Periodic Schrödinger Operators with Local Defects and Spectral Pollution (original) (raw)
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Periodic Schrödinger operator with local defects and spectral pollution
2016
This article deals with the numerical calculation of eigenvalues of perturbed periodic Schrödinger operators located in spectral gaps. Such operators are encountered in the modeling of the electronic structure of crystals with local defects, and of photonic crystals. The usual finite element Galerkin approximation is known to give rise to spectral pollution. In this article, we give a precise description of the corresponding spurious states. We then prove that the supercell model does not produce spectral pollution. Lastly, we extend results by Lewin and Séré on some no-pollution criteria. In particular, we prove that using approximate spectral projectors enables one to eliminate spectral pollution in a given spectral gap of the reference periodic Schrödinger operator.
Band formation and defects in a finite periodic quantum potential
American Journal of Physics
Periodic quantum systems often exhibit energy spectra with well-defined energy bands separated by band gaps. The formation of band structure in periodic quantum systems is usually presented in the context of Bloch's theorem or through other specialized techniques. Here we present a simple model of a finite one-dimensional periodic quantum system that can be used to explore the formation of band structure in a straightforward way. Our model consists of an infinite square well containing several evenly-spaced identical Dirac delta wells. Both attractive and repulsive delta wells are considered. We solve for the energy eigenvalues and eigenfunctions of this system directly and show the formation of band structure as the number of delta wells is increased, as well as how the size of the bands and gaps depends on the strength of the delta wells. These results are compared ot the predictions from Bloch's theorem. In addition, we use this model to investigate how the energy spectrum is altered by the introduction of two types of defects in the periodicity of the system. Strength defects, in which the strength of one delta well is changed, can result in an energy level moving from one band, through the band gap, to another band as the strength of the well is varied. Position defects, in which the location of one delta well is changed, can modulate the size of the energy bands and sufficiently large position defects can move an energy level into a gap. Band structure and defects are important concepts for understanding many properties of quantum solids and this simple model provides an elementary introduction to these ideas.
Communications in Mathematical Sciences, 2007
We are concerned with the numerical study of a simple one-dimensional Schrödinger operator − 1 2 ∂ xx + αq(x) with α ∈ R, q(x) = cos(x) + ε cos(kx), 0 ≤ ε 1 and k being irrational. This governs the quantum wave function of an independent electron within a crystalline lattice perturbed by some impurities whose dissemination induces long-range order only, which is rendered by means of the quasi-periodic potential q. We study numerically what happens for various values of k and ε; especially, increasing values of k correspond to a stronger disorder in the medium and we expect to observe a mobility edge. However, it turns out that for k > 1, that is to say, in case more than one impurity shows up inside an elementary cell of the original lattice, "impurity bands" appear and seem to be k-periodic.
Analysis of periodic Schrödinger operators: Regularity and approximation of eigenfunctions
Journal of Mathematical Physics, 2008
Let V be a real valued potential that is smooth everywhere on R 3 , except at a periodic, discrete set S of points, where it has singularities of the Coulomb type Z/r. We assume that the potential V is periodic with period lattice L. We study the spectrum of the Schrödinger operator H = −∆ + V acting on the space of Bloch waves with arbitrary, but fixed, wavevector k. Let T := R 3 /L. Let u be an eigenfunction of H with eigenvalue λ and let ǫ > 0 be arbitrarily small. We show that the classical regularity of the eigenfunction u is u ∈ H 5/2−ǫ (T) in the usual Sobolev spaces, and u ∈ K m 3/2−ǫ (T S) in the weighted Sobolev spaces. The regularity index m can be as large as desired, which is crucial for numerical methods. For any choice of the Bloch wavevector k, we also
On absence of embedded eigenvalues for schrÖdinger operators with perturbed periodic potentials
Communications in Partial Differential Equations, 2000
The problem of absence of eigenvalues imbedded into the continuous spectrum is considered for a Schrödinger operator with a periodic potential perturbed by a sufficiently fast decaying "impurity" potential. Results of this type have previously been known for the one-dimensional case only. Absence of embedded eigenvalues is shown in dimensions two and three if the corresponding Fermi surface is irreducible modulo natural symmetries. It is conjectured that all periodic potentials satisfy this condition. Separable periodic potentials satisfy it, and hence in dimensions two and three Schrödinger operator with a separable periodic potential perturbed by a sufficiently fast decaying "impurity" potential has no embedded eigenvalues.
Schr "odinger operator with a junction of two 1-dimensional periodic potentials
Asymptotic Analysis, 2005
The spectral properties of the Schrödinger operator Tty=−y′′+qtyT_ty= -y''+q_tyTty=−y′′+qty in L2(R)L^2(\R)L2(R) are studied, with a potential qt(x)=p1(x),x<0,q_t(x)=p_1(x), x<0, qt(x)=p1(x),x<0, and qt(x)=p(x+t),x>0,q_t(x)=p(x+t), x>0, qt(x)=p(x+t),x>0, where p1,pp_1, pp1,p are periodic potentials and tinRt\in \RtinR is a parameter of dislocation. Under some conditions there exist simultaneously gaps in the continuous spectrum of T0T_0T0 and eigenvalues in these gaps. The main goal of this paper is to study the discrete spectrum and the resonances of TtT_tTt. The following results are obtained: i) In any gap of TtT_tTt there exist 0,10,10,1 or 2 eigenvalues. Potentials with 0,1 or 2 eigenvalues in the gap are constructed. ii) The dislocation, i.e. the case p1=pp_1=pp1=p is studied. If tto0t\to 0tto0, then in any gap in the spectrum there exist both eigenvalues ($ \le 2 )andresonances() and resonances ()andresonances( \le 2 )of) of )ofT_t$ which belong to a gap on the second sheet and their asymptotics as tto0t\to 0 tto0 are determined. iii) The eigenvalues of the half-solid, i.e. p1=rmconstantp_1={\rm constant}p1=rmconstant, are also studied. iv) We prove that for any even 1-periodic potential ppp and any sequences dn1iy\{d_n\}_1^{\iy}dn1iy, where dn=1d_n=1dn=1 or dn=0d_n=0dn=0 there exists a unique even 1-periodic potential p1p_1p1 with the same gaps and dnd_ndn eigenvalues of T_0T_0T_0 in the n-th gap for each nge1.n\ge 1.nge1.
Perturbation Theory for the Multidimensional Schrodinger Operator with a Periodic Potential
2005
In this paper we obtain asymptotic formulas of arbitrary order for the Bloch eigenvalue and the Bloch function of the periodic Schrodinger operator − ∆ + q(x), of arbitrary dimension, when corresponding quasimomentum lies near a diffraction hyperplane. Moreover, we estimate the measure of the isoenergetic surfaces in the high energy region. Bisides, writing the asymptotic formulas for the Bloch eigenvalue and the Bloch function, when corresponding quasimomentum lies far from the diffraction hyperplanes, obtained in my previous papers in improved and enlarged form, we obtain the complete perturbation theory for the multidimensional Schrodinger operator with a periodic potential. 1
Periodic Schrödinger operators with large gaps and Wannier-Stark ladders
Physical review letters, 1994
We describe periodic, one dimensional Schr odinger operators, with the property that the widths of the forbidden gaps increase at large energies and the gap to band ratio is not small. Such systems can be realized by periodic arrays of geometric scatterers, e.g., a necklace of rings. Small, multichannel scatterers lead (for low energies) to the same band spectrum as that of a periodic array of (singular) point interactions known as 0. We consider the Wannier{Stark ladder of 0 and show that the corresponding Schr odinger operator has no absolutely continuous spectrum.