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Open Physics

formerly Central European Journal of Physics

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Volume 16, Issue 1

Issues

Volume 13 (2015)

Sturm-Liouville difference equations having Bessel and hydrogen atom potential type

Erdal Bas / Ramazan Ozarslan / Dumitru Baleanu
  • Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Ankara, Magurele-Bucharest, TurkeyInstitute of Space Sciences, Magurele-Bucharest, Romania
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Published Online: 2018-12-26 | DOI: https://doi.org/10.1515/phys-2018-0100

Abstract

In this work, we bring a different approach for Sturm-Liouville problems having Bessel and hydrogen atom type and we provide a basis for direct and inverse problems. From this point of view, we find representations of solutions and asymptotic expansions for eigenfunctions. Furthermore, some numerical estimations are given to illustrate the necessity of the Sturm-Liouville difference equations with the potential function for the convenience to the spectral theory. The behavior of eigenfunctions for the Sturm-Liouville problem having Bessel and hydrogen atom potential type is analyzed and compared to each other. And then, comparisons are showed by tables and figures.

Keywords: Sturm-Liouville; Difference equation; Bessel potential; Hydrogen atom potential; Schrödinger equation

PACS: 02.30.Tb; 02.90.+p; 02.30.Zz

1 Introduction

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+(λp214x2q(x))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+(λ(+1)x2+2xq(x))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.

d2Rdr2+2rdRdr(+1)r2R+(E+2r)R=0(0<r<),

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=yxand E = λ is taken in the equation above, it forms of

y+(λ(+1)x2+2xq(x))y=0.

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

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

The Bessel functions of the first kind
Figure 2

The Bessel functions of the first kind

The Bessel functions of the second 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

Δ2u(n1)+(λ+q(n)p214n2)u(n)=0,n=1,,b,(1)

where p ϵ R, u (n) , q (n) ϵ l2 [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) − p214n2are called potential function, n is a finite integer. Then, let’s introduce hydrogen atom difference equation

Δ2v(n1)+(λq(n)+2n(+1)n2)v(n)=0,n=1,,b,(2)

where l is the orbital quantum number and a positive integer, v (n) ϵ l2 [0, b] , q (n) , b, λ and n is as defined above, q(n)+2n(+1)n2are called potential function. Our object is to give discrete analogue results to the studies mentioned before [2], [3], [26]–[28], [30].

2 Preliminaries

Definition 1 [25] The matrix of Casoratian is defined by

w(n)=(u1(n)u2(n)ur(n)u1(n+1)u2(n+1)ur(n+1)u1(n+r1)u2(n+r1)ur(n+r1))

where u1 (n) , u2 (n) , ..., ur (n) are known functions. The determinant

W(n)=detw(n)

is called Casoratian

Theorem 1

[16] (Wronskian-Type Identity) Assume y and y are linearly independent solutions of Sturm-Liouville difference equation, mentioned in [26]. Then, for anb

W[y,z](n)=[y(n)Δz(n1)y(n)Δz(n1)]=[y(n)z(n1)y(n1)z(n)](3)

is a constant (Particularly equal to W [r, u] (a))

Definition 2 Let’s define Sturm-Liouville problem having Bessel potential type (1) as following, n 2 [1, b]

L1u(n)=Δ2u(n1)+(q(n)p214n2)u(n)=λu(n),(4)

with initial conditions

u(1)=h,u(2)=1,(5)

where L1 is a self-adjoint difference operator, h = cot α, α ϵ R.

Definition 3 Let’s define Sturm-Liouville problem having hydrogen atom potential (2) as following, n 2 [1, b],

L2v(n)=Δ2v(n1)+(q(n)+2n(+1)n2)v(n)=λv(n),(6)

with initial conditions

v(1)=h,v(2)=1,(7)

where L2 is a self-adjoint difference operator, h = cot α, α ϵ R.

Theorem 2

[25] (Summation by parts) Assume m < n, then

k=mn1u(k)Δr(k)=[u(k)r(k)]mnk=mn1Δu(k)r(k+1).(8)

Theorem 3

[25] Assume zn is an indefinite sum of un, then

k=mn1z(k)=u(n)u(m).(9)

3 Main results

In this paper, we are concerned with the discrete analogue of Sturm-Liouville problem with Bessel and hydrogen atom potentials (6\)−(7) , (4)−(5), and we provide a basis for direct and inverse problems. From this point of view, we obtain the representation of solutions, asymptotic estimations of eigenfunction and some numerical estimations about behaviors of eigenfunctions and eigenvalues. The numerical results for 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. Our main aim is to apply the spectral theory of these type of differential equations to the discrete case.

