Mourre Estimate Research Articles

Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon

In this paper we consider linear, time dependent Schrödinger equations of the form i∂tψ=K0ψ+V(t)ψ, where K0 is a strictly positive selfadjoint operator with discrete spectrum and constant spectral gaps, and V(t) a smooth in time periodic potential. We give sufficient conditions on V(t) ensuring that K0+V(t) generates unbounded orbits. The main condition is that the resonant average of V(t), namely the average with respect to the flow of K0, has a nonempty absolutely continuous spectrum and fulfills a Mourre estimate. These conditions are stable under perturbations. The proof combines pseudodifferential normal form with dispersive estimates in the form of local energy decay.We apply our abstract construction to the Harmonic oscillator on R and to the half-wave equation on T; in each case, we provide large classes of potentials which are transporters.

Spectral asymptotics for the Schrödinger operator with a non-decaying potential

We study Schrödinger operators H ( h ) = − h 2 Δ + V ( x ) acting in L 2 ( R n ) for non-decaying potentials V. We give a full asymptotic expansion of the spectral shift function for a pair of such operators in the high energy limit. In particular for asymptotically homogeneous potentials W at infinity of degree zero, we also study the semiclassical asymptotics to give a Weyl formula of the spectral shift function above the threshold max W and Mourre estimates in the range of W except at its critical values.

Absence of embedded eigenvalues for non-local Schrödinger operators

We consider non-local Schr\"odinger operators with kinetic terms given by several different types of functions of the Laplacian and potentials decaying to zero at infinity, and derive conditions ruling embedded eigenvalues out. Our goal in this paper is to advance techniques based on virial theorems, Mourre estimates, and an extended version of the Birman-Schwinger principle, previously developed for classical Schr\"odinger operators but thus far not used for non-local operators. We also present a number of specific cases by choosing particular classes of kinetic and potential terms, and discuss existence/non-existence of at-edge eigenvalues in a basic model case in function of the coupling parameter.

Spectral and Scattering Properties of Quantum Walks on Homogenous Trees of Odd Degree

For unitary operators $U_0,U$ in Hilbert spaces ${\mathcal H}_0,{\mathcal H}$ and identification operator $J:{\mathcal H}_0\to{\mathcal H}$, we present results on the derivation of a Mourre estimate for $U$ starting from a Mourre estimate for $U_0$ and on the existence and completeness of the wave operators for the triple $(U,U_0,J)$. As an application, we determine spectral and scattering properties of a class of anisotropic quantum walks on homogenous trees of odd degree with evolution operator $U$. In particular, we establish a Mourre estimate for $U$, obtain a class of locally $U$-smooth operators, and prove that the spectrum of $U$ covers the whole unit circle and is purely absolutely continuous, outside possibly a finite set where $U$ may have eigenvalues of finite multiplicity. We also show that (at least) three different choices of free evolution operators $U_0$ are possible for the proof of the existence and completeness of the wave operators.

On the Mourre estimates for Floquet Hamiltonians

In this paper, we consider the Floquet Hamiltonian $K$ associated with a three-body Schr\"odinger operator with time-periodic pair potentials $H(t)$. By introducing a conjugate operator $A$ for $K$ in the standard Mourre theory, we prove the Mourre estimate for $K$.

Sub-exponential decay of eigenfunctions for some discrete Schrödinger operators

Following the method of Froese and Herbst, we show for a class of potentials V that an embedded eigenfunction \psi with eigenvalue E of the multi-dimensional discrete Schrödinger operator H = \Delta + V on \mathbb Z^d decays sub-exponentially whenever the Mourre estimate holds at E . In the one-dimensional case we further show that this eigenfunction decays exponentially with a rate at least of cosh ^{-1}((E-2)/(\theta_E-2)) , where \theta_E is the nearest threshold of H located between E and 2 . A consequence of the latter result is the absence of eigenvalues between 2 and the nearest thresholds above and below this value. The method of Combes–Thomas is also reviewed for the discrete Schrödinger operators.

Scattering theory for the Dirac equation in Schwarzschild–Anti–de Sitter space-time

We show asymptotic completeness for linear massive Dirac fields on the Schwarzschild-Anti-de Sitter spacetime. The proof is based on a Mourre estimate. We also construct an asymptotic velocity for this field.

