Resolvent Convergence Research Articles

An ergodic theorem with weights and applications to random measures, homogenization and hydrodynamics

We prove a multidimensional ergodic theorem with weighted averages for the action of the group Zd on a probability space. At level n weights are of the form n−dψ(j/n), j∈Zd, for real functions ψ decaying suitably fast. We discuss applications to random measures and to quenched stochastic homogenization of random walks on simple point processes with long-range random jump rates, allowing to remove the technical Assumption (A9) from Faggionato (2023) [Theorem 4.4]. This last result concerns also some semigroup and resolvent convergence particularly relevant for the derivation of the quenched hydrodynamic limit of interacting particle systems via homogenization and duality. As a consequence we show that also the quenched hydrodynamic limit of the symmetric simple exclusion process on point processes stated in Faggionato (2022) [Theorem 4.1] remains valid when removing the above mentioned Assumption (A9).

On sequences of sectorial forms converging 'from above'

We present a form convergence theorem for sequences of sectorial forms and their associated semigroups in a complex Hilbert space. Roughly speaking, the approximating forms $a_n$ are all `bounded below' by the limiting form $a$, but in contrast to the previous literature there is no monotonicity hypothesis on the sequence. Moreover, the forms are not supposed to be closed or densely defined. For a sectorial form one obtains an associated linear relation, whose negative generates a degenerate strongly continuous semigroup of linear operators. Our hypotheses on the sequence of forms imply strong resolvent convergence of the associated linear relations, which in turn implies convergence of the corresponding semigroups. The result is illustrated by two examples, one of them closely related to the Galerkin method of numerical analysis.

Resolvent Convergence for Differential–Difference Operators with Small Variable Translations

We consider general higher-order matrix elliptic differential–difference operators in arbitrary domains with small variable translations in lower-order terms. The operators are introduced by means of general higher-order quadratic forms on arbitrary domains. Each lower-order term depends on its own translation and all translations are governed by a small multi-dimensional parameter. The operators are considered either on the entire space or an arbitrary multi-dimensional domain with a regular boundary. The boundary conditions are also arbitrary and general and involve small variable translations. Our main results state that the considered operators converge in the norm resolvent sense to ones with zero translations in the best possible operator norm. Estimates for the convergence rates are established as well. We also prove the convergence of the spectra and pseudospectra.

Stability of ground state eigenvalues of non-local Schrödinger operators with respect to potentials and applications

In a first part of this paper we investigate the continuity (stability) of the spectrum of a class of non-local Schrödinger operators on varying the potentials. By imposing conditions of different strength on the convergence of the sequence of potentials, we give either direct proofs to show the strong or norm resolvent convergence of the so-obtained sequence of non-local Schrödinger operators, or via Γ-convergence of the related positive forms for more rough potentials. In a second part we use these results to show via a sequence of suitably constructed approximants that the ground states of massive or massless relativistic Schrödinger operators with spherical potential wells are radially decreasing functions.

Homogenization for operators with arbitrary perturbations in coefficients

We consider a general second order matrix operator in a multi-dimensional domain subject to a classical boundary condition. We perturbed it by a first order differential operator arbitrarily depending on a small multi-dimensional parameter and we study the existence of a limiting (homogenized) operator in the sense of the norm resolvent convergence. The first part of our main results states that the norm resolvent convergence is equivalent to the convergence of the coefficients in the perturbing operator in certain space of multipliers. The second part of our results says that the convergence in the mentioned spaces of multipliers is equivalent to the convergence of certain local mean values over small pieces of the considered domains. These results are supported by series of examples. We also provide a series of ways of generating new non-periodically oscillating perturbations for which our results are applicable.

Discrete approximations to Dirac operators and norm resolvent convergence

We consider continuous Dirac operators defined on \mathbf{R}^d , d\in\{1,2,3\} , together with various discrete versions of them. Both forward-backward and symmetric finite differences are used as approximations to partial derivatives. We also allow a bounded, Hölder continuous, and self-adjoint matrix-valued potential, which in the discrete setting is evaluated on the mesh. Our main goal is to investigate whether the proposed discrete models converge in norm resolvent sense to their continuous counterparts, as the mesh size tends to zero and up to a natural embedding of the discrete space into the continuous one. In dimension one we show that forward-backward differences lead to norm resolvent convergence, while in dimension two and three they do not . The same negative result holds in all dimensions when symmetric differences are used. On the other hand, strong resolvent convergence holds in all these cases. Nevertheless, and quite remarkably, a rather simple but non-standard modification to the discrete models, involving the mass term, ensures norm resolvent convergence in general.

