The Sturm-Liouville problem is a one-particle Schrödinger equation, and it has an important place in mathematical physics. The potential function *q*(*t*) determines the type of equation, like Bessel and hydrogen atom equations. In this study, we consider Sturm-Liouville problems with two different potentials and investigate the behaviors of its eigenfunctions. The Liouville normal form of Bessel equations, also called Sturm-Liouville equation having Bessel potential type, is studied in [1]–[9] , and it is defined as follows,

$${y}^{\u2033}+\left(\lambda -\frac{{p}^{2}-\frac{1}{4}}{{x}^{2}}-q\left(x\right)\right)y=0.$$

Bessel functions are obtained by series solutions while *q* (*x*) = 0 in the equation above, accordingly, the equation above is more general than classical Bessel equation. As *q* (*x*) changes, the structure of the equation will change and hence, a different investigation will require. Lately, the Bessel difference equation has been studied by [10], and obtained discrete Bessel functions, which are the discrete analogue of Bessel functions. The radial Schrödinger equation, also called hydrogen atom equation, is studied by [3, 7, 8, 11, 12], and it is defined as follows,

$${y}^{\u2033}+\left(\lambda -\frac{\ell \left(\ell +1\right)}{{x}^{2}}+\frac{2}{x}-q\left(x\right)\right)y=0.$$

Physical interpretation of hydrogen atom and Bessel equations:

Bessel equations seem to have a lot of application areas in physics and mathematics. For example, the solution of two-dimensional wave equation, the heat equation and the Dirichlet problem in a cylinder are obtained with the help of Bessel functions [4, 13, 14]. Also, vibrational, gravitational and electromagnetic potential problems with cylindrical symmetry, diffraction problems (astronomy) resolving power of optical instruments, heavy chain, certain subjects in chemistry and biochemistry are expressed by the means of Bessel equations [5].

The radial Schrödinger equation enables to calculate the development of quantum systems with time, also it can give analytical solutions for the non-relativistic hydrogen atom.

$$\frac{{d}^{2}R}{d{r}^{2}}+\frac{2}{r}\frac{dR}{dr}-\frac{\ell \left(\ell +1\right)}{{r}^{2}}R+\left(E+\frac{2}{r}\right)R=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(0<r<\infty \right),$$

where *R* is the distance of the mass center to the origin, *l* is the orbital quantum number and a positive integer, *E* is energy constant and *r* is the distance between the nucleus with the electron.

The hydrogen atom is a system consists of a two-particle system, and it forms of one electron and one proton. Internal motion of two particles around the center of

mass corresponds to the movement of a single particle by a reduced mass. If $R=\frac{y}{x}$and *E* = *λ* is taken in the equation above, it forms of

$${y}^{\u2033}+\left(\lambda -\frac{\ell \left(\ell +1\right)}{{x}^{2}}+\frac{2}{x}-q\left(x\right)\right)y=0.$$

Figure 1 Bessel functions are the radial part of modes of vibrations of a circular drum[9]

Figure 2 The Bessel functions of the first kind

Figure 3 The Bessel functions of the second kind

Difference equations have always been an interesting subject due to the fact that the discrete analogue of differential equations. The theory of linear ordinary difference equations was improved by [15]–[19]. Recently, spectral analysis of difference equations has attracted great attention. Especially, many scientists study on Sturm-Liouville difference equations, see [16]–[18],[20]–[25].

The zeros of Bessel and hydrogen atom problems cannot be calculated directly because of having closed form solutions, and this type of solution is called the representation of solution. Accordingly, solution function *y* (*x*), also called eigenfunction, can be found only by asymptotic estimations. Based on this, the number of eigenvalues *λ* can be found only by asymptotic estimations. Thereafter, the other spectral data, like norming constants, normalized eigenfunctions and spectral function, are found by asymptotic estimations. If we close attention, we have no knowledge about potential function *q*(*x*) except for it is a continuous function. This type of problem is named "direct problem". Viceversa, while there is knowledge about spectral data, one is made estimations about potential function *q*(*x*) and this type of problem is named "inverse problem". Additionally, Bessel functions are obtained by series solutionswhile *q*(*x*) = 0 in the equation above, accordingly the equation above is more general than classical Bessel equation.As *q*(*x*) changes, structure of the equation will change and hence, a different investigation will require. Our main aim is to apply the spectral theory of these type of differential equations to the discrete case.

In this paper, we are concerned with the discrete analogue of Sturm-Liouville equations having hydrogen atom and Bessel potentials, and we provide a basis for direct and inverse problems. From this point of view, we obtain representation of solutions, asymptotic estimations of eigenfunctions and some numerical estimations about behaviors of eigenfunctions and eigenvalues. The numerical results for the eigenfunctions corresponding to the certain significant eigenvalues for Sturm-Liouville problem having Bessel and hydrogen atom potential type are shown and compared to the each other. The integral representation and asymptotic formulae for eigenfunctions of Sturm-Liouville differential problem are found in [3]. Similarly, the sum representation and asymptotic formulae for eigenfunctions of Sturm-Liouville difference problem are acquired in [26]–[28]. Also, numerical computations of Sturm-Liouville problem are considered in [21], [29].

Now, let’s introduce Liouville normal form of Bessel’s difference equations

$${\Delta}^{2}u\left(n-1\right)+\left(\lambda +q\left(n\right)-\frac{{p}^{2}-\frac{1}{4}}{{n}^{2}}\right)u\left(n\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}b,$$(1)

where *p *^{ϵ} R, *u* (*n*) , *q* (*n*) ^{ϵ} l^{2} [0, *b*] , *b* is a finite integer, *Δ* is the forward difference operator, *Δx* (*n*) = *x* (*n* + 1)−*x* (*n*) , assume that *λ* is the positive spectral parameter, *q* (*n*) − $\frac{{p}^{2}-\frac{1}{4}}{{n}^{2}}$are called potential function, *n* is a finite integer. Then, let’s introduce hydrogen atom difference equation

$$\begin{array}{l}{\Delta}^{2}v\left(n-1\right)+\left(\lambda -q\left(n\right)+\frac{2}{n}-\frac{\ell \left(\ell +1\right)}{{n}^{2}}\right)v\left(n\right)=0,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}b,\end{array}$$(2)

where *l* is the orbital quantum number and a positive integer, *v* (*n*) ^{ϵ} l^{2} [0, *b*] , *q* (*n*) , *b*, *λ* and *n* is as defined above, $-q\left(n\right)+\frac{2}{n}-\frac{\ell \left(\ell +1\right)}{{n}^{2}}$are called potential function. Our object is to give discrete analogue results to the studies mentioned before [2], [3], [26]–[28], [30].

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.