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The three-body scattering problem in Coulombic systems is widespread, however yet unresolved problem by the mathematically rigorous methods. In this work this long term challenge has been undertaken by combining distorted waves and…

While it is known that any spherical fluid distribution may only source the spherically symmetric Schwarzschild space-time, the inverse is not true. Thus, in this manuscript, we find exact axially symmetric and static fluid (interior)…

In this thesis we present a new formalism to study linear and non-linear response in extended systems. Our approach is based on real-time solution of an effective Schr\"odinger equation. The coupling between electrons and external field is…

This article is concerned with one dimensional dispersive flows with cubic nonlinearities on the real line. In a very recent work, the authors have introduced a broad conjecture for such flows, asserting that in the defocusing case, small…

Integration of the Dirac equation with an external electromagnetic field is explored in the framework of the method of separation of variables and of the method of noncommutative integration. We have found a new type of solutions that are…

We present a linear-response formalism for a system of correlated electrons out of equilibrium, as relevant for the probe optical absorption in pump-probe experiments. We consider the time dependent optical conductivity $\sigma(\omega,t)$…

We investigate the Calder\'on problem for the fractional Schr\"odinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior…

We find infinitely many positive non-radial solutions for a system of Schr\"odinger equations with critical growth in a fully attractive or repulsive regime in presence of an external radial trapping potential.

Immersed boundary methods have attracted substantial interest in the last decades due to their potential for computations involving complex geometries. Often these cannot be efficiently discretized using boundary-fitted finite elements.…

Using a new local smoothing estimate of the first and third authors, we prove local-in-time Strichartz and smoothing estimates without a loss exterior to a large class of polygonal obstacles with arbitrary boundary conditions and…

We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time…

In this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schr\"odinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate…

Study of far-from-equilibrium thermalization dynamics in quantum materials, including the dynamics of different types of quasiparticles, is becoming increasingly crucial. However, the inherent complexity of either the full quantum…

In this paper we propose a modified Lie-type spectral splitting approximation where the external potential is of quadratic type. It is proved that we can approximate the solution to a one-dimensional nonlinear Schroedinger equation by…

A formalism based on the complex-scaling method is presented to solve the few particle scattering problem in configuration space using bound state techniques with trivial boundary conditions. Several applications to A=3,4 systems are…

We are interested in this article in investigating the smoothing effect properties of the solutions of the Schrodinger equation. We deduce global well-posedness results for the cubic Schrodinger equation in the exterior of several convex…

We present an efficient perturbative method to obtain both static and dynamic polarizabilities and hyperpolarizabilities of complex electronic systems. This approach is based on the solution of a frequency dependent Sternheimer equation,…

The integrating factor technique is widely used to solve numerically (in particular) the Schr\"odinger equation in the context of spectral methods. Here, we present an improvement of this method exploiting the freedom provided by the gauge…

We consider equations of nonlinear Schrodinger type augmented by nonlinear damping terms. We show that nonlinear damping prevents finite time blow-up in several situations, which we describe. We also prove that the presence of a quadratic…

This paper presents some absorbing boundary conditions (ABC) for simulations based on the time-dependent density-functional theory (TDDFT). The boundary conditions are expressed in terms of the elements of the density-matrix, and it is…