Generalised norm resolvent convergence: comparison of different concepts

In this paper, we show that the two concepts of generalised norm resolvent convergence introduced by Weidmann and the first named author of this paper are equivalent.We also focus on the convergence speed and provide conditions under which the convergence speed is the same for both concepts. We illustrate the abstract results by a large number of examples.

Contact interactions and strong resolvent convergence, a partly variational approach

In Dell’Antonio (Eur. Phys. J. Plus 136:392, 2021) we considered several types of contact (zero range) interactions. Their Hamiltonians are limit, in strong resolvent topology, of a sequence of potentials with decreasing support. Here we review and improve these results and provide a new analysis Bose–Einstein condensation and of the Fermi sea.

On the convergence of numerical integration as a finite matrix approximation to multiplication operator

We study the convergence of a family of numerical integration methods where the numerical integration is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, convergence has already been established using the theory of strong operator convergence. In this article, we consider unbounded functions and domains which pose several difficulties compared to the bounded case. A natural choice of method for this study is the theory of strong resolvent convergence which has previously been mostly applied to study the convergence of approximations of differential operators. The existing theory already includes convergence theorems that can be used as proofs as such for a limited class of functions and extended for a wider class of functions in terms of function growth or discontinuity. The extended results apply to all self-adjoint operators, not just multiplication operators. We also show how Jensen’s operator inequality can be used to analyse the convergence of an improper numerical integral of a function bounded by an operator convex function.

Geometric Approximation of Point Interactions in Two-Dimensional Domains for Non-Self-Adjoint Operators

We define the notion of a point interaction for general non-self-adjoint elliptic operators in planar domains. We show that such operators can be approximated in a geometric way by cutting out a small cavity around the point, at which the interaction is concentrated. On the boundary of the cavity, we impose a special Robin-type boundary condition with a nonlocal term. As the cavity shrinks to a point, the perturbed operator converges in the norm resolvent sense to a limiting one with a point interaction containing an arbitrary prescribed complex-valued coupling constant. The mentioned convergence holds in a few operator norms, and for each of these norms we establish an estimate for the convergence rate. As a corollary of the norm resolvent convergence, we prove the convergence of the spectrum.

Quantum Graphs: Coulomb-Type Potentials and Exactly Solvable Models

We study the Schr\"{o}dinger operators on a non-compact star graph with the Coulomb-type potentials having singularities at the vertex. The convergence of regularized Hamiltonians $H_\varepsilon$ with cut-off Coulomb potentials coupled with $(\alpha \delta+\beta\delta')$-like ones is investigated.The 1D Coulomb potential and the $\delta'$-potential are very sensitive to their regularization method. The conditions of the norm resolvent convergence of $H_\varepsilon$ depending on the regularization are established. The limit Hamiltonians give the Schr\"{o}dinger operators with the Coulomb-type potentials a mathematically precise meaning, ensuring the correct choice of vertex conditions. We also describe all self-adjoint realizations of the formal Coulomb Hamiltonians on the star graph.

Discrete approximations to Dirichlet and Neumann Laplacians on a half-space and norm resolvent convergence

Discrete approximations to Dirichlet and Neumann Laplacians on a half-space and norm resolvent convergence

Regular approximation of singular third-order differential operators

In this paper, regular approximation of third-order differential operators with two singular endpoints is investigated. Firstly the induced restriction operator is constructed and proved to be a self-adjoint operator in the Hilbert space. Moreover the convergence of operators and spectrum is obtained by strong resolvent convergence. In particular, norm resolvent convergence is given when the two endpoints are both limit circle.

Approximation of point interactions by geometric perturbations in two-dimensional domains

In this paper, we present a new type of approximation of a second-order elliptic operator in a planar domain with a point interaction. It is of a geometric nature that the approximating family consists of operators with the same symbol and regular coefficients on the domain with a small hole. At the boundary of it, Robin condition is imposed with the coefficient which depends on the linear size of a hole. We show that as the hole shrinks to a point and the parameter in the boundary condition is scaled in a suitable way, nonlinear and singular, the indicated family converges in the norm-resolvent sense to the operator with the point interaction. This resolvent convergence is established with respect to several operator norms and order-sharp estimates of the convergence rates are provided.

Gamma convergence and renormalization group: Two sides of a coin?

