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We prove sharp analytic regularity and decay at infinity of solutions of variable coefficients nonlinear harmonic oscillators. Namely, we show holomorphic extension to a sector in the complex domain, with a corresponding Gaussian decay,…

We introduce a theorem currently proved unique by the asymptotic behaviors of eigenvalues of a compact operator. Specifically, a problem of partitions is considered and the Neumann--Poincar\'e operator is employed as the compact linear…

Under a perturbation by a decaying electric potential, the Landau Hamiltonian acquires some discrete eigenvalues between the Landau levels. We study the perturbation by an "expanding" electric potential $V(t^{-1}x)$, $t>0$, and derive a…

We consider $N$-body Schr\"odinger operators with a virtual level at the threshold of the essential spectrum. We show that in the case of $N\geq 3$ particles in dimension $n\geq3$ virtual levels correspond to eigenvalues of the system and…

Initial-boundary value problems for the linear Zakharov-Kuznetsov equation posed on bounded rectangles are considered. Spectral properties of a stationary operator are studied in order to show that the evolution problem posed on a bounded…

Lecture notes for one of the courses at the OPSFA Summerschool 6, July 11-15, 2016. All the results in these notes have appeared in the literature. Many special functions are eigenfunctions to explicit operators, such as difference and…

The two major discrete time formulations for quantum walks, coined and scattering, are unitarily equivalent for arbitrary position dependent transition amplitudes and any topology (PRA {\bf 80}, 052301 (2009)). Although the proof explicit…

We develop the scattering theory of general conformally compact metrics. For low frequencies, the domain of the scattering matrix is shown to be frequency dependent. In particular, generalized eigenfunctions exhibit L^2 decay in directions…

It is well known that the sum of negative (positive) eigenvalues of some finite Hermitian matrix $V$ is concave (convex) with respect to $V$. Using the theory of the spectral shift function we generalize this property to self-adjoint…

In [8] we recently proved that in our model of quantum gravity the solutions to the quantized version of the full Einstein equations or to the Wheeler-DeWitt equation could be expressed as products of spatial and temporal eigenfunctions, or…

We consider the Schr\"odinger operator $H_{\eta W} = -\Delta + \eta W$, self-adjoint in $L^2({\mathbb R}^d)$, $d \geq 1$. Here $\eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study…

We study the properties of the exponential functional $\int\_0^{+ \infty} e^{- X^{\uparrow} (t)}dt$ where $X^{\uparrow}$ is a spectrally one-sided L{\'e}vy process conditioned to stay positive. In particular, we study finiteness,…

We consider the eigenvalue problem for the Laplace operator in a planar domain which can be decomposed into a bounded domain of arbitrary shape and elongated \branches" of variable cross-sectional profiles. When the eigenvalue is smaller…

I review arguments demonstrating how the concept of "particle" numbers arises in the form of equidistant energy eigenvalues of coupled harmonic oscillators representing free fields. Their quantum numbers (numbers of nodes of the wave…

We confirm rigorously the conjecture, based on numerical and asymptotic evidence, that all the eigenvalues of a certain non-self-adjoint operator are real.

It has been recently shown that complex two-dimensional (2D) potentials $V_\varepsilon(x,y)=V(y+\mathrm{i}\varepsilon\eta(x))$ can be used to emulate non-Hermitian matrix gauge fields in optical waveguides. Here $x$ and $y$ are the…

In this work, we study the effect of a moving detector on a discrete time one dimensional Quantum Random Walk where the movement is realized in the form of hopping/shifts. The occupation probability $f(x,t;n,s)$ is estimated as the number…

In this work, we consider the so-called correlated random walk system (also known as correlated motion or persistent motion system), used in biological modelling, among other fields, such as chromatography. This is a linear system which can…

We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly…

We consider the problem of embedding eigenvalues into the essential spectrum of periodic Jacobi operators, using an oscillating, decreasing potential. To do this we employ a geometric method, previously used to embed eigenvalues into the…