Periodic Schrodinger Operator Research Articles

Numerical quadrature in the Brillouin zone for periodic Schrödinger operators

As a consequence of Bloch's theorem, the numerical computation of the fermionic ground state density matrices and energies of periodic Schrodinger operators involves integrals over the Brillouin zone. These integrals are difficult to compute numerically in metals due to discontinuities in the integrand. We perform an error analysis of several widely-used quadrature rules and smearing methods for Brillouin zone integration. We precisely identify the assumptions implicit in these methods and rigorously prove error bounds. Numerical results for two-dimensional periodic systems are also provided. Our results shed light on the properties of these numerical schemes, and provide guidance as to the appropriate choice of numerical parameters.

Existence of eigenvalues embedded in the spectral bands of Schrödinger operators on carbon nanotubes with impurities

It is known in Korotyaev and Lobanov (Ann Henri Poincare 8:1151–1176, 2007) and Parchment (Commun Math Phys 275:805–826, 2007) that spectra of periodic Schrodinger operators on the metric graph corresponding to a carbon nanotube have the band-gap structure. The band-gap spectrum consists of not only infinitely many closed intervals but also the so-called flat bands (the set of eigenvalue with infinite multiplicities). In this paper, we study the spectrum of the free Schrodinger operators on a zigzag carbon nanotube with finite number of impurities expressed as $$\delta $$-interactions with strength $$\alpha \in \mathbf{R}$$. Assume that our impurities are symmetrically located in two ways. For a suitable $$\alpha $$, we construct eigenvalues embedded in the interior of spectral bands (not in spectral gaps).

Wannier functions and discrete NLS equations for nematicons$^\dagger$

We derive nonlocal discrete nonlinear Schrodinger (DNLS) equations for laser beam propagation in optical waveguide arrays that use a nematic liquid crystal substrate. We start with an NLS-elliptic model for the problem and propose a simplified version that incorporates periodicity in one of the directions transverse to the propagation of the beam. We use Wannier basis functions for an associated Schrodinger operator with periodic potential to derive discrete equations for Wannier modes and propose some possible simplified systems for interactions of modes within the first energy band of the periodic Schrodinger operator. In particular, we present the simplest generalization of a model proposed by Fratalocchi and Assanto by including a linear nonlocal term, and see evidence for parameter regimes where nonlinearity is more pronounced.

Two-Dimensional Periodic Schrödinger Operators Integrable at an Energy Eigenlevel

The main goal of the first part of the paper is to show that the Fermi curve of a two-dimensional periodic Schrodinger operator with nonnegative potential whose points parameterize the Bloch solutions of the Schrodinger equation at the zero energy level is a smooth M-curve. Moreover, it is shown that the poles of the Bloch solutions are located on the fixed ovals of an antiholomorphic involution so that each but one oval contains precisely one pole. The topological type is stable until, at some value of the deformation parameter, the zero level becomes an eigenlevel for the Schrodinger operator on the space of (anti)periodic functions. The second part of the paper is devoted to the construction of such operators with the help of a generalization of the Novikov-Veselov construction.

Topologically Protected States in One-Dimensional Systems

We study a class of periodic Schrodinger operators, which we prove have Dirac points. We then show that the introduction of an via adiabatic modulation of a periodic potential by a domain wall results in the bifurcation of spatially localized \edge states, associated with the topologically protected zero- energy mode of an asymptotic one-dimensional Dirac operator. The bound states we construct can be realized as highly robust TM- electromagnetic modes for a class of photonic waveguides with a phase-defect. Our model also captures many aspects of the phenomenon of topologically protected edge states for two-dimensional bulk structures such as the honeycomb structure of graphene.

Spectral band structure of periodic Schrödinger operators with two potentials on the degenerate zigzag nanotube

In this paper, we study the spectra of periodic Schrodinger operators with two potentials on a quantum graph, which is related to a carbon nanotube. We show that the spectrum has the band structure. Furthermore, we deal with a coexistence problem and see the difference between one-potential case and two-potentials case.

Decisiveness of the spectral gaps of periodic Schrödinger operators on the dumbbell-like metric graph

In this paper, we consider periodic Schrodinger operators on the dumbbell-like metric graph, which is a periodic graph consisting of lines and rings. Let one line and two rings be in the basic period. We see the relationship between the structure of graph and the band-gap spectrum.

On the measure of the spectrum of direct integrals

We obtain the estimate of the Lebesgue measure of the spectrum for the direct integral of matrix-valued functions. These estimates are applicable for a wide class of discrete periodic operators. For example these results give new and sharp spectral bounds for 1D periodic Jacobi matrices and 2D discrete periodic Schrodinger operators.

Non-consistent approximations of self-adjoint eigenproblems: application to the supercell method

In this article, we introduce a general theoretical framework to analyze non-consistent approximations of the discrete eigenmodes of a self-adjoint operator. We focus in particular on the discrete eigenvalues laying in spectral gaps. We first provide a priori error estimates on the eigenvalues and eigenvectors in the absence of spectral pollution. We then show that the supercell method for perturbed periodic Schrodinger operators falls into the scope of our study. We prove that this method is spectral pollution free, and we derive optimal convergence rates for the planewave discretization method, taking numerical integration errors into account. Some numerical illustrations are provided.

