Symmetrized exponential oscillator

Abstract

Several properties of bound states in potential V (x) = g 2 exp(|x|) are studied. Firstly, with the emphasis on the reliability of our arbitrary-precision construction, wave functions are considered in the two alternative (viz., asymptotically decreasing or regular) exact Bessel-function forms which obey the asymptotic or matching conditions, respectively. The merits of the resulting complementary transcendental secular equation approaches are compared and their applicability is discussed.

Morse potential, symmetric Morse potential and bracketed bound-state energies

“…Recently [7] we imagined that besides its usual connection with the half-line radial Schrödinger equation, boundary condition (5) may be equally well interpreted as a matching condition for all of the odd-parity bound states living on the whole real line. In other words, one can move from radial half-line Eq.…”

Real Discrete Spectrum of Complex PT-Symmetric Scattering Potentials

“…These results are due to the special method [12,13] of diagonalization (16) To conclude, in this paper we have found parametric evolution of eigenvalues for five complex PT-symmetric scattering potentials employing different methods. Such works are instructive and desirable especially when complex PT-symmetry makes way in to textbooks of quantum mechanics.…”

“…First, to confirm our numerical method, we take up the well known rectangular complex PT-symmetric profile [12] of width 2a.…”

“…and Using (10,11,12), we write the matrix elements h m,n = m|H|n to get the eigenvalues as det |h m,n − E δ m,n | = 0.…”

Quasi-exactly solvable symmetrized quartic and sextic polynomial oscillators

“…It was recently suggested [20,21,22,23,24] that, apart from those abovementioned interaction models, the ES status should also be attributed to less common ones, for which the real line of coordinates is splitted into subintervals wherein the potential admits different definitions, while being continuous on the whole line, and the wavefunctions remain piece-wise proportional to special functions, while being matched, as well as their first derivatives, at the subinterval limit points.…”