Analysis of periodic Schrödinger operators: Regularity and approximation of eigenfunctions

Hunsicker, Eugénie; Nistor, Victor; Sofo, Jorge O.

doi:10.1063/1.2957940

Cited by 13 publications

(23 citation statements)

References 42 publications

(24 reference statements)

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“…The definition (1.12) slightly differs from the definition of the weighted Sobolev space given in [CS15] (Equation (2.6)). However, our definition is consistent with the results that can be found in [HNS08] (see Theorem I.1) and the original paper [FSS08] (see Proposition 1) from which the definition appearing in [CS15] is taken.…”

Section: Singular Expansionsupporting

confidence: 87%

“…Although the geometry here is simple, Coulomb singularities generated by the nuclei fit perfectly in this treatment. The behavior of the electronic wave function close the nucleus has been precisely characterized using this theory [FSS08,HNS08]. Those results paved the way to the analysis of the muffin-tin and LAPW methods [CS15] and the VPAW method [Dup18].…”

Section: Singular Expansionmentioning

confidence: 99%

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Discretization Error Cancellation in the Plane-Wave Approximation of Periodic Hamiltonians with Coulomb Singularities

Dupuy

2019

J Sci Comput

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In solid-state physics, energies of molecular systems are usually computed with a plane-wave discretization of Kohn-Sham equations. A priori estimates of plane-wave convergence for periodic Kohn-Sham calculations with pseudopotentials have been proved , however in most computations in practice, plane-wave cut-offs are not tight enough to target the desired accuracy. It is often advocated that the real quantity of interest is not the value of the energy but of energy differences for different configurations. The computed energy difference is believed to be much more accurate because of "discretization error cancellation", since the sources of numerical errors are essentially the same for different configurations. In the present work, we focused on periodic linear Hamiltonians with Coulomb potentials where error cancellation can be explained by the universality of the Kato cusp condition. Using weighted Sobolev spaces, Taylor-type expansions of the eigenfunctions are available yielding a precise characterization of this singularity. This then gives an explicit formula of the first order term of the decay of the Fourier coefficients of the eigenfunctions. As a consequence, the error on the difference of discretized eigenvalues for different configurations is indeed reduced by an explicit factor. However, this error converges at the same rate as the error on the eigenvalue. Plane-wave methods for periodic Hamiltonians with Coulomb potentials are thus still inefficient to compute energy differences.

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Section: Singular Expansionsupporting

confidence: 87%

Section: Singular Expansionmentioning

confidence: 99%

Discretization Error Cancellation in the Plane-Wave Approximation of Periodic Hamiltonians with Coulomb Singularities

Dupuy

2019

J Sci Comput

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“…The simple proofs of these results are the same as that of the analogous results in [27], and follow directly from properties of b-operators [3,39,36].…”

Section: Invertibilitymentioning

confidence: 75%

“…Assumptions on the potential V . In this paper, we extend and test the results of [27] to deal with the more singular potentials that have inverse-square singularities. More precisely, we extend the results of the aforementioned paper from potentials where ρV is smooth in polar coordinates to potentials where ρ 2 V is smooth in polar coordinates and continuous on T. (Recall that ρ is a function that locally gives the distance to the singular point.)…”

Section: 2mentioning

confidence: 99%

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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“…In [5] the author investigates eigenvectors of Toeplitz matrices under higher order three term recurrence and circulant perturbations. The paper [9] deals with approximations of eigenfunctions of the periodic Schrödinger operators. The paper [16] introduces an algorithm to numerically approximate eigenfunctions of Sturm-Liouville problems corresponding to eigenvalues in a given region.…”

Section: Introduction and Notationsmentioning

confidence: 99%

Simple eigenvectors of unbounded operators of the type "normal plus compact"

Gil'

2015

OpMath

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Abstract. The paper deals with operators of the form A = S + B, where B is a compact operator in a Hilbert space H and S is an unbounded normal one in H, having a compact resolvent. We consider approximations of the eigenvectors of A, corresponding to simple eigenvalues by the eigenvectors of the operators An = S + Bn (n = 1, 2, . . .), where Bn is an n-dimensional operator. In addition, we obtain the error estimate of the approximation.

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Analysis of periodic Schrödinger operators: Regularity and approximation of eigenfunctions

Cited by 13 publications

References 42 publications

Discretization Error Cancellation in the Plane-Wave Approximation of Periodic Hamiltonians with Coulomb Singularities

Discretization Error Cancellation in the Plane-Wave Approximation of Periodic Hamiltonians with Coulomb Singularities

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

Simple eigenvectors of unbounded operators of the type "normal plus compact"

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