New class of conditionally exactly solvable potentials in quantum mechanics

PT -Symmetric Potentials from the Confluent Heun Equation

We derive exactly solvable potentials from the formal solutions of the confluent Heun equation and determine conditions under which the potentials possess PT symmetry. We point out that for the implementation of PT symmetry, the symmetrical canonical form of the Heun equation is more suitable than its non-symmetrical canonical form. The potentials identified in this construction depend on twelve parameters, of which three contribute to scaling and shifting the energy and the coordinate. Five parameters control the z(x) function that detemines the variable transformation taking the Heun equation into the one-dimensional Schrödinger equation, while four parameters play the role of the coupling coefficients of four independently tunable potential terms. The potentials obtained this way contain Natanzon-class potentials as special cases. Comparison with the results of an earlier study based on potentials obtained from the non-symmetrical canonical form of the confluent Heun equation is also presented. While the explicit general solutions of the confluent Heun equation are not available, the results are instructive in identifying which potentials can be obtained from this equation and under which conditions they exhibit PT symmetry, either unbroken or broken.

Gauge symmetry, chirality and parity effects in four-particle systems: Coulomb's law as a universal function for diatomic molecules

Following recent work in search for a universal function (Van Hooydonk, Eur. J. Inorg. Chem., (1999), 1617), we test four symmetric +/- a(n)Rn potentials for reproducing molecular potential energy curves (PECs). Classical gauge symmetry for 1/R-potentials results in generic left right asymmetric PECs. A pair of symmetric perturbed Coulomb potentials is quantitatively in accordance with observed PECs. For a bond, a four-particle system, charge inversion (a parity effect, atom chirality) is the key to explain this shape generically. A parity adapted Hamiltonian reduces from ten to two terms and to a soluble Bohr-like formula, a Kratzer (1 - Re/R)2 potential. The result is similar to the combined action of spin and wave function symmetry upon the Hamiltonian in Heitler-London theory. Analytical perturbed Coulomb functions varying with (1 - Re/R) scale attractive and repulsive branches of PECs for 13 bonds H2, HF, LiH, KH, AuH, Li2, LiF, KLi, NaCs, Rb2, RbCs, Cs2 and I2 in a single straight line. The 400 turning points for 13 bonds are reproduced with a deviation of 0.007 A at both branches. For 230 points at the repulsive side, the deviation is 0.003 A. The perturbed electrostatic Coulomb law is a universal molecular function. Ab initio zero molecular parameter functions give PECs of acceptable quality, just using atomic ionisation energies. The function can be used as a model potential for inverting levels and gives a first principle's comparison of short- and long-range interactions, important for the study of cold atoms. Wave-packet dynamics, femto-chemistry applied to the crossing of covalent and ionic curves, can provide evidence for this theory. We anticipate this scale/shape invariant scheme applies to smaller scales in nuclear and high-energy particle physics. For larger gravitational scales (Newton 1/R potentials), problems with super-unification are discussed. Reactions between hydrogen and antihydrogen, feasible in the near future, will probably produce normal H2.