1,721,003 research outputs found
Kreĭn formula and convergence of hamiltonians with scaled potentials in dimension one
No full textIn this brief report we study the convergence of the Hamiltonian hε := −(⋅)′′ + V (x∕ε)∕ε2 in dimension one as ε goes to zero. This problem has already been studied in several former works (also in the more general setting of metric graphs) and the results that we present here are not new. Aim of this work is to formulate the problem in the setting of metric graphs and to exploit an approach based on a Kreı̆n formula for the resolvent of hε. Such a formula allows to mark out the rôle of the zero eigenvalue for an auxiliary Hamiltonian. The existence of the zero eigenvalue is responsible of the coupling in the limiting Hamiltonian, otherwise hε converges in norm resolvent sense to the direct sum of two Dirichlet Laplacians on the half-line. In a forthcoming paper such approach will be generalized to the study of an analogous problem on metric graphs with a small compact core
Well posedness of the nonlinear Schrödinger equation with isolated singularities
No full textWe study the well posedness of the nonlinear Schrödinger (NLS) equation with a point interaction and power nonlinearity in dimension two and three. Behind the autonomous interest of the problem, this is a model of the evolution of so called singular solutions that are well known in the analysis of semilinear elliptic equations. We show that the Cauchy problem for the NLS considered enjoys local existence and uniqueness of strong (operator domain) solutions, and that the solutions depend continuously from initial data. In dimension two well posedness holds for any power nonlinearity and global existence is proved for powers below the cubic. In dimension three local and global well posedness are restricted to low powers
On inverses of Kreın’s Q-functions
Full text linkLet AQ be the self-adjoint operator defined by the Q-function Q: z 7→ Qz through the Kreın-like resolvent formula (−AQ + z)−1 = (−A0 + z)−1 + GzWQ−z1V G∗z ̄ , z ∈ ZQ , where V and W are bounded operators and ZQ := {z ∈ ρ(A0): Qz and Qz ̄ have a bounded inverse}. We show that ZQ 6= ∅ = ZQ = ρ(A0) ∩ ρ(AQ) . We do not suppose that Q is represented in terms of a uniformly strict, operator-valued Nevanlinna function (equivalently, we do not assume that Q is associated to an ordinary boundary triplet), thus our result extends previously known ones. The proof relies on simple algebraic computations stemming from the first resolvent identity
The semiclassical limit on a star-graph with Kirchhoff conditions
No full textWe consider the dynamics of a quantum particle of mass m on a n-edges star-graph with Hamiltonian HK=−(2m)−1ħ2Δ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kreĭn’s theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators for the couple (HK,H⊕D), where H⊕D is the Hamiltonian with Dirichlet conditions in the vertex
Scattering Theory for Delta-Potentials Supported by Locally Deformed Planes
No full textWe give a self-contained description of the main results from the paper (Cacciapuoti et al., J Math Anal Appl 473(1):215–257, 2019). We focus on the fundamental concepts and on the chief achievements, omitting some auxiliary results and a number of technical details given in the original paper. We discuss the scattering problem for a quantum particle in dimension three in the presence of a semitransparent unbounded obstacle, modelled by a δ-interaction supported on a surface obtained through a local, Lipschitz continuous deformation of a flat plane. We discuss existence and asymptotic completeness of the wave operators with respect to a suitable reference dynamics. Additionally, we provide an explicit expression for the related scattering matrix and show that it converges to the identity as the deformation goes to zero (giving a quantitative estimates on the rate of convergence)
Three-Body Hamiltonian with Regularized Zero-Range Interactions in Dimension Three
No full textWe study the Hamiltonian for a system of three identical bosons in dimension three interacting via zero-range forces. In order to avoid the fall to the center phenomenon emerging in the standard Ter-Martirosyan-Skornyakov (TMS) Hamiltonian, known as Thomas effect, we develop in detail a suggestion given in a seminal paper of Minlos and Faddeev in 1962 and we construct a regularized version of the TMS Hamiltonian which is self-adjoint and bounded from below. The regularization is given by an effective three-body force, acting only at short distance, that reduces to zero the strength of the interactions when the positions of the three particles coincide. The analysis is based on the construction of a suitable quadratic form which is shown to be closed and bounded from below. Then, domain and action of the corresponding Hamiltonian are completely characterized and a regularity result for the elements of the domain is given. Furthermore, we show that the Hamiltonian is the norm resolvent limit of Hamiltonians with rescaled non-local interactions, also called separable potentials, with a suitably renormalized coupling constant
Relative-zeta and casimir energy for a semitransparent hyperplane selecting transverse modes
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