An Efficient Finite Element Method and Error Analysis for Schr¨odinger Equation with Inverse Square Singular Potential

We provide in this study an eﬀective ﬁnite element method of the Schr¨odinger equation with inverse square singular potential on circular domain. By introducing proper polar condition and weighted Sobolev space, we overcome the diﬃculty of singularity caused by polar coordinates’ transformation and singular potential, and the weak form and the corresponding discrete scheme based on the dimension reduction scheme are established. Then, using the approximation properties of the interpolation operator, we prove the error estimates of approximation solutions. Finally, we give a large number of numerical examples, and the numerical results show the eﬀectiveness of the algorithm and the correctness of the theoretical results.


Introduction
Schrödinger equation with the inverse square or centrifugal potential plays an important role in quantum mechanics, quantum cosmology, nuclear physics, molecular physics, and so on [1][2][3][4][5][6][7][8]. e potential has the same differential order as the Laplacian operator near the origin, which usually leads to strong singularities and cannot be treated as a lower-order perturbation term [9][10][11][12][13][14]. Li et al. [15] proposed an efficient finite element method to discuss the numerical solution of time-fractional Schrö dinger equations. us, we need to develop some new numerical methods to solve Schrödinger equation with inverse square singular potential.
In recent years, more and more attention has been paid to the numerical methods of the schrödinger equations with similar singular potential [1,[16][17][18][19][20][21]. However, many numerical methods are based on low-order finite element methods. If we solve these problems directly in two-dimensional domain, it will cost a lot of computing time and memory capacity to obtain high-precision numerical solutions [22][23][24]. In practice, we usually need to solve the Schrödinger equation with inverse square singular potential on circular domain. As far as we know, there are few reports on an effective numerical method for the Schrödinger equation with inverse square potential in circular domain.
us, the purpose of this paper is to propose an effective finite element method of the Schrödinger equation with inverse square singular potential on circular domain. By introducing proper polar condition and weighted Sobolev space, we overcome the difficulty of singularity caused by polar coordinates transformation and singular potential and establish the weak form and corresponding discrete scheme based on the dimension reduction format. en, using the approximation properties of interpolation operator, we prove the error estimates of approximation solutions. Finally, we give a large number of numerical examples, and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results [25,26]. e rest of this paper is organized as follows. In Section 2, we derive an equivalent scheme based on variable separation. In Section 3, we prove the existence and uniqueness of the solution. In Section 4, we prove the error estimation of approximation solutions. In Section 5, we describe the details for an efficient implementation of the algorithm. In Section 6, we provide some numerical experiments to show the accuracy and efficiency of our algorithm. Finally, in Section 7, we give some concluding remarks.

An Equivalent Scheme Based on Variable Separation
We are interested in studying the following Schrödinger equation with inverse square singular potential: y). en, the Laplace operator in polar coordinates is as follows: We can rewrite (1) and (2) as follows: Since U(r, θ) and F(r, θ) are 2π periodic in θ, then we have where u m and f m are the Fourier coefficients of U(r, θ) and F(r, θ), respectively. We can derive from (3) and (6) that To make (7) and (8) meaningful, we need introduce the following essential pole conditions: e pole condition (9) can be further reduced to Using the orthogonal properties of Fourier basis functions and polar condition (10), we can reduce (4) and (5) to a series of equivalent one-dimensional Schrödinger equations as follows: −rL m u m (r) + β(r) r u m (r) � rf m (r), r ∈ (0, R), (11) where

Existence and Uniqueness of the Solution
For convenience, we use the expression a≲b to mean that a ≤ Cb, where C is a positive constant. In order to derive the weak form and corresponding discrete scheme of equations (11) and (12), we need to introduce the usual weighted Sobolev space: with the corresponding inner product and norm, and the nonuniformly weighted Sobolev space: with the corresponding inner product and norm, where ω � r is a weight function. en, the weak form of equations (11) and (12) is to find where Define approximation space X h � P h ∩ H 1 0,ω (0, R), where P h is a piecewise linear interpolation polynomial space. en, the corresponding discrete scheme of (18) is to find u mh ∈ X h , such that

Lemma 1. a m (u m , v m ) is a continuous and coercive bilinear functional on
Proof. We derive from the Cauchy-Schwarz inequality that Proof. From Cauchy-Schwarz inequality, we have □ is finishes our proof. (18) and (20) have unique solutions u m and u mh , respectively.
is a bounded and positive definite bilinear functional defined on . en, from Lax-Milgram lemma, we know that equations (18) and (20)

Error Estimation of Approximation Solutions
In this section, we will present the error estimates of approximate solutions. Define the piecewise linear interpolation operator I h : where p mi (r) is the linear interpolation polynomial of u m on interval en, from error formula of linear interpolating remainder term, we have where ξ i ∈ I i is a function depending on r.
Proof. Since then we derive that Mathematical Problems in Engineering en, we obtain us, we derive that □ e proof of eorem 2 is complete.

