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Register a new userResonances and antibound states of Poschl-Teller potential: Ladder operators and SUSY partners
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We analyze the one dimensional scattering produced by all variations of the Poschl-Teller potential, i.e., potential well, low and high barriers. We show that the Poschl-Teller well and low barrier potentials have no resonance poles, but an infinite number of simple poles along the imaginary axis corresponding to bound and antibound states. A quite different situation arises on the Poschl-Teller high barrier potential, which shows an infinite number of resonance poles and no other singularities. We have obtained the explicit form of their associated Gamow states. We have also constructed ladder operators connecting wave functions for bound and antibound states as well as for resonance states. Finally, using wave functions of Gamow and antibound states in the factorization method, we construct some examples of supersymmetric partners of the Poschl-Teller Hamiltonian.
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Pairs of SUSY partner Hamiltonians are studied which are interrelated by usual (linear) or polynomial supersymmetry. Assuming the model of one of the Hamiltonians as exactly solvable with known propagator, expressions for propagators of partner models are derived. The corresponding general results are applied to a particle in a box, the Harmonic oscillator and a free particle (i.e. to transparent potentials).
In recent years, many natural Hamiltonian systems, classical and quantum, with constants of motion of high degree, or symmetry operators of high order, have been found and studied. Most of these Hamiltonians, in the classical case, can be included in the family of extended Hamiltonians, geometrically characterized by the structure of warped manifold of their configuration manifold. For the extended manifolds, the characteristic constants of motion of high degree are polynomial in the momenta of determined form. We consider here a different form of the constants of motion, based on the factorization procedure developed by S. Kuru, J. Negro and others. We show that an important subclass of the extended Hamiltonians admits factorized constants of motion and we determine their expression. The classical constants may be non-polynomial in the momenta, but the factorization procedure allows, in a type of extended Hamiltonians, their quantization via shift and ladder operators, for systems of any finite dimension.
We examine time dependent Schru007fodinger equation with oscillating boundary condition. More specifically, we use separation of variable technique to construct time dependent rationally extended Pu007foschl-Teller potential (whose solutions are given by in terms of X1 Jacobi exceptional orthogonal polynomials) and its supersymmetric partner, namely the Pu007foschl-Teller potential. We have obtained exact solutions of the Schru007fodinger equation with the above mentioned potentials subjected to some boundary conditions of the oscillating type. A number of physical quantities like the average energy, probability density, expectation values etc. have also been computed for both the systems and compared with each other.
The fast computation of the Gauss hypergeometric function 2F1 with all its parameters complex is a difficult task. Although the 2F1 function verifies numerous analytical properties involving power series expansions whose implementation is apparently immediate, their use is thwarted by instabilities induced by cancellations between very large terms. Furthermore, small areas of the complex plane are inaccessible using only 2F1 power series formulas, thus rendering 2F1 evaluations impossible on a purely analytical basis. In order to solve these problems, a generalization of R.C. Forreys transformation theory has been developed. The latter has been successful in treating the 2F1 function with real parameters. As in real case transformation theory, the large canceling terms occurring in 2F1 analytical formulas are rigorously dealt with, but by way of a new method, directly applicable to the complex plane. Taylor series expansions are employed to enter complex areas outside the domain of validity of power series analytical formulas. The proposed algorithm, however, becomes unstable in general when |a|,|b|,|c| are moderate or large. As a physical application, the calculation of the wave functions of the analytical Poschl-Teller-Ginocchio potential involving 2F1 evaluations is considered.
We obtain the quantized momentum solutions, $mathcal{P}_{n}$, of the Feinberg-Horodecki equation. We study the space-like coherent states for the space-like counterpart of the Schrodinger equation with trigonometric Poschl-Teller potential which is constructed by temporal counterpart of the spatial Poschl-Teller potential.
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