1,720,998 research outputs found
On the derivation of the Schrödinger equation with point-like nonlinearity
Full text linkIn this report we discuss the problem of approximating nonlinear delta-interactions in dimensions one and three with regular, local or non-local nonlinearities. Concerning the one dimensional case, we discuss a recent result proved in [10], on the derivation of nonlinear delta-interactions as limit of scaled, local nonlinearities.
For the three dimensional case, we consider an equation with scaled, non-local nonlinearity. We conjecture that such an equation approximates the nonlinear delta-interaction, and give an heuristic argument to support our conjecture
Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core
No full textWe consider a compact metric graph of size ε and attach to it several edges (leads) of length of order one (or of infinite length). As ε goes to zero, the graph G ε obtained in this way looks like the star-graph formed by the leads joined in a central vertex. On G ε we define an Hamiltonian H ε , properly scaled with the parameter ε . We prove that there exists a scale invariant effective Hamiltonian on the star-graph that approximates H ε (in a suitable norm resolvent sense) as ε → 0 . The effective Hamiltonian depends on the spectral properties of an auxiliary ε -independent Hamiltonian defined on the compact graph obtained by setting ε = 1 . If zero is not an eigenvalue of the auxiliary Hamiltonian, in the limit ε → 0 , the leads are decoupled
Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide
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The Riemann zeta in terms of the dilogarithm
No full textWe give a representation of the classical Riemann ζ-function in the half plane Res > 0 in terms of a Mellin transform involving the real part of the dilogarithm function with an argument on the unit circle (associated Clausen Gl2-function). We also derive corresponding representations involving the derivatives of the Gl2-function. A generalized symmetrized Müntz-type formula is also derived. For a special choice of test functions it connects to our integral representation of the ζ-function, providing also a computation of a concrete Mellin transform. Certain formulae involving series of zeta functions and gamma functions are also derived
Bounds for the Stieltjes transform and the density of states of Wigner matrices
No full textWe consider ensembles of Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. We show the convergence of the Stieltjes transform towards the Stieltjes transform of the semicircle law on optimal scales and with the optimal rate. Our bounds improve previous results, in particular from Erdős et al. (Adv Math 229(3):1435–1515, 2012; Electron J Probab 18(59):1–58, 2013), by removing the logarithmic corrections. As applications, we establish the convergence of the eigenvalue counting functions with the rate (Formula presented.) and the rigidity of the eigenvalues of Wigner matrices on the same scale. These bounds improve the results of Erdős et al. (Adv Math 229(3):1435–1515, 2012; Electron J Probab 18(59):1–58, 2013), Götze and Tikhomiro
The semiclassical limit on a star-graph with Kirchhoff conditions
Full text linkWe consider the dynamics of a quantum particle of mass m on a n-edges star-graph with Hamiltonian HK= - (2 m) - 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,HD⊕), where HD⊕ is the Hamiltonian with Dirichlet conditions in the vertex
Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph
No full textIn this paper we give the complete classification of solitons for a cubic nonlinear Schrödinger equation on the simplest network with a nontrivial topology: the tadpole graph, i.e., a ring with a half line attached to it and free boundary conditions at the junction. This is a step toward the modelization of condensate propagation and confinement in quasi-one-dimensional traps. The model, although simple, exhibits a surprisingly rich behavior and in particular we show that it admits: (i) a denumerable family of continuous branches of embedded solitons vanishing on the half line and bifurcating from linear eigenstates and threshold resonances of the system; (ii) a continuous branch of edge solitons bifurcating from the previous families at the threshold of the continuous spectrum with a pitchfork bifurcation; and (iii) a finite family of continuous branches of solitons without linear analog. All the solutions are explicitly constructed in terms of elliptic Jacobian functions. Moreover we show that families of nonlinear bound states of the above kind continue to exist in the presence of a uniform magnetic field orthogonal to the plane of the ring when a well definite flux quantization condition holds true. In this sense the magnetic field acts as a control parameter. Finally we highlight the role of resonances in the linearization as a signature of the occurrence of bifurcations of solitons from the continuous spectrum
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