Lemma 3.
For any u m (r) ∈ H 1 0,ω (0, R), the following inequality holds: Proof. Since u m (r) ∈ H 1 0,ω (0, R), then we have us, we derive that □ Theorem 3. Let u m and u mh be the solutions of problems (18) and (20), respectively. en, the following inequality holds: Proof. We derive from (18) and (20) that en, we have We derive from Lemma 1 and (40) that en, we obtain en, we have We derive from Lemma 3 that Combining with eorem 2, we can obtain the desired result.

Implementation of the Algorithm
To solve the discrete scheme (20), we need to construct a set of basis functions of approximation space. Let where i � 1, . . . , N − 1. It is clear that 4 Mathematical Problems in Engineering Substituting expression (48) into (20) and taking v mh through all the basis functions in X h , we obtain the following linear system: where A � a ij , From the properties of the basis functions, we know that the stiff matrices and mass matrix in (49) are all tridiagonal sparse matrices.

Numerical Experiments
We will perform some numerical tests in order to show the convergence and the effectiveness of our algorithm. We operate our programs in MATLAB 2015b.
Let ψ Mh (x, y) be the approximation solution of exact solution ψ(x, y).
By using the following polar coordinate transformation, x � r cos θ, we obtain (52) We define the error between the exact solution ψ(x, y) and the approximation solution ψ Mh (x, y) as follows: Example 1. We take β(r) � r 2 + 4, R � 1, ψ(x, y) � xy(x 2 + y 2 − 1). It is clear that ψ(x, y) satisfies the boundary condition (2). e f(x, y) can be obtained by substituting ψ(x, y) into equation (1). Next, we solve problems (1) and (2) by using the algorithm proposed in this paper. We list the errors between exact solution and approximation solutions in Table 1 for different M and h. In order to further show the accuracy and convergence of our algorithm, we present the figures and their error figures of exact solution and approximation solution in Figures 1 and 2, respectively.
We observe from Table 1 that the error e(ψ(x, y), ψ Mh (x, y)) achieve about 10 − 4 with h ≤ (1/32) and M � 6. In addition, we can see from Figures 1 and 2 that the numerical solution converges to exact solution with the decrease of h. Example 2. We take β(r) � (2/3), R � 1, and u � (x 2 + y 2 − 1)sin(x 2 + y 2 ). It is obvious that ψ(x, y) satisfies the boundary condition (2). Similarly, f(x, y) can be obtained by substituting ψ(x, y) into equation (1). We list the errors between exact solution and approximation solutions in Table 2 for different M and h. In order to further show the accuracy and convergence of our algorithm, we present the figures and their error figures of exact solution and approximation solution in Figures 3 and 4, respectively.
We observe from Table 2 that the error e(ψ(x, y), ψ Mh (x, y)) achieves about 10 − 4 with h ≤ (1/32) and M � 6. In addition, we can see from Figures 3 and 4 that the numerical solution converges to exact solution with the decrease of h.

Example
3. We take β � 1/2, R � 1, and ψ(x, y) � (x 2 + y 2 − 1)xe x+y . We list the errors between exact solution and approximation solutions in Table 3 for different M and h. We also present the figures and their error figures of exact solution and approximation solution in Figures 5 and 6, respectively.

Mathematical Problems in Engineering
We observe from Table 3 that the error e(ψ(x, y), ψ Mh (x, y)) achieves about 10 − 4 with h ≤ (1/64) and M � 6. In addition, we can see from Figures 5 and 6 that the numerical solution converges to exact solution with the decrease of h.

Conclusions
We present in this paper an efficient finite element method for the Schrödinger equation with the inverse square potential on the circular domain. By using polar coordinate transformation and Fourier basis function expansion, we reduce the original problem into a series of equivalent onedimensional problems. By introducing polar conditions, we overcome not only the difficulty brought by the singular potential but also the degree of freedom which is greatly reduced by dimension reduction. us, we only spend less computing time and memory capacity to obtain high-precision numerical solutions. Numerical results show that our algorithm is very effective. We mainly focus on, in this paper, the Schrödinger equation with the inverse square potential on the circular domain. In fact, we can extend our method to the Schrödinger equation with more complex potentials.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.