Raffaele Carlone

On the spectral theory of Gesztesy–Šeba realizations of 1-D Dirac operators with point interactions on a discrete set

Abstract We investigate spectral properties of Gesztesy–Seba realizations D X , α and D X , β of the 1-D Dirac differential expression D with point interactions on a discrete set X = { x n } n = 1 ∞ ⊂ R . Here α : = { α n } n = 1 ∞ and β : = { β n } n = 1 ∞ ⊂ R . The Gesztesy–Seba realizations D X , α and D X , β are the relativistic counterparts of the corresponding Schrodinger operators H X , α and H X , β with δ- and δ ′ -interactions, respectively. We define the minimal operator D X as the direct sum of the minimal Dirac operators on the intervals ( x n − 1 , x n ) . Then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator D X ⁎ in the case d ⁎ ( X ) : = inf { | x i − x j | , i ≠ j } = 0 . It turns out that the boundary operators B X , α and B X , β parameterizing the realizations D X , α and D X , β are Jacobi matrices. These matrices substantially differ from the ones appearing in spectral theory of Schrodinger operators with point interactions. We show that certain spectral properties of the operators D X , α and D X , β correlate with the corresponding spectral properties of the Jacobi matrices B X , α and B X , β , respectively. Using this connection we investigate spectral properties (self-adjointness, discreteness, absolutely continuous and singular spectra) of Gesztesy–Seba realizations. Moreover, we investigate the non-relativistic limit as the velocity of light c → ∞ . Most of our results are new even in the case d ⁎ ( X ) > 0 .

Spin-dependent point potentials in one and three dimensions

We consider a system realized with one spinless quantum particle and an array of N spins 1/2 in dimensions 1 and 3. We characterize all the Hamiltonians obtained as point perturbations of assigned free dynamics in terms of some generalized boundary conditions. For every boundary condition, we give the explicit formula for the resolvent of the corresponding Hamiltonian. We discuss the problem of locality and give two examples of spin-dependent point potentials that could be of interest as multi-component solvable models.

A solvable model of a tracking chamber

We analyse a quantum system consisting of a particle travelling in a model-environment made up of a net of localised two-level subsystems. We examine the dynamics of the entanglement through which the environment acquires information about the direction of propagation of the particle. Our study reproduces, within a nonperturbative investigation, an old result obtained by N. F. Mott in the early days of quantum mechanics.

Resonances in Models of Spin Dependent Point Interactions

In dimension d = 1, 2, 3 we define a family of two-channel Hamiltonians obtained as point perturbations of the generator of the free decoupled dynamics. Within the family we choose two Hamiltonians, and , giving rise respectively to the unperturbed and to the perturbed evolution. The Hamiltonian does not couple the channels and has an eigenvalue embedded in the continuous spectrum. The Hamiltonian is a small perturbation, in resolvent sense, of and exhibits a small coupling between the channels. We take advantage of the complete solvability of our model to prove with simple arguments that the embedded eigenvalue of shifts into a resonance for . In dimension three we analyze details of the time behavior of the projection onto the region of the spectrum close to the resonance.

Two-dimensional time-dependent point interactions

The Chernoff √ n-Lemma is revised. This concerns two aspects: an improvement of the Chernoff estimate in the strong operator topol-ogy and an operator-norm estimate for quasi-sectorial contractions. Applications to the Lie-Trotter product formula approximation for semigroups is presented.We study Riesz means of eigenvalues of the Heisenberg Laplacian with Dirichlet boundary conditions on a cylinder in dimension three. We obtain an inequality with a sharp leading term and an additional lower order term.We revisit and connect several notions of algebraic multiplicities of zeros of analytic operator-valued functions and discuss the concept of the index of meromorphic operator-valued functions in complex, separable Hilbert spaces. Applications to abstract perturbation theory and associated Birman-Schwinger-type operators and to the operator-valued Weyl-Titchmarsh functions associated to closed extensions of dual pairs of closed operators are provided.

Dynamics of an electron confined to a “hybrid plane” and interacting with a magnetic field

We discuss spectral and resonance properties of a Hamiltonian describing motion of an electron moving on a “hybrid surface” consisting on a halfline attached by its endpoint to a plane under influence of a constant magnetic field which interacts with its spin through a Rashba-type term.

Perturbations of eigenvalues embedded at threshold: I. One- and three-dimensional solvable models

We examine perturbations of eigenvalues and resonances for a class of multi-channel quantum mechanical model Hamiltonians describing a particle interacting with a localized spin in dimension d = 1, 3. We consider unperturbed Hamiltonians showing eigenvalues and resonances at the threshold of the continuous spectrum and we analyze the effect of various types of perturbations on the spectral singularities. We provide algorithms to obtain convergent series expansions for the coordinates of the singularities.

Well-posedness of the two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity

Abstract We consider a two-dimensional nonlinear Schrodinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well as energy and mass conservation. In addition, we prove that this implies global existence in the defocusing case, irrespective of the power of the nonlinearity, while in the focusing case blowing-up solutions may arise.

THE ONE-DIMENSIONAL DIRAC EQUATION WITH CONCENTRATED NONLINEARITY

We define and study the Cauchy problem for a one-dimensional (1-D) nonlinear Dirac equation with nonlinearities concentrated at one point. Global well-posedness is provided and conservation laws for mass and energy are shown. Several examples, including nonlinear Gesztesy--Seba models and the concentrated versions of the Bragg resonance and 1-D Soler (also known as massive Gross--Neveu) type models, all within the scope of the present paper, are given. The key point of the proof consists in the reduction of the original equation to a nonlinear integral equation for an auxiliary, space-independent variable.

The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions

Abstract For kernels ν which are positive and integrable we show that the operator g ↦ J ν g = ∫ 0 x ν ( x − s ) g ( s ) d s on a finite time interval enjoys a regularizing effect when applied to Holder continuous and Lebesgue functions and a “contractive” effect when applied to Sobolev functions. For Holder continuous functions, we establish that the improvement of the regularity of the modulus of continuity is given by the integral of the kernel, namely by the factor N ( x ) = ∫ 0 x ν ( s ) d s . For functions in Lebesgue spaces, we prove that an improvement always exists, and it can be expressed in terms of Orlicz integrability. Finally, for functions in Sobolev spaces, we show that the operator J ν “shrinks” the norm of the argument by a factor that, as in the Holder case, depends on the function N (whereas no regularization result can be obtained). These results can be applied, for instance, to Abel kernels and to the Volterra function I ( x ) = μ ( x , 0 , − 1 ) = ∫ 0 ∞ x s − 1 / Γ ( s ) d s , the latter being relevant for instance in the analysis of the Schrodinger equation with concentrated nonlinearities in R 2 .