This paper is related to the spectral stability of traveling wave solutions of partial differential equations. In the first part of the paper we use the Gohberg-Rouche Theorem to prove equality of the algebraic multiplicity of an isolated eigenvalue of an abstract operator on a Hilbert space, and the algebraic multiplicity of the eigenvalue of the corresponding Birman-Schwinger type operator pencil. In the second part of the paper we apply this result...

We describe a joint work with C.E. Kenig and G. Uhlmann [9] where we improve an earlier result by Bukhgeim and Uhlmann [1], by showing that in dimension $n\ge 3$, the knowledge of the Cauchy data for the Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of [1] but use a richer set of solutions to the Dirichlet problem.

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.