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.Comment: 26 page

We consider in a Hilbert space a self-adjoint operator H and a family Φ ≡ (Φ1, . . . , Φ d ) of mutually commuting self-adjoint operators. Under some regularity properties of H with respect to Φ, we propose two new formulae for a time operator for H and prove their equality. One 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.

International audienceWe 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 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.

Although many physical arguments account for using a modified definition of time delay in multichannel-type scattering processes, one can hardly find rigorous results on that issue in the literature. We try to fill in this gap by showing, both in an abstract setting and in a short-range case, the identity of the modified time delay and the Eisenbud-Wigner time delay in waveguides. In the short-range case we also obtain limiting absorption principles, state spectral properties of the total Hamiltonian, prove the existence of the wave operators and show an explicit formula for the S-matrix. The proofs rely on stationary and commutator methods.