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We consider the eigenvalue problem for the restricted fractional Laplacian in a bounded domain with homogeneous Dirichlet boundary conditions. We introduce the notion of fractional capacity for compact subsets, with the property that the…
In this paper we deal with the problems of the weak localization of the eigenfunction expansions related to Laplace-Beltrami operator on unit sphere. The conditions for weak localization of Fourier-Laplace series are investigated by…
We study the level spacing distribution for the spectrum of a point scatterer on a flat torus. In the 2-dimensional case, we show that in the weak coupling regime the eigenvalue spacing distribution coincides with that of the spectrum of…
By means of molecular dynamics simulations, we study the stationary points of the potential energy in a Lennard-Jones liquid, giving a purely geometric characterization of the energy landscape of the system. We find a linear relation…
Lee-Wick-like scalar model near a Dirichlet plate is considered in this work. The modified propagator for the scalar field due to the presence of a Dirichlet boundary is computed, and the interaction between the plate and a point-like…
We investigate several aspects of the nodal geometry and topology of Laplace eigenfunctions, with particular emphasis on the low frequency regime. This includes investigations in and around the Payne property, opening angle estimates of…
The self-similar Lorentz billiard channel is a spatially extended deterministic dynamical system which consists of an infinite one-dimensional sequence of cells whose sizes increase monotonically according to their indices. This special…
Particles interacting through long-range attraction and short-range repulsion given by power-laws have been widely used to model physical and biological systems, and to predict or explain many of the patterns they display. Apart from rare…
We study the set of Quantum Limits, and more generally, of semiclassical measures of sequences of eigenfunctions of perturbations of the Laplacian on the spheres $\mathbb{S}^{2}$ and $\mathbb{S}^{3}$ by point-scatterers. In the unperturbed…
A microscopic model of the effect of unbinding in diffusion limited aggregation based on a cellular automata approach is presented. The geometry resembles electrochemical deposition - ``ions'' diffuse at random from the top of a container…
The low-energy properties of the SU(4) spin-orbital model on a two-leg ladder are studied by a variety of analytical and numerical techniques. Like in the case of SU(2) models, there is a singlet-multiplet gap in the spectrum, but the…
This paper concerns the concentration of Dirichlet eigenfunctions of the Laplacian on a compact two-dimensional Riemannian manifold with strictly geodesically concave boundary. We link three inequalities which bound the concentration in…
We consider spaces of high-energy quasimodes for the Laplacian on a compact hyperbolic surface, and show that when the spaces are large enough, one can find quasimodes that exhibit strong localization phenomena. Namely, take any constant c,…
Scattering on the energy shell is viewed here as the relation between the bound states of the Hamiltonian, restricted to sections on leads that are asymptotically independent, far away from the interaction region. The decomposition is…
We prove (i) a simple sufficient geometric condition for localisation of a sequence of first Dirichlet eigenfunctions provided the corresponding Dirichlet Laplacians satisfy a uniform Hardy inequality, and (ii) localisation of a sequence of…
This paper is concerned with the location of nodal sets of eigenfunctions of the Dirichlet Laplacian in thin tubular neighbourhoods of hypersurfaces of the Euclidean space of arbitrary dimension. In the limit when the radius of the…
We study the behaviour of Laplace-type operators H on a complex vector bundle E $\rightarrow$ M in the adiabatic limit of the base space. This space is a fibre bundle M $\rightarrow$ B with compact fibres and the limit corresponds to…
Spectral boundary conditions for Laplace-type operators, of interest in string and brane theory, are partly Dirichlet, partly Neumann-type conditions, partitioned by a pseudodifferential projection. We give sufficient conditions for…
We present a new design of dielectric microcavities supporting modes with large quality factors and highly directional light emission. The key idea is to place a point scatterer inside a dielectric circular microdisk. We show that,…
The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential…