Claudio Cacciapuoti

Coupling in the singular limit of thin quantum waveguides

We analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve, consisting of one vertex and two infinite, straight edges, i.e., a broken line. We discuss the convergence of the Laplacian, with Dirichlet boundary conditions on the strip, in a suitable sense and we obtain two possible limits: the Laplacian on the line with Dirichlet boundary conditions in the origin and a nontrivial family of point perturbations of the Laplacian on the line. The first case generically occurs and corresponds to the decoupling of the two components of the limit curve, while in the second case a coupling takes place. We present also two families of curves which give rise to coupling.

On the structure of critical energy levels for the cubic focusing NLS on star graphs

We provide information on a non-trivial structure of phase space of the cubic nonlinear Schrodinger (NLS) on a three-edge star graph. We prove that, in contrast to the case of the standard NLS on the line, the energy associated with the cubic focusing Schrodinger equation on the three-edge star graph with a free (Kirchhoff) vertex does not attain a minimum value on any sphere of constant L2-norm. We moreover show that the only stationary state with prescribed L2-norm is indeed a saddle point.

Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide

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Local Marchenko-Pastur law at the hard edge of sample covariance matrices

Let XN be a N × N matrix whose entries are independent identically distributed complex random variables with mean zero and variance 1N. We study the asymptotic spectral distribution of the eigenvalues of the covariance matrix XN*XN for N → ∞. We prove that the empirical density of eigenvalues in an interval [E, E + η] converges to the Marchenko-Pastur law locally on the optimal scale, Nη/E≫(logN)b, and in any interval up to the hard edge, (logN)bN2≲E≤4−κ, for any κ > 0. As a consequence, we show the complete delocalization of the eigenvectors.

Bounds for the Stieltjes transform and the density of states of Wigner matrices

We 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

Graph-like models for thin waveguides with Robin boundary conditions

(\log N)/N

The point-like limit for a NLS equation with concentrated nonlinearity in dimension three

(logN)/N 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 Tikhomirov (2013).

A solvable model of a tracking chamber

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.