# Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well

## Abstract

This paper is a direct offspring of the work of Garbaczewski and Stephanovich [“Lévy flights and nonlocal quantum dynamics,” J. Math. Phys. 54, 072103 (2013)] where basic tenets of the nonlocally induced random and quantum dynamics were analyzed. A number of mentions were made with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so-called fractional quantum mechanics spectral problems. Presently, we give a decisive computer-assisted proof, for an exemplary finite and ultimately infinite Cauchy well problem, that spectral solutions proposed so far were plainly wrong. As a constructive input, we provide an explicit spectral solution of the finite Cauchy well. The infinite well emerges as a limiting case in a sequence of deepening finite wells. The employed numerical methodology (algorithm based on the Strang splitting method) has been tested for an exemplary Cauchy oscillator problem, whose analytic solution is available. An impact of the inherent spatial nonlocality of motion generators upon computer-assisted outcomes (potentially defective, in view of various cutoffs), i.e., detailed eigenvalues and shapes of eigenfunctions, has been analyzed.

- Authors:

- Institute of Physics, University of Opole, 45-052 Opole (Poland)

- Publication Date:

- OSTI Identifier:
- 22306032

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 55; Journal Issue: 9; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; ANALYTICAL SOLUTION; EIGENFUNCTIONS; EIGENVALUES; OSCILLATORS; QUANTUM MECHANICS

### Citation Formats

```
Żaba, Mariusz, and Garbaczewski, Piotr.
```*Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well*. United States: N. p., 2014.
Web. doi:10.1063/1.4894057.

```
Żaba, Mariusz, & Garbaczewski, Piotr.
```*Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well*. United States. doi:10.1063/1.4894057.

```
Żaba, Mariusz, and Garbaczewski, Piotr. Mon .
"Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well". United States. doi:10.1063/1.4894057.
```

```
@article{osti_22306032,
```

title = {Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well},

author = {Żaba, Mariusz and Garbaczewski, Piotr},

abstractNote = {This paper is a direct offspring of the work of Garbaczewski and Stephanovich [“Lévy flights and nonlocal quantum dynamics,” J. Math. Phys. 54, 072103 (2013)] where basic tenets of the nonlocally induced random and quantum dynamics were analyzed. A number of mentions were made with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so-called fractional quantum mechanics spectral problems. Presently, we give a decisive computer-assisted proof, for an exemplary finite and ultimately infinite Cauchy well problem, that spectral solutions proposed so far were plainly wrong. As a constructive input, we provide an explicit spectral solution of the finite Cauchy well. The infinite well emerges as a limiting case in a sequence of deepening finite wells. The employed numerical methodology (algorithm based on the Strang splitting method) has been tested for an exemplary Cauchy oscillator problem, whose analytic solution is available. An impact of the inherent spatial nonlocality of motion generators upon computer-assisted outcomes (potentially defective, in view of various cutoffs), i.e., detailed eigenvalues and shapes of eigenfunctions, has been analyzed.},

doi = {10.1063/1.4894057},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 9,

volume = 55,

place = {United States},

year = {2014},

month = {9}

}