Lie algebras for systems with mixed spectra. I. The scattering Pöschl–Teller potential (original) (raw)

Lie algebraic treatment of scattering by the Pöschl-Teller potential

Starting from an N-body quantum space, we consider the Lie-algebraic framework where the Poschl-Teller Hamiltonian,-! a; + c sech 2 X + s csch 2 X' is the single sp(2,R) Casimir operator. The spectrum of this system is mixed: it contains a finite number of negative-energy bound states and a positive-energy continuum of free states; it is identified with the Clebsch-Gordan series of the .fj!+ X.fj!-representation coupling. The wave functions are the sp(2,R) Clebsch-Gordan coefficients of that coupling in the parabolic basis. Using only Lie-algebraic techniques, we find the asymptotic behavior of these wave functions; for the special pure-trough potential (s = 0) we derive thus the transmission and reflection amplitUdes of the scattering matrix.

Lie Algebras for Potential Scattering

Physical Review Letters, 1984

We consider a new Lie-algebraic framework for the Poschl-Teller potential, in which a single [sp(2, R) Casimir] operator has both bound and scattering states. This approach allows the determination of the Smatrix by purely algebraic means.

A Lie Algebraic Approach to the Schrödinger Equation for Bound States of Pöschl-Teller Potential

Electronic Journal of …, 2010

The application of Group theoretical techniques to physical problems has a long and fruitful history. Lie algebraic methods have been useful in the study of problems in physics ever since Lie algebras were introduced by M.Sophus Lie (1842-1899) at the end of the 19th century, especially after the development of quantum mechanics. This is because quantum mechanics makes use of commutators [x, P x ] = i , which are the defining ingredients of Lie algebras. The theory of Lie groups and Lie algebras has become important not only in explaining the behaviour of various physical systems but also in constructing new physical theories. By identifying the suitable Spectrum Generating Algebra (SGA) the problem of interest can be approached. A Spectrum Generating Algebra exists when the Hamiltonian H can be expressed in terms of generators of the algebra. As a consequence the solution of the Schrödinger equation then becomes an algebraic problem which can be attacked using the tools of group theory. Here in this paper we derive the Schrödinger equation for the bound states of Pöschl-Teller potential using Lie algebra.

$\Script P$$\Script T$-symmetric potentials and the so(2, 2) algebra

Journal of Physics A: Mathematical and General, 2002

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 Schrödinger equation. In the latter case we consider problems for which so(2, 2) appears as a potential algebra, 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 Pöschl-Teller II potentials. We also analyse scattering states, transmission and reflection coefficients for these potentials.

Application of SU(1,1) Lie algebra in connection with Bound States of P¨oschl-Teller Potential

Exactly solvable quantum mechanical potentials have attracted much attention since the early days of quantum mechanics and the Schrödinger equation has been solved for a large number of potentials by employing a variety of methods. Here we consider a specific realization of SU(1,1) algebra and use it to describe the bound states of Pöschl-Teller potential without solving the Schrödinger equation for the mentioned potential.

Green's functions through so(2, 1) Lie algebra in nonrelativistic quantum mechanics

Annals of Physics, 1991

We discuss an algebraic technique to construct the Green's function for systems described by the noncompact so(2, 1) Lie algebra. We show that this technique solves the one-dimensional linear oscillator and Coulomb potentials and also generates particular solutions for other one-dimensional potentials. Then we construct explicitly the Green's function for the three-dimensional oscillator and the three-dimensional Coulomb potential, which are generalizations of the one-dimensional cases, and the Coulomb plus an Aharonov-Bohm potential. We discuss the dynamical algebra involved in each case and also find their wave functions and bound state spectra. Finally we introduce a point canonical transformation in the generators of so(2, 1) Lie algebra, show that this procedure permits us to solve the one-dimensional Morse potential in addition to the previous cases, and construct its Green's function and find its energy spectrum and wave functions. I(')

Dynamical algebras of general two-parametric Pöschl–Teller Hamiltonians

Annals of Physics, 2012

A class of operators connecting general two-parametric Pöschl-Teller Hamiltonians is found. These operators include the so-called ''shift'' (changing only the potential parameters) and ''ladder'' (changing also the energy eigenvalue) operators. The explicit action on eigenfunctions is computed within a simple and symmetric three-subindex notation. It is shown that the whole set of operators close an su(2, 2) ≈ so(4, 2) dynamical Lie algebra. A unitary irreducible representation of this so(4, 2) differential realization is characterized.

Resonances and antibound states for the Pöschl–Teller potential: Ladder operators and SUSY partners

Physics Letters A, 2016

We analyze the one dimensional scattering produced by all variations of the Pöschl-Teller potential, i.e., potential well, low and high barriers. We show that the Pöschl-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 Pöschl-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 Pöschl-Teller Hamiltonian.

Scattering and bound state Green’s functions on a plane via so(2,1) Lie algebra

Journal of Mathematical Physics, 2006

We calculate the Green's functions for the particle-vortex system, for two anyons on a plane with and without a harmonic regulator and in a uniform magnetic field. These Green's functions which describe scattering or bound states (depending on the specific potential in each case) are obtained exactly using an algebraic method related to the SO(2,1) Lie group. From these Green's functions we obtain the corresponding wave functions and for the bound states we also find the energy spectra. * pborges@cefet-rj.br †

Algebraic Approach to the Scattering Matrix

Physical Review Letters, 1984

A purely algebraic procedure is presented for calculating belonging to problems associated with the group SU(l, l). menting a recently introduced group-theoretic approach to work that characterizes asymptotic behavior. Our formula group, which contains the symmetry transformations for a Poschl-Teller potentials are discussed to illustrate our meth PACS numbers: 03.80.+r, 11. 20.Dj Dynamical groups have proved to be useful in describing bound-state spectra in nuclei' and molecules. Recently the group-theoretic approach has been extended to the continuum. The methods were illustrated for one-dimensional Poschl-Teller and Morse potentials, whose scattering eigenstates were shown to form a basis for certain representations