Mourre Theory Research Articles

Mourre theory and spectral analysis of energy-momentum operators in relativistic quantum field theory

A central task of theoretical physics is to analyse spectral properties of quantum mechanical observables. In this endeavour, Mourre’s conjugate operator method emerged as an effective tool in the spectral theory of Schrödinger operators. This paper introduces a novel class of examples from relativistic quantum field theory that are amenable to Mourre’s method. By assuming Lorentz covariance and the spectrum condition, we derive a limiting absorption principle for the energy-momentum operators and provide new proofs of the absolute continuity of the energy-momentum spectra. Moreover, under the assumption of dilation covariance, we show that the spectrum of the relativistic mass operator is purely absolutely continuous in (0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(0,\\infty )$$\\end{document}.

Scattering theory for Dirac fields near an extreme Kerr–de Sitter black hole

In this paper, we construct a scattering theory for classical massive Dirac fields near the horizon of an extreme Kerr-de Sitter blackhole. Our main tool is the existence of a conjugate operator in the sense of Mourre theory. Additionally, despite the fact that effects of the rotation are 'amplified' near the double horizon, we show that one can still reduce our study to a 1-dimensional problem through an appropriate decomposition of the Hilbert space.

Dynamics of Open Quantum Systems I, Oscillation and Decay

We develop a framework to analyze the dynamics of a finite-dimensional quantum systemSin contact with a reservoirR. The full, interactingSRdynamics is unitary. The reservoir has a stationary state but otherwise dissipative dynamics. We identify a main part of the full dynamics, which approximates it for small values of theSRcoupling constant, uniformly for all timest≥0. The main part consists of explicit oscillating and decaying parts. We show that the reduced system evolution is Markovian for all times. The technical novelty is a detailed analysis of the link between the dynamics and the spectral properties of the generator of theSRdynamics, based on Mourre theory. We allow forSRinteractions with little regularity, meaning that the decay of the reservoir correlation function only needs to be polynomial in time, improving on the previously required exponential decay.In this work we distill the structural and technical ingredients causing the characteristic features of oscillation and decay of theSRdynamics. In the companion paper [27] we apply the formalism to the concrete case of anN-level system linearly coupled to a spatially infinitely extended thermal bath of non-interacting Bosons.

Limiting Absorption Principle and Equivalence of Feynman Propagators on Asymptotically Minkowski Spacetimes

In this paper, we shall show that the limiting absorption principle for the wave operator on the asymptotically Minkowski spacetime. This problem was previously considered by [A. Vasy, J. Spect. Theory, 10,439-461 , (2020)]. Here, we employ a more transparent tool, the Mourre theory and removes an additional condition which is imposed in his paper. Moreover, we also prove that the anti-Feynman propagator defined by G\'erard and Wrochna coincides with the outgoing resolvent.

Limiting Absorption Principle for Discrete Schrödinger Operators with a Wigner–von Neumann Potential and a Slowly Decaying Potential

We consider discrete Schr{\"o}dinger operators on ${\mathbb{Z}}^d$ for which the perturbation consists of the sum of a long-range type potential and a Wigner-von Neumann type potential. Still working in a framework of weighted Mourre theory, we improve the limiting absorption principle (LAP) that was obtained in [Ma1]. To our knowledge, this is a new result even in the one-dimensional case. The improvement consists in a weakening of the assumptions on the long-range potential and better LAP weights. The improvement relies only on the fact that the generator of dilations (which serves as conjugate operator) is bounded from above by the position operator. To exploit this, Loewner's theorem on operator monotone functions is invoked.

One-boson scattering processes in the massive Spin-Boson model

The Spin-Boson model describes a two-level quantum system that interacts with a second-quantized boson scalar field. Recently the relation between the integral kernel of the scattering matrix and the resonance in this model has been established in [18] for the case of massless bosons. In the present work, we treat the massive case. On the one hand, one might rightfully expect that the massive case is easier to handle since, in contrast to the massless case, the corresponding Hamiltonian features a spectral gap. On the other hand, it turns out that the non-zero boson mass introduces a new complication as the spectrum of the complex dilated, free Hamiltonian exhibits lines of spectrum attached to every multiple of the boson rest mass energy starting from the ground and excited state energies. This leads to an absence of decay of the corresponding complex dilated resolvent close to the real line, which, in [18], was a crucial ingredient to control the time evolution in the scattering regime. With the new strategy presented here, we provide a proof of an analogous formula for the scattering kernel as compared to the massless case and use the opportunity to provide the required spectral information by a Mourre theory argument combined with a suitable application of the Feshbach-Schur map instead of complex dilation.

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 $K$.

Mourre theory for time-periodic magnetic fields

We study the Mourre theory for the Floquet Hamiltonian Hˆ=−i∂t+H(t) generated by the time-periodic Hamiltonian H(t) with H(t+T)=H(t) which describes the Schrödinger equations with general time-periodic magnetic fields.

Propagation Estimates for One Commutator Regularity

In the abstract framework of Mourre theory, the propagation of states is understood in terms of a conjugate operator A. A powerful estimate has long been known for Hamiltonians having a good regularity with respect to A thanks to the limiting absorption principle (LAP). We study the case where H has less regularity with respect to A, specifically in a situation where the LAP and the absence of singularly continuous spectrum have not yet been established. We show that in this case the spectral measure of H is a Rajchman measure and we derive some propagation estimates. One estimate is an application of minimal escape velocities, while the other estimate relies on an improved version of the RAGE formula. Based on several examples, including continuous and discrete Schrödinger operators, it appears that the latter propagation estimate is a new result for multi-dimensional Hamiltonians.

