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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., Comment: Revised version (with new title and more general results) to appear in Annales Henri Poincar\'e

We determine the low-energy behaviour of the scattering operator of two-dimensional Schrodinger operators with any type of obstructions at 0-energy. We also derive explicit formulas for the wave operators in the absence of p-resonances, and outline in this case a topological version of Levinson’s theorem.

We perform the scattering analysis of the evolution operator of quantum walks with an anisotropic coin, and we prove a weak limit theorem for their asymptotic velocity. The quantum walks that we consider include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. Our analysis is based on an abstract framework for the scattering theory of unitary operators in a two-Hilbert spaces setting, which is of independent interest., 23 pages

International audience; For the scattering system given by the Laplacian in a half-space with a periodic boundary condition,we derive resolvent expansions at embedded thresholds, we prove the continuity of the scatteringmatrix, and we establish new formulas for the wave operators.

We present a detailed study of the scattering system given by the Neumann Laplacian on the discrete half-space perturbed by a periodic potential at the boundary. We derive asymptotic resolvent expansions at thresholds and eigenvalues, we prove the continuity of the scattering matrix, and we establish new formulas for the wave operators. Along the way, our analysis puts into evidence a surprising relation between some properties of the potential, like the parity of its period, and the behaviour of the integral kernel of the wave operators.

We study the spectral and scattering theory of light transmission in a system consisting of two asymptotically periodic waveguides, also known as one-dimensional photonic crystals, coupled by a junction. Using analyticity techniques and commutator methods in a two-Hilbert spaces setting, we determine the nature of the spectrum and prove the existence and completeness of the wave operators of the system., Comment: 31 pages, 1 figure. Keywords: Spectral theory, scattering theory, Maxwell operators, commutator methods. [v3] revised version to appear in Reviews in Mathematical Physics

We present a general account on the stationary scattering theory for unitary operators in a two-Hilbert spaces setting. For unitary operators U 0 , U in Hilbert spaces H 0 , H and an identification operator J : H 0 → H , we give the definitions and collect properties of the stationary wave operators, the strong wave operators, the scattering operator and the scattering matrix for the triple ( U , U 0 , J ) . In particular, we exhibit conditions under which the stationary wave operators and the strong wave operators exist and coincide, and we derive representation formulas for the stationary wave operators and the scattering matrix. As an application, we show that these representation formulas are satisfied for a class of anisotropic quantum walks recently introduced in the literature.

We present a suitable framework for the definition of quantum time delay in terms of sojourn times for unitary operators in a two-Hilbert spaces setting. We prove that this time delay defined in terms of sojourn times (time-dependent definition) exists and coincides with the expectation value of a unitary analogue of the Eisenbud-Wigner time delay operator (time-independent definition). Our proofs rely on a new summation formula relating localisation operators to time operators and on various tools from functional analysis such as Mackey's imprimititvity theorem, Trotter-Kato Formula and commutator methods for unitary operators. Our approach is general and model-independent., 38 pages, to appear in Rev. Math. Phys

Let X and G be compact Lie groups, F 1: X → X the time-one map of a C ∞ measure-preserving flow, ϕ: X → G a continuous function and π a finite-dimensional irreducible unitary representation of G. Then, we prove that the skew products $${T_\phi }:X \times G \to X \times G,(x,g) \mapsto ({F_1}(x),g\phi (x))$$ , have purely absolutely continuous spectrum in the subspace associated to π if π po ϕ has a Dini-continuous Lie derivative along the flow and if a matrix multiplication operator related to the topological degree of πpoϕ has nonzero determinant. This result provides a simple, but general, criterion for the presence of an absolutely continuous component in the spectrum of skew products of compact Lie groups. As an illustration, we consider the cases where F 1 is an ergodic translation on T d and X × G = T d × T dʹ , X × G = T d × SU(2) and X × G = T d × U(2). Our proofs rely on recent results on positive commutator methods for unitary operators.

We perform the spectral analysis of the evolution operator U of quantum walks with an anisotropic coin, which include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. In particular, we determine the essential spectrum of U, we show the existence of locally U-smooth operators, we prove the discreteness of the eigenvalues of U outside the thresholds, and we prove the absence of singular continuous spectrum for U. Our analysis is based on new commutator methods for unitary operators in a two-Hilbert spaces setting, which are of independent interest., 26 pages

We prove new and explicit formulas for the wave operators of Schrodinger operators in $${\mathbb{R}^3}$$ . These formulas put into light the very special role played by the generator of dilations and validate the topological approach of Levinson’s theorem introduced in a previous publication. Our results hold for general (not spherically symmetric) potentials decaying fast enough at infinity, without any assumption on the absence of eigenvalue or resonance at 0-energy.

