Exact analytical solutions of the wave function for some \(q\)-deformed potentials (Q362115)

The authors derive analytic solutions of the 1-D stationary Schrödinger equation with a potential \(V(x)\) NEWLINE\[NEWLINE \frac{d^2 \Psi(x)}{dx^2} +\left( E - V(x)\right)\Psi(x) = 0. NEWLINE\]NEWLINE They succeed in the case when the potential \(V(x)\) has a specific form. Namely when \(V(x)\) is a \(q\)-deformed version of \(\cosh(x)\), \(\sinh(x)\), \(\tanh(x)\), \((\mathrm{csch}(x))^2\), \((\sec(x))^2\) or \(\sin(x)\). They are the generalized versions of the ordinary trigonometric and hyperbolic functions. Every \(q\)-deformed function coincides with its ordinary analogue when \(q=1\). For example \(q\)-deformed hyperbolic cosine \(\cosh_q(x) = \frac{1}{2}(e^x+q e^{-x})\), by setting \(q=1\) turns in to the ordinary \(\cosh(x)\).NEWLINENEWLINEIn all the cases studied, the general solution to the given equation can be exactly represented in terms of the special functions \(\mathrm{HeunD}(\alpha,\beta,\gamma,\delta,z)\) and \({}_2 F_1 (a,b,c,z)\).