Exactness of SWKB for shape invariant potentials

The supersymmetry-based semiclassical method (SWKB) is known to produce exact spectra for conventional shape invariant potentials. In this paper we prove that this exactness follows from their additive shape invariance.     

Inter-relations between additive shape invariant superpotentials

All known additive shape invariant superpotentials in nonrelativistic quantum mechanics belong to one of two categories: superpotentials that do not explicitly depend on ħ, and their ħ-dependent extensions. The former group themselves into two disjoint classes, depending on whether the corresponding Schrödinger equation can be reduced to a hypergeometric equation (type-I) or a confluent hypergeometric equation (type-II). All the superpotentials within each class are connected via point canonical transformations. Previous work [19] showed that type-I superpotentials produce type-II via limiting procedures. In this paper we develop a method to generate a type I superpotential from type II, thus providing a pathway to interconnect all known additive shape invariant superpotentials.     

Infinitely many shape invariant potentials and new orthogonal polynomials

Three sets of exactly solvable one-dimensional quantum mechanical potentials are presented. These are shape invariant potentials obtained by deforming the radial oscillator and the trigonometric/hyperbolic Pöschl–Teller potentials in terms of their degree ℓ polynomial eigenfunctions. We present the entire eigenfunctions for these Hamiltonians (ℓ=1,2,…) in terms of new orthogonal polynomials. Two recently reported shape invariant potentials of Quesne and Gómez-Ullate et al.'s are the first members of these infinitely many potentials.     

Scattering amplitudes for multi-indexed extensions of solvable potentials

New solvable one-dimensional quantum mechanical scattering problems are presented. They are obtained from known solvable potentials by multiple Darboux transformations in terms of virtual and pseudo virtual wavefunctions. The same method applied to confining potentials, e.g.  Pöschl–Teller and the radial oscillator potentials, has generated the multi-indexed Jacobi and Laguerre polynomials. Simple multi-indexed formulas are derived for the transmission and reflection amplitudes of several solvable potentials.     

Phase transition of directed polymer in random potentials on 4+1 dimensions

Finite-temperature-directed polymer in random potentials is described by a transfer matrix method. On 4+1 dimensions, the evidence for a finite-temperature phase transition is found at T_c≈0.18, where the free energy fluctuation grows logarithmically as a function of time t. When TT_c, the fluctuation of the free energy grows as t^ω with ω≈0.156. The phase transition of the restricted solid-on-solid model, which is closely related to the directed polymer problem through the Kardar–Parisi–Zhang equation, is also discussed.