Solutions of nonlocal Schrödinger equation via the Caputo-Fabrizio definition for some quantum systems. (English) Zbl 1527.81069
Summary: The aim of this work is to treat the time-independent one-dimensional nonlocal Schrödinger equation. The nonlocality is described by a kernel with a noninteger power \(\alpha\) between 1 and 2. At first stage, by using Caputo-Fabrizio definition and other known results, we have transformed the nonlocal Schrödinger equation to an ordinary linear differential equation. Secondly, we have applied the last result to solve two problems in nonlocal quantum mechanics: the Coulomb-type and Hulthen-type potentials in one dimension. The eigenenergies and eigenfunctions are calculated. As expected, when the power \(\alpha\) tends to two, the resulting solutions go to the standard case.
MSC:
| 81Q80 | Special quantum systems, such as solvable systems |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 34A08 | Fractional ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
Keywords:
fractional Schrödinger equation; Caputo-Fabrizio fractional derivative; Coulomb-type potential; Hulthen-type potentialReferences:
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