Norm Resolvent Convergence Research Articles

Resolvent Convergence to Dirac Operators on Planar Domains

Consider a Dirac operator defined on the whole plane with a mass term of size m supported outside a domain Omega. We give a simple proof for the norm resolvent convergence, as m goes to infinity, of this operator to a Dirac operator defined on Omega with infinite mass boundary conditions. The result is valid for bounded and unbounded domains and gives estimates on the speed of convergence. Moreover, the method easily extends when adding external matrix-valued potentials.

Norm-resolvent convergence in perforated domains

Norm-resolvent convergence in perforated domains

Approximation of Fractals by Discrete Graphs: Norm Resolvent and Spectral Convergence

We show a norm convergence result for the Laplacian on a class of pcf self-similar fractals with arbitrary Borel regular probability measure which can be approximated by a sequence of finite-dimensional weighted graph Laplacians. As a consequence other functions of the Laplacians (heat operator, spectral projections etc.) converge as well in operator norm. One also deduces convergence of the spectrum and the eigenfunctions in energy norm.

The Norm Resolvent Convergence for Elliptic Operators in Multi-Dimensional Domains with Small Holes

We consider a second order elliptic operator with variable coefficients in a multidimensional domain with a small hole and some classical boundary condition on the hole boundary. We show that the resolvent of this operator converges to the resolvent of the limit operator in the domain without holes in the sense of the norm of bounded operators acting from L2 to $$ {W}_2^1 $$ . For the convergence rate we obtain sharp estimates relative to the smallness order.

REDUCTION OF DIMENSION AS A CONSEQUENCE OF NORM‐RESOLVENT CONVERGENCE AND APPLICATIONS

This paper is devoted to dimensional reductions via the norm resolvent convergence. We derive explicit bounds on the resolvent difference as well as spectral asymptotics. The efficiency of our abstract tool is demonstrated by its application on seemingly different PDE problems from various areas of mathematical physics; all are analysed in a unified manner now, known results are recovered and new ones established.

Approximations of spectra of Schrödinger operators with complex potentials on ℝd

ABSTRACTWe study spectral approximations of Schrödinger operators T = −Δ+Q with complex potentials on Ω = ℝd, or exterior domains Ω⊂ℝd, by domain truncation. Our weak assumptions cover wide classes of potentials Q for which T has discrete spectrum, of approximating domains Ωn, and of boundary conditions on ∂Ωn such as mixed Dirichlet/Robin type. In particular, Re Q need not be bounded from below and Q may be singular. We prove generalized norm resolvent convergence and spectral exactness, i.e. approximation of all eigenvalues of T by those of the truncated operators Tn without spectral pollution. Moreover, we estimate the eigenvalue convergence rate and prove convergence of pseudospectra. Numerical computations for several examples, such as complex harmonic and cubic oscillators for d = 1,2,3, illustrate our results.

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

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.

Asymptotic spectral analysis in colliding leaky quantum layers

We consider the Schrödinger operator with a complex delta interaction supported by two parallel hypersurfaces in the Euclidean space of any dimension. We analyse spectral properties of the system in the limit when the distance between the hypersurfaces tends to zero. We establish the norm-resolvent convergence to a limiting operator and derive first-order corrections for the corresponding eigenvalues.

Norm resolvent convergence of Dirichlet Laplacian in unbounded thin waveguides

The proof of norm resolvent convergence of Dirichlet Laplacian to effective operators, as the diameter of spatial waveguides vanishes, is extended to the unbounded case.

Norm-resolvent convergence for elliptic operators in domain with perforation along curve

We consider an infinite strip perforated along a curve by small holes. In this perforated domain, we consider a scalar second-order elliptic differential operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic, we describe possible homogenized problems and prove the norm-resolvent convergence of the perturbed operator to a homogenized one. We also provide estimates for the rate of the convergence.

Magnetic Effects in Curved Quantum Waveguides

The interplay among the spectrum, geometry and magnetic field in tubular neighbourhoods of curves in Euclidean spaces is investigated in the limit when the cross section shrinks to a point. Proving a norm resolvent convergence, we derive effective, lower-dimensional models which depend on the intensity of the magnetic field and curvatures. The results are used to establish complete asymptotic expansions for eigenvalues. Spectral stability properties based on Hardy-type inequalities induced by magnetic fields are also analysed.

Resolvent convergence and spectral approximations of sequences of self-adjoint subspaces

This paper studies resolvent convergence and spectral approximations of sequences of self-adjoint subspaces (relations) in complex Hilbert spaces. Concepts of strong resolvent convergence, norm resolvent convergence, spectral inclusion, and spectral exactness are introduced. Fundamental properties of resolvents of subspaces are studied. By applying these properties, several equivalent and sufficient conditions for convergence of sequences of self-adjoint subspaces in the strong and norm resolvent senses are given. It is shown that a sequence of self-adjoint subspaces is spectrally inclusive under the strong resolvent convergence and spectrally exact under the norm resolvent convergence. A sufficient condition is given for spectral exactness of a sequence of self-adjoint subspaces in an open interval lacking essential spectral points. In addition, criteria are established for spectral inclusion and spectral exactness of a sequence of self-adjoint subspaces that are defined on proper closed subspaces.