Scattering theory for multistate Schrödinger operators

We study multistate Schrödinger operators related to molecular dynamics. We consider potentials which do not necessarily decay and prove the absence of the singular continuous spectrum and propagation estimates which mean the scattering at speed larger than a positive constant and decay of the state with potentials higher than the considered energy at infinity. We also consider the multistate Schrödinger operators with many-body structures. We obtain the Mourre estimate and the minimal velocity estimate for the many-body operators. The lower bound of the velocity is determined by the distance between the energy and thresholds below the energy.

Local Spectral Deformation

We develop an analytic perturbation theory for eigenvalues with finite multiplicities, embedded into the essential spectrum of a self-adjoint operator $H$. We assume the existence of another self-adjoint operator $A$ for which the family $H_\theta = e^{\mathrm{i}\theta A} H e^{-\mathrm{i}\theta A}$ extends analytically from the real line to a strip in the complex plane. Assuming a Mourre estimate holds for $\mathrm{i}[H,A]$ in the vicinity of the eigenvalue, we prove that the essential spectrum is locally deformed away from the eigenvalue, leaving it isolated and thus permitting an application of Kato's analytic perturbation theory.

Rellich’s theorem and N-body Schrödinger operators

We show an optimal version of Rellich’s theorem for generalized [Formula: see text]-body Schrödinger operators. It applies to singular potentials, in particular, to a model for atoms and molecules with infinite mass and finite extent nuclei. Our proof relies on a Mourre estimate [10] and a functional calculus localization technique.

The Translation Invariant Massive Nelson Model: III. Asymptotic Completeness Below the Two-Boson Threshold

We show asymptotic completeness of two-body scattering for a class of translation invariant models describing a single quantum particle (the electron) linearly coupled to a massive scalar field (bosons). Our proof is based on a recently established Mourre estimate for these models. In contrast to previous approaches, it requires no number cutoff, no restriction on the particle-field coupling strength, and no restriction on the magnitude of total momentum. Energy, however, is restricted by the two-boson threshold, admitting only scattering of a dressed electron and a single asymptotic boson. The class of models we consider include the UV-cutoff Nelson and polaron models.

Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators

We study two-dimensional magnetic Schrodinger operators with a magnetic field that is equal to b>0 for x > 0 and (-b) for x < 0. This magnetic Schrodinger operator exhibits a magnetic barrier at x=0. The unperturbed system is invariant with respect to translations in the y-direction. As a result, the Schrodinger operator admits a direct integral decomposition. We analyze the band functions of the fiber operators as functions of the wave number and establish their asymptotic behavior. Because the fiber operators are reflection symmetric, the band functions may be classified as odd or even. The odd band functions have a unique absolute minimum. We calculate the effective mass at the minimum and prove that it is positive. The even band functions are monotone decreasing. We prove that the eigenvalues of an Airy operator, respectively, harmonic oscillator operator, describe the asymptotic behavior of the band functions for large negative, respectively positive, wave numbers. We prove a Mourre estimate for perturbations of the magnetic Schrodinger operator and establish the existence of absolutely continuous spectrum in certain energy intervals. We prove lower bounds on magnetic edge currents for states with energies in the same intervals. We also prove that these lower bounds imply stable lower bounds for the asymptotic currents. We study the perturbation by slowly decaying negative potentials and establish the asymptotic behavior of the eigenvalue counting function for the infinitely-many eigenvalues below the bottom of the essential spectrum.

Scattering in twisted waveguides

We consider a twisted quantum waveguide, i.e. a domain of the form Ωθ:=rθω×R where ω⊂R2 is a bounded domain, and rθ=rθ(x3) is a rotation by the angle θ(x3) depending on the longitudinal variable x3. We investigate the nature of the essential spectrum of the Dirichlet Laplacian Hθ, self-adjoint in L2(Ωθ), and consider related scattering problems. First, we show that if the derivative of the difference θ1−θ2 decays fast enough as |x3|→∞, then the wave operators for the operator pair (Hθ1,Hθ2) exist and are complete. Further, we concentrate on appropriate perturbations of constant twisting, i.e. θ′=β−ε with constant β∈R, and ε which decays fast enough at infinity together with its first derivative. In that case the unperturbed operator corresponding to ε is an analytically fibered Hamiltonian with purely absolutely continuous spectrum. Obtaining Mourre estimates with a suitable conjugate operator, we prove, in particular, that the singular continuous spectrum of Hθ is empty.

