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We solve the quantum system with the symmetric Razavy cosine type potential and find that its exact solutions are given by the confluent Heun function. The eigenvalues are calculated numerically. The properties of the wave functions, which depend on the potential parameter

As we know, the exact solutions of quantum systems have been playing an important role since the foundation of quantum mechanics. The hydrogen atom and harmonic oscillator have been taken as typical and seminal examples to explain the classic quantum phenomena in almost all quantum mechanics textbooks [

More than thirty years ago, Razavy proposed a cosine type potential [

A plot of potential as function of the variables

This paper is organized as follows. In Section

Let us consider the one-dimensional Schrödinger equation,

Substituting potential (

Take the wave functions of the form

Choose a new variable

To make the confluent Heun function reduce to polynomials, two termination conditions have to be satisfied [

For present case, there is a problem for the first condition. That is,

Now, let us study some basic properties of the solutions as shown in Figures

Spectra of the Schrödinger equation with potential (

| | | | | | | |
---|---|---|---|---|---|---|---|

| -1.25000 | 1.79206 | 3.88549 | 5.52599 | 7.571 | 10.2587 | 13.4768 |

| -3.85555 | 0.105572 | 3.35555 | 5.88783 | 8.10716 | 10.6369 | 13.7448 |

| -6.5289 | -1.86011 | 2.22966 | 5.68013 | 8.54924 | 11.2207 | 14.2107 |

| -9.2426 | -3.97095 | 0.793349 | 5.00238 | 8.62128 | 11.7636 | 14.8280 |

| -11.9842 | -6.17622 | -0.831098 | 4.01497 | 8.31808 | 12.0716 | 15.4571 |

| -14.7467 | -8.44991 | -2.58550 | 2.81773 | 7.72170 | 12.0880 | 15.9545 |

| -17.5255 | -10.7764 | -4.43719 | 1.46786 | 6.90679 | 11.8388 | 16.2407 |

| -20.3178 | -13.1452 | -6.36553 | 0.00034 | 5.92502 | 11.3718 | 16.3004 |

| -23.1212 | -15.5492 | -8.35647 | -1.56169 | 4.81114 | 10.7299 | 16.1532 |

| -25.9342 | -17.9829 | -10.3999 | -3.20197 | 3.58925 | 9.94554 | 15.8289 |

| -28.7554 | -20.4422 | -12.4881 | -4.90854 | 2.27678 | 9.04264 | 15.3551 |

The characteristics of wave functions as a function of the position

The variation of the energy spectra

In this work we have studied the quantum system with the Razavy cosine type potential and found that its exact solutions are given by confluent Heun function

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by project 20180677-SIP-IPN, COFAA-IPN, Mexico, and partially by the CONACYT project under grant No. 288856-CB-2016.