Related papers: Generic controllability properties for the bilinea…
This paper aims at completing an earlier work of Russell and Zhang to study internal control problems for the distributed parameter system described by the Korteweg-de Vries equation on a periodic domain T^1. In their article, Russell and…
For the first time, the general nonlinear Schr\"odinger equation is investigated, in which the chromatic dispersion and potential are specified by two arbitrary functions. The equation in question is a natural generalization of a wide class…
We study spectral approximations of Schr\"odinger operators $T=-\Delta+Q$ with complex potentials on $\Omega=\mathbb{R}^d$, or exterior domains $\Omega\subset \mathbb{R}^d$, by domain truncation. Our weak assumptions cover wide classes of…
In this paper we consider the two-dimensional Schr\"odinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region, whose thickness is varying and is determined by the…
This work is concerned with the possibility of proving the boundary null controllability for the degenerate wave equation, developing the asymptotic analysis of a suitable family of state-control pairs $((u_\varepsilon ,…
This manuscript is concerned with the approximate controllability of fractional nonlinear differential equations with nonlocal conditions of order $1<q<2$ in Banach spaces. As far as we know, few articles have investigated this issue. The…
We study Schr\"odinger operators on compact finite metric graphs subject to $\delta$-coupling and standard boundary conditions. We compare the $n$-th eigenvalues of those self-adjoint realizations and derive an asymptotic result for the…
We show that the measure of the spectrum of Schr\"odinger operator with potential defined by non-constant function over any minimal aperiodic finite subshift tends to zero, as the coupling constant tends to infinity. We also obtained a…
We consider a derivative nonlinear Schr\"odinger equation with a general nonlinearity. This equation has a two parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the…
In this article we discuss which controllability properties of classical Hamiltonian systems are preserved after quantization. We discuss some necessary and some sufficient conditions for small-time controllability of classical systems and…
We relax the regularity condition on potentials of the Schr\"odinger equation in uniqueness results on the inverse boundary value problem which were recently proved in [11] and [5].
This work investigates the regularity of Schr\"odinger eigenfunctions and the solvability of Schr\"odinger equations in spectral Barron space $\mathcal{B}^{s}(\mathbb{R}^{nN})$, where neural networks exhibit dimension-free approximation…
In this paper, we study the small-time global null controllability of the generalized Burgers' equations $y_t + \gamma |y|^{\gamma-1}y_x-y_{xx}=u(t)$ on the segment $[0,1]$. The scalar control $u(t)$ is uniform in space and plays a role…
Schr\"odinger operators with potentials generated by primitive substitutions are simple models for one dimensional quasi-crystals. We review recent results on their spectral properties. These include in particular an algorithmically…
Consider a two-dimensional domain shaped like a wire, not necessarily of uniform cross section. Let $V$ denote an electric potential driven by a voltage drop between the conducting surfaces of the wire. We consider the operator ${\mathcal…
By mean of generalized Fourier series and Parseval's equality in weighted $L^{2}$--spaces, we derive a sharp energy estimate for the wave equation in a bounded interval with a moving endpoint. Then, we show the observability, in a sharp…
We provide a class of necessary and sufficient conditions for the discreteness of spectrum of Schr\"odinger operators with scalar potentials which are semibounded below. The classical discreteness of spectrum criterion by A.M.Molchanov…
Given a potential $V$ and the associated Schr\"odinger operator $-\Delta+V$, we consider the problem of providing sharp upper and lower bound on the energy of the operator. It is known that if for example $V$ or $V^{-1}$ enjoys suitable…
We establish necessary and sufficient conditions for the discreteness of spectrum and strict positivity of magnetic Schr\"odinger operators with a positive scalar potential. They are expressed in terms of Wiener's capacity and the local…
We study the internal controllability of the semilinear wave equation $$v_{tt}(x,t)-\Delta v(x,t) + f(x,v(x,t))= \Un_{\omega} u(x,t)$$ for some nonlinearities $f$ which can produce several non-trivial steady states. One of the usual…