Continuum limit for lattice Schrödinger operators

  title={Continuum limit for lattice Schr{\"o}dinger operators},
  author={Hiroshi Isozaki and Arne Jensen},
  journal={Reviews in Mathematical Physics},
We study the behavior of solutions of the Helmholtz equation [Formula: see text] on a periodic lattice as the mesh size [Formula: see text] tends to 0. Projecting to the eigenspace of a characteristic root [Formula: see text] and using a gauge transformation associated with the Dirac point, we show that the gauge transformed solution [Formula: see text] converges to that for the equation [Formula: see text] for a continuous model on [Formula: see text], where [Formula: see text]. For the case… 

Figures from this paper

Continuum limit of the lattice quantum graph Hamiltonian
. We consider the quantum graph Hamiltonian on the square lattice in Euclidean space, and we show that the spectrum of the Hamiltonian converges to the corresponding Schr(cid:127)odinger operator on
Continuum limits for discrete Dirac operators on 2D square lattices
A natural and simple embedding of l(Z h ) into L(R) that enables us to compare the discreteDirac operators with the continuum Dirac operators in the same Hilbert space L( R) is proposed and strong resolvent convergence is proved.
Semiclassical analysis and the Agmon-Finsler metric for discrete Schr\"odinger operators
The Agmon estimate for multi-dimensional discrete Schrödinger operators is studied with emphasis on the microlocal analysis on the torus. We first consider the semiclassical setting where
Resonances for quasi-one-dimensional discrete Schr\"odinger operators
Consider a multichannel Laplace type operator H0 on ` (Z) ⊗ G, where G is an auxiliary separable Hilbert space. For suitable compact perturbations V , we study the distribution of resonances of the
We discuss the continuum limit of discrete Dirac operators on the square lattice in R as the mesh size tends to zero. To this end, we propose a natural and simple embedding of l(Z h ) into L(R) that
Discrete approximations to Dirac operators and norm resolvent convergence
We consider continuous Dirac operators defined on Rd, d ∈ {1, 2, 3}, together with various discrete versions of them. Both forward-backward and symmetric finite differences are used as approximations
Norm Resolvent Convergence of Discretized Fourier Multipliers
We prove norm estimates for the difference of resolvents of operators and their discrete counterparts, embedded into the continuum using biorthogonal Riesz sequences. The estimates are given in the


Numerical Dispersive Schemes for the Nonlinear Schrödinger Equation
The convergence of the two-grid method for nonlinearities that cannot be handled by energy arguments and which, even in the continuous case, require Strichartz estimates is proved.
Localization for gapped Dirac Hamiltionians with random perturbations: Application to graphene antidot lattices
In this paper we study random perturbations of first order elliptic operators with periodic potentials. We are mostly interested in Hamiltonians modeling graphene antidot lattices with impurities.
Inverse Scattering for Schrödinger Operators on Perturbed Lattices
We study the inverse scattering for Schrödinger operators on locally perturbed periodic lattices. We show that the associated scattering matrix is equivalent to the Dirichlet-to-Neumann map for a
Spectral Properties of Schrödinger Operators on Perturbed Lattices
We study the spectral properties of Schrödinger operators on perturbed lattices. We shall prove the non-existence or the discreteness of embedded eigenvalues, the limiting absorption principle for
The paper deals with the asymptotic behaviour (as ) of the solutions of some non-stationary problems in mathematical physics. The main aim of the paper is to clarify conditions under which stationary
On a continuum limit of discrete Schrödinger operators on square lattice
The norm resolvent convergence of discrete Schrodinger operators to a continuum Schrodinger operator in the continuum limit is proved under relatively weak assumptions. This result implies, in
Strong Convergence for Discrete Nonlinear Schrödinger equations in the Continuum Limit
We consider discrete nonlinear Schrodinger equations (DNLS) on the lattice $h\Bbb{Z}^d$ whose linear part is determined by the discrete Laplacian which accounts only for nearest neighbor interactio...
Spectral properties of Schrödinger operators and scattering theory
© Scuola Normale Superiore, Pisa, 1975, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze »
Absence of singular continuous spectrum for certain self-adjoint operators
We give a sufficient condition for a self-adjoint operator to have the following properties in a neighborhood of a pointE of its spectrum:a)its point spectrum is finite;b)its singular continuous
Spectral Gaps in Graphene Antidot Lattices
We consider the gap creation problem in an antidot graphene lattice, i.e. a sheet of graphene with periodically distributed obstacles. We prove several spectral results concerning the size of the gap