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We study standing waves for the nonlinear Schr\"odinger equation on a discrete graph. We characterize for a self-adjoint realizations of Schr\"odinger operators conditions related with the geometry of the graph that guarantee discreteness…
We apply both the theory of boundary triples and perturbation theory to the setting of semi-bounded Sturm-Liouville operators with two limit-circle endpoints. For general boundary conditions we obtain refined and new results about their…
We study the (H\"older-)continuous behavior of the spectra belonging to a family of linear bounded operators $(A_t)_{t\in T}$ indexed by a topological space $T$. For the cases of self-adjoint, unitary and normal operators, a…
We explore a prototypical two-dimensional model of the nonlinear Dirac type and examine its solitary wave and vortex solutions. In addition to identifying the stationary states, we provide a systematic spectral stability analysis,…
We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schr\"odinger equation with a point interaction and a focusing power nonlinearity. The Schr\"odinger operator with a point interaction…
Understanding, predicting, and controlling physical processes often relies on the analysis of the dynamics of partial differential equations (PDEs). In this context, the present study offers an in-depth investigation into the nonlinear…
We consider a class of self-adjoint Sturm-Liouville problems with rational functions of the spectral parameter in the boundary conditions. The uniform stability for direct and inverse spectral problems is proved for the first time for…
This study is devoted to the asymptotic spectral analysis of multiscale Schr\"odinger operators with oscillating and decaying electric potentials. Different regimes, related to scaling considerations, are distinguished. By means of a normal…
This paper is to study some conditions on semigroups, generated by some class of non-densely defined operators in the closure of its domain, in order that certain bounded perturbations preserve some regularity properties of the semigroup…
We consider a periodic system of domains coupled by small windows. In such domain we study the band spectrum of a Schroedinger operator subject to Neumann condition. We show that near each isolated eigenvalue of the similar operator but in…
This is mostly a survey paper, where we collect results concerning the spectral bounds of deterministic and random Schr\"odinger operators with complex potentials, both on \(\mathbb{R}^d\) and on compact manifolds. The survey part is…
We review the recent developments in the spectral theory of discrete one-dimensional Schr\"odinger operators with potentials generated by substitutions and circle maps. We discuss how occurrences of local repetitive structures allow for…
We consider the nonlinear Schr{\"o}dinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. We assume that the larger power is defocusing, and the smaller power is focusing. Such a…
This paper presents the spectral analysis of 1-dimensional Schroedinger operator on the half-line whose potential is a linear combination of the Coulomb term 1/r and the centrifugal term 1/r^2. The coupling constants are allowed to be…
The present work shows that essentially all small-amplitude periodic traveling waves of the electronic Euler-Poisson system are spectrally unstable. This instability is neither modulational nor co-periodic, and thus requires an unusual…
The present paper is devoted to the study of spectral properties of random Schroedinger operators. Using a finite section method for Toeplitz matrices, we prove a Wegner estimate for some alloy type models where the single site potential is…
We investigate the stability of ground states to a nonlinear focusing Schr\"odinger equation in presence of a Kirchhoff term. Through a spectral analysis of the linearized operator about ground states, we show a modulation stability…
We consider the stability and instability of periodic travling waves for Korteweg-de Vries type equations with fractional dispersion and other nonlinear dispersive equations. We establish that a constrained minimizer for the related…
The research explores a high irregularity, commonly referred to as intermittency, of the solution to the non-stationary parabolic Anderson problem: \begin{equation*} \frac{\partial u}{\partial t} = \varkappa \mathcal{L}u(t,x) +…
We investigate the stability and nonlinear local dynamics of spectrally stable wave trains in reaction-diffusion systems. For each $N\in\mathbb{N}$, such $T$-periodic traveling waves are easily seen to be nonlinearly asymptotically stable…