Extending the class of solvable potentials: III. The hyperbolic single wave

Abstract

A new solvable hyperbolic single wave potential is found by expanding the regular solution of the 1D Schrödinger equation in terms of square integrable basis. The main characteristic of the basis is in supporting an infinite tridiagonal matrix representation of the wave operator. However, the eigen-energies associated with this potential cannot be obtained using traditional procedures. Hence, a new approach (the "potential parameter" approach) has been adopted for this eigenvalue problem. For a fixed energy, the problem is solvable for a set of values of the potential parameters (the "parameter spectrum"). Subsequently, the map that associates the parameter spectrum with the energy is inverted to give the energy spectrum. The bound states wavefunction is written as a convergent series involving products of the ultra-spherical Gegenbauer polynomial in space and a new polynomial in energy, which is a special case of the "dipole polynomial" of the second kind.

Solutions of the Schrödinger equation in the tridiagonal representation with the noncentral electric dipole plus a novel angle-dependent component

“…The main objective and motivation of this program is to find solutions of new problems that could not be solved by the traditional methods (the diagonal program). For example, the electric quadruple potential in two dimensions [32], the hyperbolic single wave potential in one dimension [33], and the Yukawa potential [34], etc. As expected, of course, the tridiagonal program does give the traditional solutions automatically [35].…”

“…Therefore, the action of the radial wave operator (H c − E) on the basis element (33) gives the following…”

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

“…The main objective and motivation of this program are to find solutions of new problems that could not be solved by the traditional methods (the diagonal 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].…”

“…Introduction I n quantum mechanics, exactly analytical solutions of the wave equation with solvable potentials are still playing an important role because they contain all the necessary information about the quantum system under consideration [1,2]. Moreover, these solutions are valuable means for checking and improving models and numerical methods introduced for solving complicated physical systems [3]. In recent years, many efforts have been devoted to the exactly solvable problems by many authors.…”

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

“…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].…”