Géza Lévai

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.

Algebraic and scattering aspects of a -symmetric solvable potential

We study a particular solvable potential and analyse the effect of symmetry on its bound state as well as scattering solutions. We determine the transmission and reflection coefficients for the -symmetric case and also formulate the problem in terms of an SU(1,1) potential group, which allows unified treatment of the discrete and the continuous spectra in a natural way. We find that (bound and scattering) states of the -symmetric problem supply a basis for the unitary irreducible representations of the SU(1,1) potential group, and this gives a straightforward group theoretical interpretation of the fact that the (complex) -invariant potential has a real energy spectrum.

Phase-Equivalent Complex Potentials

Potentials providing the same complex phase shifts as a given complex potential but with a shallower real part are constructed with supersymmetric transformations. Successive pairs of transformations eliminate normalizable solutions corresponding to complex eigenvalues of the Schrodinger equation with the full complex potential. With respect to real potentials, a new feature is the occurrence of normalizable solutions with complex energies presenting a positive real part. Removing such solutions provides a way of suppressing narrow resonances but may lead to complicated equivalent potentials with little physical interest. We discuss the singularity of the transformed potential and its relation with the Levinson theorem, the transformation of the Jost function, and the link with the Marchenko approach. The technique is tested with the solvable Poschl-Teller potential. As physical applications, deep optical potentials for the α + 16O and 16O + 16O scatterings are transformed into l-dependent phase-equivalent shallow optical potentials.

Conditions for complex spectra in a class of PT symmetric potentials

We study a wide class of solvable symmetric potentials in order to identify conditions under which these potentials have regular solutions with complex energy. Besides confirming previous findings for two potentials, most of our results are new. We demonstrate that the occurrence of conjugate energy pairs is a natural phenomenon for these potentials and that the present method can readily be extended to further potential classes.

PT-Symmetric Potentials and the so(2,2) Algebra

Starting from a differential realization of the generators of the so(2, 2) algebra we connect the eigenvalue equation of the Casimir invariant either with the hypergeometric equation, or the Schr¨ odinger equation. In the latter case we consider problems for whichso(2, 2) appears as a potentia la lgebra, connecting states with the same energy in different potentials. We analyse the role of the two so(2, 1) subalgebras and point out their importance for PT -symmetric problems, where the doubling of bound states is known to occur. We present two mechanisms for this and illustrate them with the example of the Scarf and the P¨ oschl–Teller II potentials. We also analyse scattering states, transmission

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.

The sextic oscillator as a γ -independent potential

The sextic oscillator is proposed as a two-parameter solvable {gamma}-independent potential in the Bohr Hamiltonian. It is shown that closed analytical expressions can be derived for the energies and wave functions of the first few levels and for the strength of electric quadrupole transitions between them. Depending on the parameters this potential has a minimum at {beta}=0 or at {beta}>0, and might also have a local maximum before reaching its minimum. A comparison with the spectral properties of the infinite square well and the {beta}{sup 4} potential is presented, together with a brief analysis of the experimental spectrum and E2 transitions of the {sup 134}Ba nucleus.

PT symmetry breaking and explicit expressions for the pseudo-norm in the Scarf II potential

Closed expressions are derived for the pseudo-norm, norm and orthogonality relations for arbitrary bound states of the PT symmetric and the Hermitian Scarf II potential for the first time. The pseudo-norm is found to have indefinite sign in general. Some aspects of the spontaneous breakdown of PT symmetry are analyzed.

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.

An exactly solvable Schrödinger equation with finite positive position-dependent effective mass

The solution of the one-dimensional Schrodinger equation is discussed in the case of position-dependent mass. The general formalism is specified for potentials that are solvable in terms of generalized Laguerre polynomials and mass functions that are positive and bounded on the whole real x axis. The resulting four-parameter potential is shown to belong to the class of “implicit” potentials. Closed expressions are obtained for the bound-state energies and the corresponding wave functions, including their normalization constants. The constant mass case is obtained by a specific choice of the parameters. It is shown that this potential contains both the harmonic oscillator and the Morse potentials as two distinct limiting cases and that the original potential carries several characteristics of these two potentials. Possible generalizations of the method are outlined.