Abstract. The paper is devoted to operators given formally by the expressionThis expression is homogeneous of degree minus 2. However, when we try to realize it as a self-adjoint operator for real α, or closed operator for complex α, we find that this homogeneity can be broken. This leads to a definition of two holomorphic families of closed operators on L 2 (R+), which we denote Hm,κ and H ν 0 , with m 2 = α, −1 < Re(m) < 1, and where κ, ν ∈ C ∪{∞} specify the boundary condition at 0. We study these operators using their explicit solvability in terms of Bessel-type functions and the Gamma function. In particular, we show that their point spectrum has a curious shape: a string of eigenvalues on a piece of a spiral. Their continuous spectrum is always [0, ∞[. Restricted to their continuous spectrum, we diagonalize these operators using a generalization of the Hankel transformation. We also study their scattering theory. These operators are usually non-self-adjoint. Nevertheless, it is possible to use concepts typical for the self-adjoint case to study them. Let us also stress that −1 < Re(m) < 1 is the maximal region of parameters for which the operators Hm,κ can be defined within the framework of the Hilbert space L 2 (R+).

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 review the spectral and the scattering theory for the Aharonov-Bohm model on R 2 . New formulae for the wave operators and for the scattering operator are presented. The asymptotics at high and at low energy of the scattering operator are computed.

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.

In this note Levinson theorems for Schrödinger operators in R n with one point interaction at 0 are derived using the concept of winding numbers. These results are based on new expressions for the associated wave operators.