27; International audience; We present an inversion formula which can be used to obtain resolvent expansions near embedded thresholds. As an application, we prove for a class of quantum waveguides the absence of accumulation of eigenvalues and the continuity of the scattering matrix at all thresholds.

14 pages; Journal of Functional Analysis; International audience; We consider operators $H_\mu$ of convolution with measures $\mu$ on locally compact groups. We characterize the spectrum of $H_\mu$ by constructing auxiliary operators whose kernel contain the pure point and singular subspaces of $H_\mu$, respectively. The proofs rely on commutator methods.

We present new criteria, based on commutator methods, for the strong mixing property of discrete flows $\{U^N\}_{N\in\mathbb Z}$ and continuous flows $\{{\rm e}^{-itH}\}_{t\in\mathbb R}$ induced by unitary operators $U$ and self-adjoint operators $H$ in a Hilbert space $\mathcal H$. Our approach put into evidence a general definition for the topological degree of the curves $N\mapsto U^N$ and $t\mapsto{\rm e}^{-itH}$ in the unitary group of $\mathcal H$. Among other examples, our results apply to skew products of compact Lie groups, time changes of horocycle flows and adjacency operators on graphs., 15 pages, 1 figure

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.

We consider in a Hilbert space a self-adjoint operator H and a family Φ ≡ (Φ1,…,Φd) of mutually commuting self-adjoint operators.Unde r some regularity properties of H with respect to Φ, we propose two new formulae for a time operator for H and prove their equality.O ne of the expressions is based on the time evolution of an abstract localisation operator defined in terms of Φ while the other one corresponds to a stationary formula.Under the same assumptions, we also conduct the spectral analysis of H by using the method of the conjugate operator. Among other examples, our theory applies to Friedrichs Hamiltonians, Stark Hamiltonians, some Jacobi operators, the Dirac operator, convolution operators on locally compact groups, pseudodifferential operators, adjacency operators on graphs and direct integral operators.

We present a method, based on commutator methods, for the spectral analysis of uniquely ergodic dynamical systems. When applicable, it leads to the absolute continuity of the spectrum of the corresponding unitary operators. As an illustration, we consider time changes of horocycle flows, skew products over translations and Furstenberg transformations. For time changes of horocycle flows, we obtain absolute continuity under assumptions weaker than the ones to be found in the literature., Comment: corrections and simplifications in section 4

We consider on a manifold M equipped with a Poisson bracket { ?, ?} a Hamiltonian H with complete flow and a family ? ? (?1, ?, ?d) of abstract position observables satisfying the condition {{?j, H}, H} = 0 for each j. Under these assumptions, we prove a new formula relating sojourn times in dilated regions defined in terms of ? to the time of arrival of classical orbits. The correspondence between this formula and a formula established recently in the framework of quantum mechanics is put into evidence. Among other examples, our theory applies to Stark Hamiltonians, homogeneous Hamiltonians, purely kinetic Hamiltonians, the repulsive harmonic potential, central force systems, the Poincar? ball model, the wave equation, the nonlinear Schr?dinger equation, the Korteweg?de Vries equation and quantum Hamiltonians defined via expectation values.

International audience; We prove that the existence of time delay defined in terms of sojourn times, as well as its identity with Eisenbud-Wigner time delay, is a common feature of two-Hilbert spaces quantum scattering theory. All statements are model-independent.

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., 30 pages

We put into evidence graphs with adjacency operator whose singular subspace is prescribed by the kernel of an auxiliary operator. In particular, for a family of graphs called admissible, the singular continuous spectrum is absent and there is at most an eigenvalue located at the origin. Among other examples, the one-dimensional XY model of solid-state physics is covered. The proofs rely on commutators methods., 16 pages, 9 figures

We advocate for the systematic use of a symmetrized definition of time delay in scattering theory. In two-body scattering processes, we show that the symmetrized time delay exists for arbitrary dilated spatial regions symmetric with respect to the origin. It is equal to the usual time delay plus a new contribution, which vanishes in the case of spherical spatial regions. We also prove that the symmetrized time delay is invariant under an appropriate mapping of time reversal. These results are also discussed in the context of classical scattering theory., Comment: 18 pages

We define, prove the existence and obtain explicit expressions for classical time delay defined in terms of sojourn times for abstract scattering pairs (H_0,H) on a symplectic manifold. As a by-product, we establish a classical version of the Eisenbud-Wigner formula of quantum mechanics. Using recent results of V. Buslaev and A. Pushnitski on the scattering matrix in Hamiltonian mechanics, we also obtain an explicit expression for the derivative of the Calabi invariant of the Poincar\'e scattering map. Our results are applied to dispersive Hamiltonians, to a classical particle in a tube and to Hamiltonians on the Poincar\'e ball., Comment: 22 pages