Nonlinear spectral problem for H\"ormander vector fields

Based on variational methods, we study a nonlinear eigenvalue problem for a quasilinear operator arising from smooth H\"ormander vector fields. We derive the smallest eigenvalue, prove its simplicity and isolatedness, establish the positivity of the first eigenfunction and show H\"older regularity of eigenfunctions. Moreover, we determine the best constant for the $L^{p}$-Poincar\'e inequality as a byproduct.Comment: 12 page

Finite element approximation for the fractional eigenvalue problem

The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that the discrete eigenvalue problem converges to the continuous one and we show the order of such convergence. Finally, we perform some numerical experiments and compare our results with previous work by other authors.Comment: 20 pages, 6 figure

Thermal Ionization

In the context of an idealized model describing an atom coupled to black-body radiation at a sufficiently high positive temperature, we show that the atom will end up being ionized in the limit of large times. Mathematically, this is translated into the statement that the coupled system does not have any time-translation invariant state of positive (asymptotic) temperature, and that the expectation value of an arbitrary finite-dimensional projection in an arbitrary initial state of positive (asymptotic) temperature tends to zero, as time tends to infinity. These results are formulated within the general framework of $W^*$-dynamical systems, and the proofs are based on Mourre's theory of positive commutators and a new virial theorem. Results on the so-called standard form of a von Neumann algebra play an important role in our analysis

Regularity for eigenfunctions of Schr\"odinger operators

We prove a regularity result in weighted Sobolev spaces (or Babuska--Kondratiev spaces) for the eigenfunctions of a Schr\"odinger operator. More precisely, let K_{a}^{m}(\mathbb{R}^{3N}) be the weighted Sobolev space obtained by blowing up the set of singular points of the Coulomb type potential V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}, x in \mathbb{R}^{3N}, b_j, c_{ij} in \mathbb{R}. If u in L^2(\mathbb{R}^{3N}) satisfies (-\Delta + V) u = \lambda u in distribution sense, then u belongs to K_{a}^{m} for all m \in \mathbb{Z}_+ and all a \le 0. Our result extends to the case when b_j and c_{ij} are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a<3/2.Comment: to appear in Lett. Math. Phy

Regularity for eigenfunctions of Schr\"odinger operators

We prove a regularity result in weighted Sobolev spaces (or Babuska--Kondratiev spaces) for the eigenfunctions of a Schr\"odinger operator. More precisely, let K_{a}^{m}(\mathbb{R}^{3N}) be the weighted Sobolev space obtained by blowing up the set of singular points of the Coulomb type potential V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}, x in \mathbb{R}^{3N}, b_j, c_{ij} in \mathbb{R}. If u in L^2(\mathbb{R}^{3N}) satisfies (-\Delta + V) u = \lambda u in distribution sense, then u belongs to K_{a}^{m} for all m \in \mathbb{Z}_+ and all a \le 0. Our result extends to the case when b_j and c_{ij} are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a<3/2.Comment: to appear in Lett. Math. Phy

Analysis of Schr\"odinger operators with inverse square potentials I: regularity results in 3D

Let $V$ be a 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 $Z$ continuous on \RR^3 with $Z(p) > -1/4$ for p \in \maS. Also assume that $\rho$ and $Z$ are smooth outside \maS and $Z$ is smooth in polar coordinates around each singular point. We either assume that $V$ is periodic or that the set \maS is finite and $V$ extends to a smooth function on the radial compactification of \RR^3 that is bounded outside a compact set containing \maS. In the periodic case, we let $\Lambda$ be the periodicity lattice and define \TT := \RR^3/ \Lambda. We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schr\"odinger-type operator $H = -\Delta + V$ acting on L^2(\TT), as well as for the induced \vt k--Hamiltonians \Hk obtained by restricting the action of $H$ to Bloch waves. Under some additional assumptions, we extend these regularity and solvability results to the non-periodic case. We sketch some applications to approximation of eigenfunctions and eigenvalues that will be studied in more detail in a second paper.Comment: 15 pages, to appear in Bull. Math. Soc. Sci. Math. Roumanie, vol. 55 (103), no. 2/201

The electron density is smooth away from the nuclei

We prove that the electron densities of electronic eigenfunctions of atoms and molecules are smooth away from the nuclei.Comment: 16 page