On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity

Felli, Veronica; Marchini, Elsa M.; Terracini, Susanna

doi:10.3934/dcds.2008.21.91

Cited by 25 publications

(45 citation statements)

References 12 publications

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“…Author details 1 College of Economics, Shenzhen University, 518060 Shenzhen, China. 2 School of Insurance, Southwestern University of Finance and Economics, 610074, Chendu, China.…”

Section: Fundingmentioning

confidence: 99%

“…The Schrödinger eigenvalue problem with the inverse-square (IS) or centrifugal potential is widely used in nuclear physics, quantum physics, nonlinear optics, and so on [1][2][3][4][5][6]. The potential in many electronic equations produces singularity and can describe the attraction or repulsion between objects, which usually leads to strong singularities of the eigenfunctions, and this cannot be simply regarded as a perturbation term [7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

“…In the last decade, increasing attention is paid to the numerical approximation of Schrödinger operators with similar singular potentials [1,[27][28][29][30]. If we solve the Schrödinger eigenvalue problem directly in a three-dimensional domain, it will take a lot of computing time and memory capacity to obtain high-precision numerical solutions [11,31,32].…”

Section: Introductionmentioning

confidence: 99%

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An efficient finite element method and error analysis for eigenvalue problem of Schrödinger equation with an inverse square potential on spherical domain

et al. 2020

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We provide an effective finite element method to solve the Schrödinger eigenvalue problem with an inverse potential on a spherical domain. To overcome the difficulties caused by the singularities of coefficients, we introduce spherical coordinate transformation and transfer the singularities from the interior of the domain to its boundary. Then by using orthogonal properties of spherical harmonic functions and variable separation technique we transform the original problem into a series of one-dimensional eigenvalue problems. We further introduce some suitable Sobolev spaces and derive the weak form and an efficient discrete scheme. Combining with the spectral theory of Babuška and Osborn for self-adjoint positive definite eigenvalue problems, we obtain error estimates of approximation eigenvalues and eigenvectors. Finally, we provide some numerical examples to show the efficiency and accuracy of the algorithm.

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“…Author details 1 College of Economics, Shenzhen University, 518060 Shenzhen, China. 2 School of Insurance, Southwestern University of Finance and Economics, 610074, Chendu, China.…”

Section: Fundingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An efficient finite element method and error analysis for eigenvalue problem of Schrödinger equation with an inverse square potential on spherical domain

et al. 2020

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“…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].…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Finite Element Method and Error Analysis for Schrödinger Equation with Inverse Square Singular Potential

Zhang

Lin

Cao

2021

Mathematical Problems in Engineering

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We provide in this study 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 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.

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Analysis of Schrödinger operators with inverse square potentials II: FEM and approximation of eigenfunctions in the periodic case

Hunsicker

Nistor

et al. 2014

Numerical Methods Partial

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Abstract. Let V be a periodic potential on R 3 that is smooth everywhere except at a discrete set S of points, where it has singularities of the form Z/ρ 2 , with ρ(x) = |x − p| for x close to p and Z is continuous, Z(p) > −1/4 for p ∈ S. We also assume that ρ and Z are smooth outside S and Z is smooth in polar coordinates around each singular point. Let us denote by Λ the periodicity lattice and set T := R 3 /Λ. In the first paper of this series [20], we obtained regularity results in weighted Sobolev space for the eigenfunctions of the Schrödinger-type operator H = −∆ + V acting on L 2 (T), as well as for the induced k-Hamiltonians H k obtained by resticting the action of H to Bloch waves. In this paper we present two related applications: one to the Finite Element approximation of the solution of (L + H k )v = f and one to the numerical approximation of the eigenvalues, λ, and eigenfunctions, u, of H k . We give optimal, higher order convergence results for approximation spaces defined piecewise polynomials. Our numerical tests are in good agreement with the theoretical results.

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On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity

Cited by 25 publications

References 12 publications

An efficient finite element method and error analysis for eigenvalue problem of Schrödinger equation with an inverse square potential on spherical domain

An efficient finite element method and error analysis for eigenvalue problem of Schrödinger equation with an inverse square potential on spherical domain

An Efficient Finite Element Method and Error Analysis for Schrödinger Equation with Inverse Square Singular Potential

Analysis of Schrödinger operators with inverse square potentials II: FEM and approximation of eigenfunctions in the periodic case

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