Ari Laptev

Hardy inequalities for magnetic Dirichlet forms

It is known that the classical Hardy inequality fails in ℝ.We show that under certain non-degeneracy conditions on vector potentials, the Hardy inequality becomes possible for the corresponding magnetic Dirichlet form.

Lieb–Thirring Inequalities for Schrödinger Operators with Complex-valued Potentials

Inequalities are derived for power sums of the real part and the modulus of the eigenvalues of a Schrödinger operator with a complex-valued potential.

New bounds on the Lieb-Thirring constants

Abstract.Improved estimates on the constants Lγ,d, for 1/2<γ<3/2, d∈N, in the inequalities for the eigenvalue moments of Schrödinger operators are established.

Lieb-Thirring inequalities with improved constants

Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one-dimension. This allows us to improve on the known estimates of best constants in Lieb-Thirring inequalities for the sum of the negative eigenvalues for multi-dimensional Schrodinger operators

Nonlinear eigenvalues and analytic hypoellipticity

Motivated by the problem of analytic hypoellipticity, we show that a special family of compact non-self-adjoint operators has a nonzero eigenvalue. We recover old results obtained by ordinary differential equations techniques and show how it can be applied to the higher dimensional case. This gives in particular a new class of hypoelliptic, but not analytic hypoelliptic operators.

Geometrical versions of improved Berezin-Li-Yau inequalities

We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in

Solitons and the removal of eigenvalues for fourth-order differential operators

\R^d

Discrete spectrum of the perturbed Dirac operator

,

Hardy inequalities for Robin Laplacians

d \geq 2

SPECTRAL ESTIMATES ON THE SPHERE

. In particular, we derive upper bounds on Riesz means of order