Related papers: Recurrence of singularities for second order isotr…

We investigate forward and backward problems associated with abstract time-fractional Schr\"odinger equations $\mathrm{i}^\nu \partial_t^\alpha u(t) + A u(t)=0$, $\alpha \in (0,1)\cup (1,2)$ and $\nu\in\{1,\alpha\}$, where $A$ is a…

Self-adjoint Schr\"odinger operators with $\delta$ and $\delta'$-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity…

We introduce a class of (possibly) degenerate dispersive equations with a drift. We prove that, under the H\"ormander hypoellipticity condition, the relevant Cauchy problem can be uniquely solved in the Schwartz class, and the solution…

We establish that the initial value problem for the quadratic non-linear Schr\"odinger equation $$ iu_t - \Delta u = u^2$$ where $u: \R^2 \times \R \to \C$, is locally well-posed in $H^s(\R^2)$ when $s > -1$. The critical exponent for this…

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 asymptotic behavior of large data solutions to Schr\"odinger equations $i u_t + \Delta u = F(u)$ in $\R^d$, assuming globally bounded $H^1_x(\R^d)$ norm (i.e. no blowup in the energy space), in high dimensions $d \geq 5$ and…

We study the approximation of the spectrum of a second-order elliptic differential operator by the Hybrid High-Order (HHO) method. The HHO method is formulated using cell and face unknowns which are polynomials of some degree $k\geq0$. The…

We discuss the form of the propagator $U(t)$ for the time-dependent Schr\"odinger equation on an asyptotically Euclidean, or, more generally, asymptotically conic, manifold with no trapped geodesics. In the asymptotically Euclidean case, if…

We study the linear fractional Schr\"odinger equation on a Hilbert space, with a fractional time derivative of order $0<\alpha<1,$ and a self-adjoint generator $A.$ Using the spectral theorem we prove existence and uniqueness of strong…

The initial value problem for the cubic defocusing nonlinear Schr\"odinger equation $i \partial_t u + \Delta u = |u|^2 u$ on the plane is shown to be globally well-posed for initial data in $H^s (\R^2)$ provided $s>1/2$. The proof relies…

We prove that a solution of the Schr\"odinger-type equation $\mathrm{i}\partial_t u= Hu$, where $H$ is a Jacobi operator with asymptotically constant coefficients, cannot decay too fast at two different times unless it is trivial.

We investigate the quantitative unique continuation properties of solutions to second order elliptic equations with singular lower order terms. The main theorem presents a quantification of the strong unique continuation property for…

In this paper we will study the existence of fundamental solutions for the explicit and implicit backward time dependent Schodinger equation, via discrete Fourier transform and its symbol for the Laplace operator. In both cases we will…

Let $V$ be a {\em periodic} potential on $\RR^3$ that is smooth everywhere except at a discrete set $\maS$ of points, where it has singularities of the form $Z/\rho^2$, with $\rho(x) = |x - p|$ for $x$ close to $p$ and $Z$ is continuous,…

The focussing anisotropic nonlinear Schr\"odinger equation \begin{align*} \mathrm{i} u_t-\partial_{xx} u + (-\partial_{yy})^s u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R} \times \mathbb{R}^2 \end{align*} is considered for $0<s<1$ and $p>2$.…

The Schr\"odinger equation $i \partial_t^\rho u(x,t)-u_{xx}(x,t) = p(t)q(x) + f(x,t)$ ( $0<t\leq T, \, 0<\rho<1$), with the Riemann-Liouville derivative is considered. An inverse problem is investigated in which, along with $u(x,t)$, also a…

We propose a q-deformation of the su(2)-invariant Schrodinger equation of a spinless particle in a central potential, which allows us not only to determine a deformed spectrum and the corresponding eigenstates, as in other approaches, but…

Let $H$ denote the harmonic oscillator Hamiltonian on $\mathbb{R}^d,$ perturbed by an isotropic pseudodifferential operator of order $1.$ We consider the Schr\"odinger propagator $U(t)=e^{-itH},$ and find that while $\operatorname{singsupp}…

For an elliptic, semilinear differential operator of the form $S(u) = A : D^2 u + b(x, u , Du)$, consider the functional $E_\infty(u) = \mathop{\mathrm{ess \, sup}}_\Omega |S(u)|$. We study minimisers of $E_\infty$ for prescribed boundary…

We investigate the following fractional order in time Cauchy problem \begin{equation*} \begin{cases} \mathbb{D}_{t}^{\alpha }u(t)+Au(t)=f(u(t)), & 1<\alpha <2, \\ u(0)=u_{0},\,\,\,u^{\prime }(0)=u_{1}. & \end{cases}% \end{equation*}% where…