A few results on Mourre theory in a two-Hilbert spaces setting

We introduce a natural framework for dealing with Mourre theory in an abstract two-Hilbert spaces setting. In particular a Mourre estimate for a pair of self-adjoint operators \((H,A)\) is deduced from a similar estimate for a pair of self-adjoint operators \((H_0,A_0)\) acting in an auxiliary Hilbert space. A new criterion for the completeness of the wave operators in a two-Hilbert spaces setting is also presented.

Dispersion estimates for 2D Dirac equation

We obtain a dispersion long-time decay in weighted norms for solutions of the 2D Dirac equation with generic potentials. For high energy component we develop the approach of Boussaid relying on the Mourre estimates and the minimal escape velocity estimates of Hunziker, Sigal and Soffer. For the low energy component we obtain the asymptotics of the resolvent near the thresholds in the nonsingular case and apply 2D version of Jensen-Kato lemma on one-and-half partial integration. The results can be applied to the study of asymptotic stability and scattering of solitons for nonlinear 2D Dirac equations.

Mourre estimates for compatible Laplacians on complete manifolds with corners of codimension 2

We apply Mourre theory to compatible Laplacians on complete manifolds with corners of codimension 2 in order to prove absence of singular continuous spectrum, that non-threshold eigenvalues have finite multiplicity and can accumulate only at thresholds or infinity. It turns out that we need Mourre estimates on manifolds with cylindrical ends where the results are both expected and consequences of more general theorems. In any case, we also provide a description, interesting in its own, of Mourre theory in this context that makes our text complete and suggests generalizations to manifolds with corners of higher order codimension. We use theorems of functional analysis that are suitable for these geometric applications.

Asymptotic Completeness in Quantum Field Theory: Translation Invariant Nelson Type Models Restricted to the Vacuum and One-Particle Sectors

Time-dependent scattering theory for a large class of translation invariant models, including the Nelson and Polaron models, restricted to the vacuum and one-particle sectors is studied. We formulate and prove asymptotic completeness for these models. The translation invariance imply that the Hamiltonians considered are fibered with respect to the total momentum. On the way to asymptotic completeness we determine the spectral structure of the fiber Hamiltonians, establish a Mourre estimate and derive a geometric asymptotic completeness statement as an intermediate step.

Regularity of eigenstates in regular Mourre theory

The present paper gives an abstract method to prove that possibly embedded eigenstates of a self-adjoint operator H lie in the domain of the kth power of a conjugate operator A. Conjugate means here that H and A have a positive commutator locally near the relevant eigenvalue in the sense of Mourre. The only requirement is C k + 1 ( A ) regularity of H. Regarding integer k, our result is optimal. Under a natural boundedness assumption of the multiple commutators we prove that the eigenstate ‘dilated’ by exp ( i θ A ) is analytic in a strip around the real axis. In particular, the eigenstate is an analytic vector with respect to A. Natural applications are ‘dilation analytic’ systems satisfying a Mourre estimate, where our result can be viewed as an abstract version of a theorem due to Balslev and Combes (1971) [3]. As a new application we consider the massive Spin-Boson Model.

On the spectral analysis of many-body systems

We describe the essential spectrum and prove the Mourre estimate for quantum particle systems interacting through k-body forces and creation–annihilation processes which do not preserve the number of particles. For this we compute the “Hamiltonian algebra” of the system, i.e. the C ∗ -algebra C generated by the Hamiltonians we want to study, and show that, as in the N-body case, it is graded by a semilattice. Hilbert C ∗ -modules graded by semilattices are involved in the construction of C . For example, if we start with an N-body system whose Hamiltonian algebra is C N and then we add field type couplings between subsystems, then the many-body Hamiltonian algebra C is the imprimitivity algebra of a graded Hilbert C N -module.

Limiting absorption principle at low energies for a mathematical model of weak interaction: The decay of a boson

We study the spectral properties of a Hamiltonian describing the weak decay of spin 1 massive bosons into the full family of leptons. We prove that the considered Hamiltonian is self-adjoint, with a unique ground state and we derive a Mourre estimate and a limiting absorption principle above the ground state energy and below the first threshold, for a sufficiently small coupling constant. As a corollary, we prove absence of eigenvalues and absolute continuity of the energy spectrum in the same spectral interval. To cite this article: J.-M. Barbaroux, J.-C. Guillot, C. R. Acad. Sci. Paris, Ser. I 347 (2009).