We discuss, both from the point of view of Gamma convergence and from the point of view of the renormalization Group, the zero range strong contact interaction of three non-relativistic massive particles. Formally, the potential term is g (delta (x_3-x_1) + delta (x_3 -x_2)), ;, g < 0 and is the limit epsilon rightarrow 0 of approximating potentials V_epsilon (|x_i -x_3|) = epsilon ^{-3} V ( frac{|x_i - x_3|}{epsilon }) , V( x) in L^1(R^3) cap L^2 (R^3) . The presence of a delta function in the limit does not allow the use of standard tools of functional analysis. In the first approach (European Phys. J. Plus 136-363, 2021), (European Phys. J. Plus 1136-1161, 2021), we introduced a map mathcal{K}, called Krein Map , from L^2 (R^9) to a space (Minlos space) mathcal{M}) of more singular functions. In { mathcal M}, the system is represented by a one parameter family of self-adjoint operators. In the topology of L^2 (R^9), the system is an ordered family of weakly closed quadratic forms. By Gamma convergence, the infimum is a self-adjoint operator, the Hamiltonian H of the system. Gamma convergence implies resolvent convergence (An Introduction to Gamma Convergence Springer 1993) but not operator convergence!. This approach is variational and non-perturbative. In the second approach, perturbation theory is used. At each order of perturbation theory, divergences occur when epsilon rightarrow 0. A finite renormalized Hamiltonian H_R is obtained by redefining mass and coupling constant at each order of perturbation theory. In this approach, no distinction is made between self-adjoint operators and quadratic forms. One expects that H = H_R , i.e., that “renormalization” amounts to the difference between the Hamiltonian obtained by quadratic form convergence and the one obtained by Gamma convergence. We give some hints, but a formal proof is missing. For completeness, we discuss briefly other types of zero-range interactions.

Tight-binding reduction and topological equivalence in strong magnetic fields

Topological insulators (TIs) are a class of materials which are insulating in their bulk form yet, upon introduction of an a boundary or edge, e.g. by abruptly terminating the material, may exhibit spontaneous current along their boundary. This property is quantified by topological indices associated with either the bulk or the edge system. In the field of condensed matter physics, tight binding (discrete) approximate models, parametrized by hopping coefficients, have been used successfully to capture the topological behavior of TIs in many settings. However, whether such tight binding models capture the same topological features as the underlying continuum models of quantum physics has been an open question.We resolve this question in the context of the archetypal example of topological behavior in materials, the integer quantum Hall effect. We study a class of continuum Hamiltonians, Hλ, which govern electron motion in a two-dimensional crystal under the influence of a perpendicular magnetic field. No assumption is made on translation invariance of the crystal. We prove, in the regime where both the magnetic field strength and depth of the crystal potential are sufficiently large, λ≫1, that the low-lying energy spectrum and eigenstates (and corresponding large time dynamics) of Hλ are well-described by a scale-free discrete Hamiltonian, HTB; we show norm resolvent convergence. The relevant topological index is the Hall conductivity, which is expressible as a Fredholm index. We prove that for large λ the topological indices of Hλ and HTB agree. This is proved separately for bulk and edge geometries. Our results justify the principle of using discrete models in the study of topological matter.

Operator-Norm Resolvent Asymptotic Analysis of Continuous Media with High-Contrast Inclusions

Using a generalisation of the classical notion of Dirichlet-to-Neumann map and the related formulae for the resolvents of boundary-value problems, we analyse the asymptotic behaviour of solutions to a "transmission problem" for a high-contrast inclusion in a continuous medium, for which we prove the operator-norm resolvent convergence to a limit problem of "electrostatic" type for functions that are constant on the inclusion. In particular, our results imply the convergence of the spectra of high-contrast problems to the spectrum of the limit operator, with order-sharp convergence estimates.

Norm Resolvent Convergence of Elliptic Operators in Domains with Thin Spikes

Norm Resolvent Convergence of Elliptic Operators in Domains with Thin Spikes

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 operator norm for operators on square integrable functions, and depend explicitly on the mesh size for the discrete operators. The operators are a sum of a Fourier multiplier and a multiplicative potential. The Fourier multipliers include the fractional Laplacian and the pseudo-relativistic free Hamiltonian. The potentials are real, bounded, and Hölder continuous. As a side-product, the Hausdorff distance between the spectra of the resolvents of the continuous and discrete operators decays with the same rate in the mesh size as for the norm resolvent estimates. The same result holds for the spectra of the original operators in a local Hausdorff distance.

On a continuum limit of discrete Schrödinger operators on square lattice

The norm resolvent convergence of discrete Schrödinger operators to a continuum Schrödinger operator in the continuum limit is proved under relatively weak assumptions. This result implies, in particular, the convergence of the spectrum with respect to the Hausdorff distance.