In l2 (1, b) , the Hilbert space of sequences of complex numbers u (1) , ..., u (b) with the inner product,

<u(n),r(n)>=n=1bu(n)r(n),

for every u 2 D L let’s define as follows

DL={u(n),v(n)2(1,b):L1u(n),L2v(n)2(1,b),u(1)=v(1)=h,u(2)=v(2)=1.}

In this section, we show the sum representations of solutions of Sturm-Liouville problem having Bessel and hydrogen atom potential by the variation of parameters method and prove that the solutions obtained hold for the problems.

Theorem 4

Sturm-Liouville problem having Bessel potential type is defined as follows

Δ2φ(n1)+(q(n)p214n2)φ(n)=λφ(n),(10)

φ(1)=h,φ(2)=1,(11)

then the problem (10) − (11) has a unique solution for φ (n) as follows

φ(n,λ)=(hsin2θsinθ[1+h(q(1)+p214)]sinθ)cosnθ+(cosθ[1+h(q(1)+p214)]+hcos2θsinθ)sinnθ+1sinθi=1n1(q(i)+p214i2)φ(i)sin(ni)θ,

where i=ab.=0ifa>b.

Proof We restrict the eigenvalues with 0 < λ < 4 to be able to obtain more advantageous results for the spectral theory. Let u1 (n) and u2 (n) be linearly idependent solutions for homogeneous part of (10) , so by characteristic equation, we can find characteristic roots, and since |λ − 2| < 2, we can take

λ=22cosθ,

from here, the solution of the homogeneous part of (10) is

φh(n)=c1cosnθ+c2sinnθ.

By means of variation of parameters method for difference equations [2, 17], we get

φp(n)=c1(n)cosnθ+c2(n)sinnθ,(12)

if we insert the equation (12) in the equation (10) and make necessary operations, then we can find the parameters as,

c1(n)=i=1n[q(i)+p214i2]φ(i)φ2(i)W,

c2(n)=i=1n[q(i)+p214i2]φ(i)φ1(i)W,

where W = sin θ by Theorem 1.

Eventually, we get the general solution

φ(n,λ)=c1cosnθ+c2sinnθ+1sinθi=1n[q(i)+p214i2]φ(i)sin(ni)θ,

by the initial conditions, we get the sum representation of solution, c1 ≠60, c2 ≠60,

φ(n,λ)=(hsin2θ-sinθ[1+h(q(1)+p214)]sinθ)cosnθ+(cosθ[1+h(q(1)+p214)]+hcos2θsinθ)sinnθ++1sinθi=1n1(q(i)+p214i2)φ(i)sin(ni)θ.

We can show that the solution holds for the problem (10)− (11) by inserting in (10) by virtue of Theorem 2 and Theorem 3.

The proof completes.

Theorem 5

Sturm-Liouville problem having hydrogen atom potential type is defined as follows

Δ2ψ(n1)+(q(n)+2n(+1)n2)ψ(n)=λψ(n),(14)

ψ(1)=h,ψ(2)=1,(15)

then the problem (14) − (15)has a unique solution for ψ (n) as follows

ψ(n,λ)=(hsin2θsinθ[1+h(q(1)+2+(+1))]sinθ)cosnθ+(cosθ[1+h(q(1)+2+(+1))]+hcos2θsinθ)sinθ+1sinθi=1n1(q(i)+2i+(+1)i2)ψ(i)sin(ni)θ.

where i=ab=0if a > b.

Proof Similar arguments used in the proof of Theorem 5 are applied to easily obtain the conclusion.

4 Asymptotic formulas for Bessel and hydrogen atom difference equations

In this section, we give the asymptotic formulas for the eigenfunctions of Bessel and hydrogen atom potential, respectively.

Theorem 6

Sturm-Liouville problem having Bessel potential (10) − (11) has the estimate

φ(n)=O(e|τ|n),

and more precisely

φ(n)=(hsin 2θ-sinθ[1+h(q(1)+p214)]sinθ)cosnθ+O(e|τ|nsinθ2θ),

where n 2 Z+, {φ (n)} is a complex sequence i=0i|q(i)|1

Proof Let’s insert φ(n)=f(n)e|τ|nin (10) . Hence, we get

f(n)=(hsin2θ-sinθ[1+h(q(1)+p214)]sinθ)e|τ|n+(cosθ[1+h(q(1)+p214)]+hcos2θsinθ)e|τ|n+1sinθi=1n1(q(i)+p214i2)f(i)e|τ|(ni)sin(ni)θ.

