Norm resolvent convergence of Dirichlet Laplacian in unbounded thin waveguides

2015 ◽  
Vol 46 (1) ◽  
pp. 139-158 ◽  
Author(s):  
César R. de Oliveira ◽  
Alessandra A. Verri
Keyword(s):  
Dirichlet Laplacian ◽  
Resolvent Convergence ◽  
Norm Resolvent Convergence

scholarly journals Regular approximation of singular Sturm–Liouville problems with eigenparameter dependent boundary conditions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Maozhu Zhang ◽  
Kun Li ◽  
Hongxiang Song
Keyword(s):  
Boundary Conditions ◽  
Operator Theory ◽  
Resolvent Convergence ◽  
Spectral Inclusion ◽  
Regular Approximation ◽  
Special Cases ◽  
Abstract Operator ◽  
Norm Resolvent Convergence ◽  
Sturm Liouville ◽  
Spectral Exactness

AbstractIn this paper we consider singular Sturm–Liouville problems with eigenparameter dependent boundary conditions and two singular endpoints. The spectrum of such problems can be approximated by those of the inherited restriction operators constructed. Via the abstract operator theory, the strongly resolvent convergence and norm resolvent convergence of a sequence of operators are obtained and it follows that the spectral inclusion of spectrum holds. Moreover, spectral exactness of spectrum holds for two special cases.

scholarly journals Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve

2016 ◽  
Vol 146 (6) ◽  
pp. 1115-1158 ◽  
Author(s):  
Denis Borisov ◽  
Giuseppe Cardone ◽  
Tiziana Durante
Keyword(s):  
Boundary Conditions ◽  
Elliptic Operator ◽  
Rate Of Convergence ◽  
Elliptic Operators ◽  
Resolvent Convergence ◽  
Order Elliptic Operator ◽  
Classical Boundary ◽  
Norm Resolvent Convergence ◽  
Operator Norms ◽  
Infinite Planar

We consider an infinite planar straight strip perforated by small holes along a curve. In such a domain, we consider a general second-order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic and satisfies rather weak assumptions, we describe all possible homogenized problems. Our main result is the norm-resolvent convergence of the perturbed operator to a homogenized one in various operator norms and the estimates for the rate of convergence. On the basis of the norm-resolvent convergence, we prove the convergence of the spectrum.

scholarly journals THE EFFECTIVE HAMILTONIAN IN CURVED QUANTUM WAVEGUIDES UNDER MILD REGULARITY ASSUMPTIONS

2012 ◽  
Vol 24 (07) ◽  
pp. 1250018 ◽  
Author(s):  
DAVID KREJČIŘÍK ◽  
HELENA ŠEDIVÁKOVÁ
Keyword(s):  
Cross Section ◽  
Parallel Transport ◽  
Three Dimensional ◽  
Reference Curve ◽  
Dirichlet Laplacian ◽  
Resolvent Convergence ◽  
Uniform Cross Section ◽  
Reference Curves ◽  
Twisting Angle ◽  
Quantum Waveguides

The Dirichlet Laplacian in a curved three-dimensional tube built along a spatial (bounded or unbounded) curve is investigated in the limit when the uniform cross-section of the tube diminishes. Both deformations due to bending and twisting of the tube are considered. We show that the Laplacian converges in a norm-resolvent sense to the well-known one-dimensional Schrödinger operator whose potential is expressed in terms of the curvature of the reference curve, the twisting angle and a constant measuring the asymmetry of the cross-section. Contrary to previous results, we allow the reference curves to have non-continuous and possibly vanishing curvature. For such curves, the distinguished Frenet frame standardly used to define the tube need not exist and, moreover, the known approaches to prove the result for unbounded tubes do not work. To establish the norm-resolvent convergence under the minimal regularity assumptions, we use an alternative frame defined by a parallel transport along the curve and a refined smoothing of the curvature via the Steklov approximation.

NORM RESOLVENT CONVERGENCE TO MAGNETIC SCHRÖDINGER OPERATORS WITH POINT INTERACTIONS

2001 ◽  
Vol 13 (04) ◽  
pp. 465-511 ◽  
Author(s):  
HIDEO TAMURA
Keyword(s):  
Magnetic Field ◽  
Boundary Conditions ◽  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Two Dimensions ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Point Interactions ◽  
Resolvent Convergence ◽  
Norm Resolvent Convergence

The Schrödinger operator with δ-like magnetic field at the origin in two dimensions is not essentially self-adjoint. It has the deficiency indices (2, 2) and each self-adjoint extension is realized as a differential operator with some boundary conditions at the origin. We here consider Schrödinger operators with magnetic fields of small support and study the norm resolvent convergence to Schrödinger operator with δ-like magnetic field. We are concerned with the boundary conditions realized in the limit when the support shrinks. The results obtained heavily depend on the total flux of magnetic field and on the resonance space at zero energy, and the proof is based on the analysis at low energy for resolvents of Schrödinger operators with magnetic potentials slowly falling off at infinity.

scholarly journals Potential Approximations to δ': An Inverse Klauder Phenomenon with Norm-Resolvent Convergence

2001 ◽  
Vol 224 (3) ◽  
pp. 593-612 ◽  
Author(s):  
Pavel Exner ◽  
Hagen Neidhardt ◽  
Valentin A. Zagrebnov
Keyword(s):  
Resolvent Convergence ◽  
Norm Resolvent Convergence

On the Neumann Laplacian in nonuniformly collapsing strips

2019 ◽  
Vol 22 (04) ◽  
pp. 1950021
Author(s):  
César R. de Oliveira ◽  
Alessandra A. Verri
Keyword(s):  
Boundary Conditions ◽  
Smooth Function ◽  
Robin Boundary Conditions ◽  
Effective Operator ◽  
Resolvent Convergence ◽  
Norm Resolvent Convergence ◽  
Neumann Laplacian ◽  
Robin Boundary ◽  
Effective Operators

Consider the Neumann Laplacian in the region below the graph of [Formula: see text], for a positive smooth function [Formula: see text] with both [Formula: see text] and [Formula: see text] bounded. As [Formula: see text] such region collapses to [Formula: see text] and an effective operator is found, which has Robin boundary conditions at [Formula: see text]. Then, we recover (under suitable assumptions in the case of unbounded [Formula: see text]) such effective operators through uniformly collapsing regions; in such approach, we have (roughly) got norm resolvent convergence for [Formula: see text] diverging less than exponentially.

Sign in / Sign up

Export Citation Format

Share Document