construct the factorized creation and annihilation operators. We show how the real though, generically, non-equidistant spectrum complies with the current SUSY-type isospectrality and degeneracy in an unusual way.

hep-th↗
arXiv2001

Conservation of pseudo-norm in PT symmetric quantum mechanics

An extension of the Stone's theorem to PT symmetric, extended quantum mechanics must employ its pseudo-norm. We show that the evolution of the wave functions is then pseudo-unitary, and emphasize that their pseudo-norm remains time-independent even in the spontaneously broken symmetry regime where the energies themselves cease to be real.

math-ph↗
arXiv2001

Solvable analogue of the imaginary cubic oscillator

We prove that the purely imaginary square well generates an infinite number of bound states with real energies. In the strong-coupling limit, our exact PT symmetric solutions coincide, utterly unexpectedly, with their textbook, well known Hermitian predecessors.

quant-ph↗
arXiv2001

Exactly solvable three-body Calogero-type model with translucent two-body barriers

A new exactly solvable alternative to the Calogero three-particle model is proposed. Sharing its confining long-range part, it contains the mere zero-range two-particle barriers. Their penetrability gives rise to a tunneling, tunable via their three independent strengths. Their variability can control the removal of the degeneracy of the energy levels in an innovative, non-perturbative manner.

nucl-th↗
arXiv2001

Poeschl-Teller paradoxes

Poeschl-Teller trigonometric potential well is PT symmetrically regularized at its "impenetrable" end-point barriers. This gives the four different solvable generalizations of the model and enables us to clarify some paradoxes encountered, in the SUSY context, in the classically forbidden coupling regime.

math-ph↗
arXiv2001

Comprehensive analysis of conditionally exactly solvable models

We study a quantum mechanical potential introduced previously as a conditionally exactly solvable (CES) model. Besides an analysis following its original introduction in terms of the point canonical transformation, we also present an alternative supersymmetric construction of it. We demonstrate that from the three roots of the implicit cubic equation defining the bound-state energy eigenvalues, there is always only one that leads to a meaningful physical state. Finally we demonstrate that the present CES interaction is, in fact, an exactly solvable Natanzon-class potential.

math-ph↗
arXiv2001

PT symmetric square well

In a PT symmetrically complexified square well, bound states are constructed by the matching technique. Their energies prove real in a domain of weak non-Hermiticity, and continuous in the Hermitian limit. At a sequence of certain strong-Hermiticity thresholds, the lowest two energies merge and disappear.

quant-ph↗
arXiv2000

Spiked and PT-symmetrized decadic potentials supporting elementary N-plets of bound states

We show that the potential wells with a central spike V(x) = x^{10} + a x^8 + b x^6+ c x^4 + d x^2 + f/x^2 possess arbitrary finite multiplets of elementary exact bound states. The strong asymptotic growth of V(x) implies an ambiguity in the PT-symmetric quantization via complex boundary conditions but the three eligible recipes coincide at our exceptional solutions.

quant-ph↗
arXiv2000

New perturbation method with the matching of wave functions

We propose a new approach to the Rayleigh-Schrödinger perturbation expansions of bound states in quantum mechanics. We are inspired by the enormous flexibility of solvable interactions with several (N) discontinuities. Their standard matching solution is modified and transferred in perturbation regime. We employ the global renormalization freedom of the local wave functions and derive a compact N-dimensional matrix formula for corrections. In applications, our recipe is shown non-numerical for all polynomial perturbations of any piece-wise constant zero order potential.

quant-ph↗