Approximations of quantum-graph vertex couplings by singularly scaled potentials

We investigate the limit properties of a family of Schrödinger operators of the form acting on n-edge star graphs with the Kirchhoff interface conditions at the vertex. Here the real-valued potential Q has compact support and λ( · ) is analytic around ε = 0 with λ(0) = 1. We show that if the operator has a zero-energy resonance of order m for ε = 1 and λ(1) = 1, in the limit ε → 0 one obtains the Laplacian with a vertex coupling depending on parameters. We prove the norm-resolvent convergence as well as the convergence of the corresponding on-shell scattering matrices. The obtained vertex couplings are of scale-invariant type provided λ′(0) = 0; otherwise the scattering matrix depends on energy and the scaled potential becomes asymptotically opaque in the low-energy limit.

Norm resolvent convergence of singularly scaled Schrödinger operators and δ′-potentials

For a real-valued function V of the Faddeev–Marchenko class, we prove the norm-resolvent convergence, as ε → 0, of a family Sε of one-dimensional Schrödinger operators on the line of the form Under certain conditions, the functions ε−2V (x/ε) converge in the sense of distributions as ε → 0 to δ′ (x), and then the limit S0 of Sε may be considered as a ‘physically motivated’ interpretation of the one-dimensional Schrödinger operator with potential δ′.

A General Approximation of Quantum Graph Vertex Couplings by Scaled Schrödinger Operators on Thin Branched Manifolds

We demonstrate that any self-adjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schroedinger operators on a tubular network built over the graph. If such a manifold has a boundary, Neumann conditions are imposed at it. The procedure involves a local change of graph topology in the vicinity of the vertex; the approximation scheme constructed on the graph is subsequently `lifted' to the manifold. For the corresponding operator a norm-resolvent convergence is proved, with the natural identification map, as the tube diameters tend to zero.

1D Schrödinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

For real bounded functions \Phi and \Psi of compact support, we prove the norm resolvent convergence, as \epsilon and \nu tend to 0, of a family of one-dimensional Schroedinger operators on the line of the form S_{\epsilon, \nu}= -D^2+\alpha\epsilon^{-2}\Phi(\epsilon^{-1}x)+\beta\nu^{-1}\Psi(\nu^{-1}x), provided the ratio \nu/\epsilon has a finite or infinity limit. The limit operator S_0 depends on the shape of \Phi and \Psi as well as on the limit of ratio \nu/\epsilon. If the potential \alpha\Phi possesses a zero-energy resonance, then S_0 describes a non trivial point interaction at the origin. Otherwise S_0 is the direct sum of the Dirichlet half-line Schroedinger operators.

THE EFFECTIVE HAMILTONIAN IN CURVED QUANTUM WAVEGUIDES UNDER MILD REGULARITY ASSUMPTIONS

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.

Mathematical predominance of Dirichlet condition for the one-dimensional Coulomb potential

We restrict a quantum particle under a Coulombian potential (i.e., the Schrödinger operator with inverse of the distance potential) to three-dimensional tubes along the x axis and diameter ɛ, and study the confining limit ɛ → 0. In the repulsive case we prove a strong resolvent convergence to a one-dimensional limit operator, which presents Dirichlet boundary condition at the origin. Due to the possibility of the falling of the particle in the center of force, in the attractive case we need to regularize the potential and also prove a norm resolvent convergence to the Dirichlet operator at the origin. Thus, it is argued that, among the infinitely many self-adjoint realizations of the corresponding problem in one dimension, the Dirichlet boundary condition at the origin is the reasonable one-dimensional limit.

Impenetrability of Aharonov–Bohm Solenoids: Proof of Norm Resolvent Convergence

Consider bounded solenoids in the space or planar approximate models of infinitely long solenoids. It is proved that the solenoid impenetrability, by means of a sequence of Hamiltonians with diverging potentials, converges in the norm resolvent sense to the usual Aharonov–Bohm model with Dirichlet boundary condition. The framework is that of nonrelativistic quantum mechanics.

On norm resolvent convergence of Schrödinger operators with δ′-like potentials

For a function that is integrable and compactly supported, we prove the norm resolvent convergence, as ε → 0, of a family Sε of one-dimensional Schrödinger operators on the line of the form If the potential V satisfies the conditions then the functions ε−2V(x/ε) converge in the sense of distributions as ε → 0 to δ′(x), and the limit S0 of Sε might be considered as a ‘physically motivated’ interpretation of the one-dimensional Schrödinger operator with a potential δ′. In 1985, Šeba claimed that the limit operator S0 is the direct sum of the free Schrödinger operators on positive and negative semi-axes subject to the Dirichlet condition at x = 0, which suggested that in dimension 1 there is no non-trivial Hamiltonian with the potential δ′. In this paper, we show that in fact S0 essentially depends on V: although the above results are true generically, in the exceptional (or ‘resonant’) case, the limit S0 is non-trivial and is determined by the properties of an auxiliary Sturm–Liouville spectral problem associated with V. We then set V(ξ) = αΨ(ξ) with a fixed Ψ and show that there exists a countable set of resonances {αk}∞k = −∞ for which a partial transmission of the wave package occurs for S0.