# Exact, zero-energy, square-integrable solutions of a model related to the Maxwell's fish-eye problem

## Abstract

A model, which admits normalizable wave functions of the Schroedinger equation at the energy of E = 0, is exactly solved and the solutions are compared to the corresponding classical trajectories. The wave functions are proved to be square-integrable for discrete (quantized) values of the coupling constant of the used potential. We also show that our model is a specific version of the well-known Maxwell's fish-eye. This is performed with the help of a suitably chosen conformal mapping.

- Authors:

- Institute of Physics, Nicolaus Copernicus University, ul. Grudziadzka 5, 87-100 Torun (Poland), E-mail: amak@fizyka.umk.pl

- Publication Date:

- OSTI Identifier:
- 21336069

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Annals of Physics (New York); Journal Volume: 324; Journal Issue: 12; Other Information: DOI: 10.1016/j.aop.2009.09.012; PII: S0003-4916(09)00187-0; Copyright (c) 2009 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; CONFORMAL MAPPING; COUPLING CONSTANTS; EXACT SOLUTIONS; INTEGRAL CALCULUS; POTENTIALS; SCHROEDINGER EQUATION; WAVE FUNCTIONS

### Citation Formats

```
Makowski, Adam J.
```*Exact, zero-energy, square-integrable solutions of a model related to the Maxwell's fish-eye problem*. United States: N. p., 2009.
Web. doi:10.1016/j.aop.2009.09.012.

```
Makowski, Adam J.
```*Exact, zero-energy, square-integrable solutions of a model related to the Maxwell's fish-eye problem*. United States. doi:10.1016/j.aop.2009.09.012.

```
Makowski, Adam J. Tue .
"Exact, zero-energy, square-integrable solutions of a model related to the Maxwell's fish-eye problem". United States.
doi:10.1016/j.aop.2009.09.012.
```

```
@article{osti_21336069,
```

title = {Exact, zero-energy, square-integrable solutions of a model related to the Maxwell's fish-eye problem},

author = {Makowski, Adam J.},

abstractNote = {A model, which admits normalizable wave functions of the Schroedinger equation at the energy of E = 0, is exactly solved and the solutions are compared to the corresponding classical trajectories. The wave functions are proved to be square-integrable for discrete (quantized) values of the coupling constant of the used potential. We also show that our model is a specific version of the well-known Maxwell's fish-eye. This is performed with the help of a suitably chosen conformal mapping.},

doi = {10.1016/j.aop.2009.09.012},

journal = {Annals of Physics (New York)},

number = 12,

volume = 324,

place = {United States},

year = {Tue Dec 15 00:00:00 EST 2009},

month = {Tue Dec 15 00:00:00 EST 2009}

}

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