Applying supersymmetry to energy dependent potentials

Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials

“…The right part of figure 1 shows these solutions by means of their conventional density |Φ n | 2 for the first few values of the parameter n. We observe that the solutions spread further in horizontal direction as the associated potential opens wider upward. Before we conclude this example, let us comment on orthogonality and normalizability of the functions (26). Since the underlying potential is energy-dependent, we must resort to modified definitions of the aformentioned concepts [5].…”

“…In particular, the construction of the completeness relation and the usual L 2 -norm are affected [5]. Further theoretical work on energy-dependent potentials includes the application to confined models [14], their generation by means of the supersymmetry formalism [26] and through point transformations of hypergeometric equations [6]. Similar to the work done in the latter two references, the purpose of this research is to find quantum models featuring energy-dependent potentials.…”

Pseudo-hermitian andPT-symmetric quantum systems with energy-dependent potentials: Bound-state solutions and energy spectra

“…As before, P n denotes the Jacobi polynomial of degree n. In order to check that our boundary conditions (2) are fulfilled at ± π/(2α), we inspect the first two factors on the right side of (34). These factors vanish at −π/(2α) and π/(2α), respectively, provided the real part of their exponents remains positive.…”

“…is satisfied. This constraint leads to an interesting situation concerning the number of solutions (34) to our boundary-value problem. Since E n changes with n, so do B(E n ) and the functions (34).…”

Spin (Pseudospin) doublet in view of energy-dependent potential

“…Bound states for nonrelativistic particles in view of an energy-dependent potential have also been investigated [9]. Yekken et al recently analyzed the supersymmetry features of energy-dependent harmonic oscillators and Pöschl-Teller potentials in a one-dimensional space [10]. In order to describe the subbarrier fusion cross-section of diverse systems, the Woods-Saxon potential having energy dependency has been used in a recent paper [11].…”