The probabilistic approach to the Dirichlet boundary value problem for certain Schrödinger equations with magnetic vector potentials is examined

We will consider the following problem$$-\Delta u-\lambda \frac{u}{{\left|x\right|}^2}={\left|\nabla u\right|}^p+cf,u\>0\text{in}\Omega ,$$where $\Omega \subset {ℝ}^N$ is a domain such that $0\in \Omega$, $N\ge 3$, $c\>0$ and $\lambda \>0$. The main objective of this note is to study the precise threshold $p_{+}=p_{+}\left(\lambda \right)$ for which there is novery weak supersolutionif $p\ge p_{+}\left(\lambda \right)$. The optimality of $p_{+}\left(\lambda \right)$ is also proved by showing the solvability of the Dirichlet problem when $1\le p\<p_{+}\left(\lambda \right)$, for $c\>0$ small enough and $f\ge 0$ under some hypotheses that we will prescribe.

We establish necessary and sufficient conditions on the real- or complex-valued potential $Q$ defined on ${ℝ}^n$ for the relativistic Schrödinger operator $\sqrt{-\Delta }+Q$ to be bounded as an operator from the Sobolev space $W_2^{1/2}\left({ℝ}^n\right)$ to its dual $W_2^{-1/2}\left({ℝ}^n\right)$.

We obtain inequalities between the eigenvalues of the Schrödinger operator on a compact domain Ω of a submanifold M in $R^N$ with boundary ∂Ω, which generalize many existing inequalities for the Laplacian on a bounded domain of a Euclidean space. We also establish similar inequalities for a closed minimal submanifold in the unit sphere, which generalize and improve Yang-Yau’s result.

The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics 2000, Springer (2010)], the regularity of the solutions, which increases with the number of electrons,...

The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics2000, Springer (2010)], the regularity of the solutions, which increases with the number of electrons,...

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter $\alpha \>0$. The high-frequency (or: semi-classical) parameter is $\epsilon \>0$. We let $\epsilon$ and $\alpha$ go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption.Under these assumptions, we prove that the solution $u^{\epsilon }$ radiates in the outgoing...

This paper reports on the recent proof of the bounded $L^2$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $L^2$-norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.

We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any riemannian compact manifold. As a consequence we infer global existence results for the Cauchy problem of nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on three-manifolds in the case of quadratic nonlinearities. We also discuss the optimality of these Strichartz estimates on spheres.

We consider equivariant solutions of Schrödinger equations on C∖{0} with harmonic oscillator potentials. We determine the spaces of equivariant quantum states in three cases: for an isotropic and anisotropic harmonic oscillator potential centered at 0, and for a potential not centered at 0.