Miloslav Znojil

Systematic search for -symmetric potentials with real energy spectra

Changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions. Here we relax the standard Hermiticity requirement and find an innovative construction which leads to unusual, complex potentials. Their energy spectrum is shown to stay real after a weakening of the Hermiticity of the Schrodinger equation to its mere invariance under the combined (parity) and (time-reversal) symmetry. This ultimately results in richer bound-state spectra. Some of our new exactly solvable potentials generalize the current textbook models. Details are given for constructions based on the hypergeometric and confluent hypergeometric special functions.

Closed formula for the metric in the Hilbert space of a -symmetric model

We introduce a very simple, exactly solvable -symmetric non-Hermitian model with a real spectrum, and derive a closed formula for the metric operator which relates the problem to a Hermitian one.

Strong-coupling expansions for the -symmetric oscillators

We study the traditional problem of convergence of perturbation expansions when the hermiticity of the Hamiltonian is relaxed to a weaker symmetry. An elementary and quite exceptional cubic anharmonic oscillator is chosen as an illustrative example of such models. We describe its perturbative features paying particular attention to the strong-coupling regime. Efficient numerical perturbation theory proves suitable for such a purpose.

MHD α2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator

It is shown that the α2-dynamo of magnetohydrodynamics, the hydrodynamic Squire equation as well as an interpolation model of PT-symmetric quantum mechanics are closely related as spectral problems in Krein spaces. For the α2-dynamo and the PT-symmetric model the strong similarities are demonstrated with the help of a 2×2 operator matrix representation, whereas the Squire equation is reinterpreted as a rescaled and Wick-rotated PT-symmetric problem. Based on recent results on the Squire equation the spectrum of the PT-symmetric interpolation model is analyzed in detail and the Herbst limit is described as spectral singularity.

The Coulomb – harmonic oscillator correspondence in symmetric quantum mechanics

Abstract We show that and how the Coulomb potential V(x)=Z e 2 /x can be regularized and solved exactly at the imaginary coupling Z e 2 . The new spectrum of energies is real and bounded as expected, but its explicit form proves totally different from the usual real-coupling case.

-symmetrically regularized Eckart, Pöschl-Teller and Hulthén potentials

( = parity times time-reversal) symmetry of complex Hamiltonians with real spectra is usually interpreted as a weaker mathematical substitute for Hermiticity. Perhaps an equally important role is played by the related strengthened analyticity assumptions. In a constructive illustration we complexify a few potentials solvable only in s-wave. Then we continue their domain from the semi-axis to the whole axis and obtain new exactly solvable models. Their energies turn out to be real as expected. The new one-dimensional spectra themselves differ quite significantly from their s-wave predecessors.

Tridiagonal -symmetric N-by-N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime

A generic -symmetric Hamiltonian is assumed tridiagonalized and truncated to N < ∞ dimensions, H → H(chain model), and all its up–down symmetrized special cases with J = [N/2] real couplings are considered, H(chain model) → H(N). Using symbolic manipulation and extrapolation techniques we find out that in the strongly non-Hermitian regime the secular equation gets partially factorized at all N. This enables us to reveal a fine-tuned alignment of the dominant couplings implying an asymptotically sharply spiked shape of the boundary of the J-dimensional quasi-Hermiticity domain in which all the spectrum of energies E(N)n remains real and observable.

Maximal couplings in -symmetric chain models with the real spectrum of energies

The domain of all the coupling strengths compatible with the reality of the energies is studied for a family of non-Hermitian N × N matrix Hamiltonians H(N) with tridiagonal and -symmetric structure. At all dimensions N, the coordinates are found of the extremal points at which the boundary hypersurface touches the circumscribed sphere (for odd N = 2M + 1) or ellipsoid (for even N = 2K).

Non-Hermitian supersymmetry and singular, -symmetrized oscillators

Hermitian supersymmetric partnership between singular potentials V(q) = q2 + G/q2 breaks down and can only be restored on certain ad hoc subspaces (Das A and Pernice S 1999 Nucl. Phys. B 561 357). We show that within extended, -symmetric quantum mechanics, the supersymmetry between singular oscillators can be completely re-established in a way which is continuous near G = 0 and leads to a new form of the bosonic creation and annihilation operators.

Discrete {\cal PT} -symmetric models of scattering

One-dimensional scattering mediated by non-Hermitian Hamiltonians is studied. A schematic set of models is used which simulates two point interactions at a variable strength and distance. The feasibility of the exact construction of the amplitudes is achieved via the discretization of the coordinate. By direct construction it is shown that in all our models the probability is conserved. This feature is tentatively attributed to the space- and time-reflection symmetry (also known as -symmetry) of our specific Hamiltonians.