Timo Weidl

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.

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.

REMARKS ON VIRTUAL BOUND STATES FOR SEMI-BOUNDED OPERATORS

We calculate the number of bound states appearing below the spectrum of a semi—bounded operator in the case of a weak, indefinite perturbation. The abstract result generalizes the Birman—Schwinger principle to this case. We discuss a number of examples, in particular higher order differential operators, critical Schrodinger operators, systems of second order differential operators, Schrodinger type operators with magnetic fields and the Two—dimensional Pauli operator with a localized magnetic field.

On the Lieb-Thirring constantsLγ,1 for γ≧1/2

AbstractLetEi(H) denote the negative eigenvalues of the one-dimensional Schrödinger operatorHu≔−u″−Vu,V≧0, onL2(∝). We prove the inequality(1)

Geometrical versions of improved Berezin-Li-Yau inequalities

\mathop \sum \limits_i |E_i (H)|^{ \gamma } \leqq L_{\gamma ,1} \mathop \smallint \limits_\mathbb{R} V^{\gamma + 1/2} (x)dx,

Pólya's conjecture in the presence of a constant magnetic field

for the “limit” case γ=1/2. This will imply improved estimates for the best constantsLγ,1 in (1) as 1/2

On Lieb-Thirring inequalities for higher order operators with critical and subcritical powers

We improve the Berezin-Li-Yau inequality in dimension two by adding a positive correction term to its right-hand side. It is also shown that the asymptotical behaviour of the correction term is almost optimal. This improves a previous result by Melas, [11].

A Remark on Hardy type inequalities for critical Schrödinger operators with magnetic fields

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