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Journal of Physics A Mathematical and General

Algebraic and scattering aspects of a đť’«đť’Ż-symmetric solvable potential

GÉZA LÉVAI, F. Cannata, Alberto Ventura

Published Jan 18, 2001• 82 citations
Publisher

Abstract

We study a particular solvable potential and analyse the effect of đť’«đť’Ż symmetry on its bound state as well as scattering solutions. We determine the transmission and reflection coefficients for the đť’«đť’Ż-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 đť’«đť’Ż-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) đť’«đť’Ż-invariant potential has a real energy spectrum.

DOI
10.1088/0305-4470/34/4/310

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We study a particular solvable potential and analyse the effect of đť’«đť’Ż symmetry on its bound state as well as scattering solutions. We determine the transmission and reflection coefficients for the đť’«đť’Ż-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 đť’«đť’Ż-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) đť’«đť’Ż-invariant potential has a real energy spectrum.
A general formalism is worked out for the description of one-dimensional scattering in non-hermitian quantum mechanics and constraints on transmission and reflection coefficients are derived in the cases of P, T, or PT invariance of the Hamiltonian. Applications to some solvable PT-symmetric potentials are shown in detail. Our main original results concern the association of reflectionless potentials with asymptotic exact PT symmetry and the peculiarities of separable kernels of non-local potentials in connection with hermiticity, T invariance and PT invariance.
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 Schrödinger 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.
Scattering on the -symmetric Coulomb potential is studied along a U-shaped trajectory circumventing the origin in the complex x plane from below. This trajectory reflects symmetry, sets the appropriate boundary conditions for bound states and also allows the restoration of the correct sign of the energy eigenvalues. Scattering states are composed from the two linearly independent solutions valid for non-integer values of the 2 L parameter, which would correspond to the angular momentum in the usual Hermitian setting. The transmission and reflection coefficients are written in a closed analytic form, and it is shown that, similar to other -symmetric scattering systems, the latter exhibit the handedness effect. Bound-state energies are recovered from the poles of the transmission coefficients.
We investigate the approximate bound state solutions of the Schrödinger equation for the PT-/non-PT-symmetric and non Hermitian Hellmann potential. Exact energy eigenvalues and corresponding normalized wave functions are obtained. Numerical values of energy eigenvalues for the bound states are compared with the ones obtained before. Scattering state solutions are also studied. Phase shifts of the potential are written in terms of the angular momentum quantum number ℓ\ellℓ.