Spectrum of a q ‐deformed Schrödinger equation by means of the variational method

In this work, the ‐deformed Schrödinger equations defined in different form of the ‐Hamiltonian for ‐harmonic oscillator are considered with symmetric, asymmetric, and non‐polynomial potentials. The spectrum of the ‐Hamiltonian is obtained by using the Rayleigh‐Ritz variational method in which the discrete ‐Hermite I polynomials are taken as the basis. As applications, ‐harmonic, purely ‐quartic, and ‐quartic oscillators are examined in the class of symmetric polynomial potentials. Moreover, the ‐version of Gaussian potential for an example of a non‐polynomial symmetric potential and a specific example of ‐version of asymmetric double well potential are presented. Numerous results are given for these potentials for several values of . The limit relation as is discussed. The obtained results of ground‐ and excited‐state energies of the purely ‐quartic oscillator and the accuracy of the ground‐state energy levels are compared with the existing results. Also, the results are compared with the classical case appearing in the literature in the limiting case .