An extended class of L2-series solutions of the wave equation

Abstract

We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support an infinite tridiagonal matrix representation of the wave operator. The class of solutions obtained as such includes the discrete (for bound states) as well as the continuous (for scattering states) spectrum of the Hamiltonian. The problem translates into finding solutions of the resulting three-term recursion relation for the expansion coefficients of the wavefunction. These are written in terms of orthogonal polynomials, some of which are modified versions of known polynomials. The examples given, which are not exhaustive, include problems in one and three dimensions.

Scattering and bound states for a class of non-central potentials

“…7 the radial oscillator potential V(r) = For an introduction to the above-mentioned approach and its implementation on some examples in one and three dimensions (with spherical symmetry) one may consult the papers in Refs. [1,2]. Nonetheless, it might be useful to give, in few lines, a brief account as follows.…”

“…These are written in terms of orthogonal polynomials, some of which are well-known but some are new while others are modified versions of known polynomials. In a recent article [1], we obtained solutions of problems in one and three dimensions using this approach. The solutions of some of the classic problems such as the Coulomb and Morse were reproduced adding, however, new tridiagonal representations to the solution space.…”

Analytical arbitrary ℓ‐wave solutions of the manning–rosen potential in the tridiagonalization program

“…For example, the noncentral electric dipole potential V(r,y) ¼ cosy/r 2 [16][17][18], the electric quadrupole potential [19], the hyperbolic single wave potential [3], the screened Coulomb potential with a barrier [20], and the Yukawa potential [21]. Of course, the tridiagonal program must give the traditional solutions automatically [22][23][24].…”

Solutions of the D-Dimensional Schrödinger Equation with the Hyperbolic Pöschl-Teller Potential plus Modified Ring-Shaped Term

“…The weight function can be easily calculated using (20) in ( ) = , which gives ( ) = 2 2 7 + 6 (1 + )…”

“…Recently, Ikot et al [14][15][16] investigated the Schrödinger equation with Hulthen potential plus a new ringshaped potential [3], nonspherical harmonic and Coulomb potential [15], and pseudo-Coulomb potential in the cosmic string space-time [16]. Many authors have used different methods to obtain exact solutions of the wave equation such as the methods of Supersymmetric Quantum Mechanics (SUSY-QM) [17][18][19], the Tridiagonal Representation Approach (TRA) [20][21][22][23], and Nikiforov-Uvarov (NU) method [24][25][26][27][28].…”