On the quantum graph spectra of graphyne nanotubes

We describe explicitly the dispersion relations and spectra of periodic Schrodinger operators on a graphyne nanotube structure.

Absence of eigenvalues for the periodic Schrödinger operator with singular potential in a rectangular cylinder

We consider the periodic Schrodinger operator on a d-dimensional cylinder with rectangular section. The electric potential may contain a singular component of the form σ(x, y)δΣ(x,y), where Σ is a periodic system of hypersurfaces. We establish that there are no eigenvalues in the spectrum of this operator, provided that Σ is sufficiently smooth and σ ∈ Lp,loc(Σ), p > d − 1.

Spectral shift function for slowly varying perturbation of periodic Schrödinger operators

In this paper we study the asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic Schrodinger operators. We give a weak and pointwise asymptotic expansions in powers of h of the derivative of the spectral shift function corresponding to the pair P(h) =P0 +'(hx);P0 = - +V(x) ; where '(x) 2 C 1 (R n ;R) is a decreasing function, O(jxj - ) for some > n and h is a small positive parameter. Here the potential V is real, smooth and periodic with respect to a lattice in R n . To prove the pointwise asymptotic expansion of the spectral shift function, we establish a limiting absorption Theorem for P(h).

Trace identities for commutators, with applications to the distribution of eigenvalues

We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schrodinger operators and Schrodinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue lambda(N+1) in terms of the lower spectrum, bounds on ratios of means of eigenvalues, and universal monotonicity properties of eigenvalue moments, which imply sharp versions of Lieb-Thirring inequalities. In the case of a Schrodinger operator on an immersed manifold of dimension d, we derive a version of Reilly's inequality bounding the eigenvalue lambda(N+1) of the Laplace-Beltrami operator by a universal constant times vertical bar vertical bar h vertical bar N-2(infinity)2/d.

Eigenfunction Localization for the 2D Periodic Schrodinger Operator

We prove that for any fixed trigonometric polynomial potential satisfying a genericity condition, the spectrum of the two-dimensional periodic Schrodinger operator has finite multiplicity and the Fourier series of the eigenfunctions are uniformly exponentially localized about a finite number of frequencies. As a corollary, the L p norms of the eigenfunctions are bounded for all p > 0, which answers a question of Toth and Zelditch [15].

Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps

We prove, via an elementary variational method, 1d and 2d localization within the band gaps of a periodic Schrodinger operator for any mostly negative or mostly positive defect potential, V, whose depth is not too great compared to the size of the gap. In a similar way, we also prove sufficient conditions for 1d and 2d localization below the ground state of such an operator. Furthermore, we extend our results to 1d and 2d localization in d dimensions; for example, a linear or planar defect in a 3d crystal. For the case of D-fold degenerate band edges, we also give sufficient conditions for localization of up to D states.

Peierls substitution and the Maslov operator method

We consider a periodic Schrodinger operator in a constant magnetic field with vector potential A(x). A version of adiabatic approximation for quantum mechanical equations with rapidly varying electric potentials and weak magnetic fields is the Peierls substitution which, in appropriate dimensionless variables, permits writing the pseudodifferential equation for the new auxiliary function: Open image in new window , where Open image in new window is the corresponding energy level of some auxiliary Schrodinger operator, assumed to be nondegenerate, and µ is a small parameter. In the present paper, we use V. P. Maslov’s operator method to show that, in the case of a constant magnetic field, such a reduction in any perturbation order leads to the equation Open image in new window with the operator Open image in new window represented as a function depending only on the operators of kinetic momenta \( \hat P_j = - i\mu \partial _{x_j } + A_j \left( x \right) \).

Bloch Varieties of Higher-Dimensional, Periodic Schrödinger Operators

We relax the regularity conditions on potentials of higherdimensional periodic Schrodinger operators while their resolvents may still be defined as compact operators on L. This enables us to define the Bloch varieties locally as the zero locus of a holomorphic map in a more general setting. We also give an asymptotic description of the Fermi curve.

A criterion for Hill operators to be spectral operators of scalar type

We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schro dinger operator) H = −d 2 /dx 2 + V to be a spectral operator of scalar type. The conditions show the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V. In the course of our analysis, we also establish a functional model for periodic Schrodinger operators that are spectral operators of scalar type and develop the corresponding eigenfunction expansion. The problem of deciding which Hill operators are spectral operators of scalar type appears to have been open for about 40 years.

Quasilevels of a two-particle Schrödinger operator with a perturbed periodic potential

We consider a two-dimensional periodic Schrodinger operator perturbed by the interaction potential of two one-dimensional particles. We prove that quasilevels (i.e., eigenvalues or resonances) of the given operator exist for a fixed quasimomentum and a small perturbation near the band boundaries of the corresponding periodic operator. We study the asymptotic behavior of the quasilevels as the coupling constant goes to zero. We obtain a simple condition for a quasilevel to be an eigenvalue.

On the Spectra of Carbon Nano-Structures

An explicit derivation of dispersion relations and spectra for periodic Schrodinger operators on carbon nano-structures (including graphen and all types of single-wall nano-tubes) is provided.