Let ξ=max1n|f(n)|,then if we insert the last equation in that

ξ2(|h|+1+|h|(|q(1)|+p2+14))+ξ|sinθ|i=1ξ(|q(i)|+p2+14i2),

and thus,

ξ2(|h|+1+|h|(|q(1)|+p2+14))1=1|sinθ|i=1||q(i)|+p2+14i2|,

the proof completes under the assumption that nominator of the last inequality is large enough and

|sinθ|>i=1||q(i)|+p2+14i2|.

The proof is complete under the assumption.

Theorem 7

Sturm-Liouville problem having hydrogen atom potential (14) − (15) has the estimate

ψ(n)=O(e|τ|n),

and more precisely

ψ(n)=(hsin2θsinθ[1+h(q(1)+2+(+1))]sinθ)cosnθ+O(e|τ|nsinθ2sinθ),

where n 2 Z+, (n)} is a complex sequence i=0iqi<.

Proof Similar arguments used in the proof of Theorem 6 are applied to easily obtain the conclusion.

5 Applications

Assuming that h = p = 1, l = 2, b = 24 in all of the applications.

Application 1. Taking q1 (n)=1nin the problems (10) − (11) and (14) − (15) , then we have the eigenfunctions as in Figure 4, Figure 5, Figure 6, Table 1, Table 2, Table 3 Application 2. Taking h = p = l = 1, b = 24 and q2 (n) = 1 in the problem (10) − (11) and (14) − (15) , we have the eigenfunctions as in Table4, Table5.

Comparison of datas in Table 1
Figure 4

Comparison of datas in Table 1

Comparison of datas in Table 3
Figure 5

Comparison of datas in Table 3

Comparison of datas in Table 2
Figure 6

Comparison of datas in Table 2

Comparison of datas in Table 4
Figure 7

Comparison of datas in Table 4

Application 3. Taking h = p = 1, l = 2, b = 24 and q3 (n) = 1nin the problem (10) − (11) and (14) − (15), we have the eigenfunctions as in Fig8Fig17.

ψ (n) , n = 1, 2, ..., 5
Figure 8

ψ (n) , n = 1, 2, ..., 5

Comparison of φ (n) , n = 1, 2, ..., 5
Figure 9

Comparison of φ (n) , n = 1, 2, ..., 5

Comparison of φ (n) and ψ(n) ,n=1,2,...,5
Figure 10

Comparison of φ (n) and ψ(n) ,n=1,2,...,5

Comparison of datas in Table 5
Figure 11

Comparison of datas in Table 5

Comparison of varphi (n) for different p values,λ = 1, q = 1n$\frac{1}{\sqrt{n}}$
Figure 12

Comparison of varphi (n) for different p values,λ = 1, q = 1n

Comparison of φ (n) for different potentials, λ = 1
Figure 13

Comparison of φ (n) for different potentials, λ = 1

Comparison of  ψ (n) for different eigenvalues, q=1n$q=\frac{1}{\sqrt{n}}$
Figure 14

Comparison of ψ (n) for different eigenvalues, q=1n

Comparison of ψ(n) for different potentials λ=2+3,  l=$\lambda =2+\sqrt{3},\,\,l=$1
Figure 15

Comparison of ψ(n) for different potentials λ=2+3,l=1

Comparison of ψ (n) for different l orbital quantum numbers, λ=2+3,  q=1n$\lambda =2+\sqrt{3},\,\,q=\frac{1}{\sqrt{n}}$
Figure 16

Comparison of ψ (n) for different l orbital quantum numbers, λ=2+3,q=1n

φ (n) , n = 1, 2, ..., 5
Figure 17

φ (n) , n = 1, 2, ..., 5

Table 1
Table 2
Table 3
Table 4
Table 5

6 Conclusion

In this study, we are concerned with discrete analogue of Sturm-Liouville problem with Bessel and hydrogen atom potentials (6)−(7) , (4)−(5), and we provide a basis for direct and inverse problems. From this point of view, we obtain representation of solutions, asymptotic estimations of eigenfunction and some numerical estimations about behaviors of eigenfunctions and eigenvalues. The numerical results for 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. Our main aim is to apply the spectral theory of these type of differential equations to the discrete case.

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About the article

Received: 2018-01-27

Accepted: 2018-07-31

Published Online: 2018-12-26


Citation Information: Open Physics, Volume 16, Issue 1, Pages 801–809, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0100.

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