Uniform resolvent and Strichartz estimates for Schrödinger equations with critical singularities

This paper deals with global dispersive properties of Schrödinger equations with real-valued potentials exhibiting critical singularities, where our class of potentials is more general than inverse-square type potentials and includes several anisotropic potentials. We first prove weighted resolvent estimates, which are uniform with respect to the energy, with a large class of weight functions in Morrey–Campanato spaces. Uniform Sobolev inequalities in Lorentz spaces are also studied. The proof employs the iterated resolvent identity and a classical multiplier technique. As an application, the full set of global-in-time Strichartz estimates including the endpoint case, is derived. In the proof of Strichartz estimates, we develop a general criterion on perturbations ensuring that both homogeneous and inhomogeneous endpoint estimates can be recovered from resolvent estimates. Finally, we also investigate uniform resolvent estimates for long range repulsive potentials with critical singularities by using an elementary version of the Mourre theory.

Spectral and scattering theory for Schrödinger operators on perturbed topological crystals

In this paper, we investigate the spectral and the scattering theory of Schrödinger operators acting on perturbed periodic discrete graphs. The perturbations considered are of two types: either a multiplication operator by a short-range or a long-range function, or a short-range type modification of the measure defined on the vertices and on the edges of the graph. Mourre theory is used for describing the nature of the spectrum of the underlying operators. For short-range perturbations, existence and asymptotic completeness of local wave operators are also proved.

The limiting absorption principle for the discrete Wigner–von Neumann operator

We apply weighted Mourre commutator theory to prove the limiting absorption principle for the discrete Schrödinger operator perturbed by the sum of a Wigner–von Neumann and long-range type potential. In particular, this implies a new result concerning the absolutely continuous spectrum for these operators even for the one-dimensional operator. We show that methods of classical Mourre theory based on differential inequalities and on the generator of dilation cannot apply to the aforementioned Schrödinger operators.

LOCAL DECAY FOR THE DAMPED WAVE EQUATION IN THE ENERGY SPACE

We improve a previous result about the local energy decay for the damped wave equation on $\mathbb{R}^{d}$. The problem is governed by a Laplacian associated with a long-range perturbation of the flat metric and a short-range absorption index. Our purpose is to recover the decay ${\mathcal{O}}(t^{-d+\unicode[STIX]{x1D700}})$ in the weighted energy spaces. The proof is based on uniform resolvent estimates, given by an improved version of the dissipative Mourre theory. In particular, we have to prove the limiting absorption principle for the powers of the resolvent with inserted weights.

Spectral stability of unitary network models

We review various unitary network models used in quantum computing, spectral analysis or condensed matter physics and establish relationships between them. We show that symmetric one-dimensional quantum walks are universal, as are CMV matrices. We prove spectral stability and propagation properties for general asymptotically uniform models by means of unitary Mourre theory.

A Remark on the Mourre theory for two body Schrödinger operators

On this short note, we apply the Mourre theory of the limiting absorption with difference type conditions on the potential, instead of conditions on the derivatives. In order that we modify the definition of the conjugate operator, and we apply the standard abstract Mourre theory. We also discuss examples to which the method applies.

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.

SPECTRAL ANALYSIS AND TIME-DEPENDENT SCATTERING THEORY ON MANIFOLDS WITH ASYMPTOTICALLY CYLINDRICAL ENDS

We review the spectral analysis and the time-dependent approach of scattering theory for manifolds with asymptotically cylindrical ends. For the spectral analysis, higher order resolvent estimates are obtained via Mourre theory for both short-range and long-range behaviors of the metric and the perturbation at infinity. For the scattering theory, the existence and asymptotic completeness of the wave operators is proved in a two-Hilbert spaces setting. A stationary formula as well as mapping properties for the scattering operator are derived. The existence of time delay and its equality with the Eisenbud–Wigner time delay is finally presented. Our analysis mainly differs from the existing literature on the choice of a simpler comparison dynamics as well as on the complementary use of time-dependent and stationary scattering theories.

Local energy decay of massive Dirac fields in the 5D Myers–Perry metric

We consider massive Dirac fields evolving in the exterior region of a five-dimensional Myers–Perry black hole and study their propagation properties. Our main result states that the local energy of such fields decays in a weak sense at late times. We obtain this result in two steps: first, using the separability of the Dirac equation, we prove the absence of a pure point spectrum for the corresponding Dirac operator; second, using a new form of the equation adapted to the local rotations of the black hole, we show by a Mourre theory argument that the spectrum is absolutely continuous. This leads directly to our main result.

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.

Spectral Theory for a Mathematical Model of the Weak Interaction: The Decay of the Intermediate Vector Bosons W ±, II

We do the spectral analysis of the Hamiltonian for the weak leptonic decay of the gauge bosons W+/-. Using Mourre theory, it is shown that the spectrum between the unique ground state and the first threshold is purely absolutely continuous. Neither sharp neutrino high energy cutoff nor infrared regularization are assumed.