Related papers: Schroedinger operators involving singular potentia…

We establish the Hopf boundary point lemma for the Schr\"odinger operator $-\Delta + V$ involving potentials $V$ that merely belong to the space $L^{1}_{loc}(\Omega)$. More precisely, we prove that among all supersolutions $u$ of $-\Delta +…

We investigate the Dirichlet problem associated to the Schr\"odinger operator $\mathcal L=-\Delta_{\mathbb{H}^n}+V$ on Heisenberg group $\mathbb H^n$: \begin{align*} \begin{cases} \partial_{ss}u(g,s)-\mathcal L u(g,s)=0\,,\quad &{\rm in \,\…

We study equations driven by Schr\"odinger operators consisting of a self-adjoint Dirichlet operator and a singular potential, which belongs to a class of positive Borel measures absolutely continuous with respect to a capacity generated by…

We consider the steady fractional Schr\"odinger equation $L u + V u = f$ posed on a bounded domain $\Omega$; $L$ is an integro-differential operator, like the usual versions of the fractional Laplacian $(-\Delta)^s$; $V\ge 0$ is a potential…

In this paper, we are interested in the nonlinear Schr\"odinger problem $-\Delta u + Vu = \abs{u}^{p-2}u$ submitted to the Dirichlet boundary conditions. We consider $p>2$ and we are working with an open bounded domain $\Omega\subset\IR^N$…

We study the Dirichlet problem for the stationary Schr\"odinger fractional Laplacian equation $(-\Delta)^s u + V u = f$ posed in bounded domain $ \Omega \subset \mathbb R^n$ with zero outside conditions. We consider general nonnegative…

We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schr\"odinger operator $- \Delta + V$ with a nonnegative potential $V$ which merely belongs to $L_{\mathrm{loc}}^1(\Omega)$. More precisely, if $u…

Motivated mainly by the localization over an open bounded set $\Omega$ of $\mathbb R^n$ of solutions of the Schr\"odinger equations, we consider the Schr\"odinger equation over $\Omega$ with a very singular potential $V(x) \ge C d (x,…

Let $\Omega$ be a compact smooth domain containing zero in the Poincar\'e ball model of the Hyperbolic space $\mathbb{B}^{n}$ ($n \geq 3$) and let $-\Delta_{\mathbb{B}^{n}}$ be the Laplace-Beltrami operator on $\mathbb{B}^{n}$, associated…

Consider the Dirichlet problem with respect to an elliptic operator \[ A = - \sum_{k,l=1}^d \partial_k \, a_{kl} \, \partial_l - \sum_{k=1}^d \partial_k \, b_k + \sum_{k=1}^d c_k \, \partial_k + c_0 \] on a bounded Wiener regular open set…

In the present paper we consider spectral optimization problems involving the Schr\"odinger operator $-\Delta +\mu$ on $\R^d$, the prototype being the minimization of the $k$ the eigenvalue $\lambda_k(\mu)$. Here $\mu$ may be a capacitary…

In this paper we analyze the asymptotic behaviour as $p\to 1^+$ of solutions $u_p$ to $$ \left\{ \begin{array}{rclr} -\Delta_p u_p&=&\frac{\lambda}{|x|^p}|u_p|^{p-2}u_p+f&\quad \mbox{ in } \Omega,\\ u_p&=&0 &\quad \mbox{ on }\partial\Omega,…

For the the Schr\"odinger operator $H=-\Delta+ V(x)\cdot$, acting in the space L_2(\R^d)\,(d\ge 3), with V(x)\ge 0 and V(\cdot)\in L_{1,loc}(\R^d), we obtain some constructive conditions for discreteness of its spectrum. Basing on the…

In this paper we prove existence and uniqueness results for nonlinear parabolic problems with Dirichlet boundary values whose model is \[ \left\{ \begin{aligned} &b(u)_t-\Delta_{p}u=\mu\;\mbox{in }(0,T)\times\Omega,\\…

We study the time-dependent Schr\"odinger operator $P = D_t + \Delta_g + V$ acting on functions defined on $\mathbb{R}^{n+1}$, where, using coordinates $z \in \mathbb{R}^n$ and $t \in \mathbb{R}$, $D_t$ denotes $-i \partial_t$, $\Delta_g$…

The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In…

We study the interior regularity of solutions to the Dirichlet problem $Lu=g$ in $\Omega$, $u=0$ in $\R^n\setminus\Omega$, for anisotropic operators of fractional type $$ Lu(x)= \int_{0}^{+\infty}\,d\rho \int_{S^{n-1}}\,da(\omega)\, \frac{…

Let $(X,d,\mu)$ be a metric measure space satisfying a $Q$-doubling condition, $Q>1$, and an $L^2$-Poincar\'{e} inequality. Let $\mathscr{L}=\mathcal{L}+V$ be a Schr\"odinger operator on $X$, where $\mathcal{L}$ is a non-negative operator…

We consider a Schr\"odinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive…

We study the Dirichlet problem for the weighted Schr\"odinger operator \[-\Delta u +Vu = \lambda \rho u,\] where $\rho$ is a positive weighting function and $V$ is a potential. Such equations appear naturally in conformal geometry and in…