Combining the methods of Cuenin [2019] and Borichev-Golinskii-Kupin [2009, 2018], we obtain the so-called Lieb-Thirring inequalities for non-selfadjoint perturbations of an effective Hamiltonian for bilayer graphene.
We prove a Lieb-Thirring type inequality for potentials such that the associated Schr{o}dinger operator has a pure discrete spectrum made of an unbounded sequence of eigenvalues. This inequality is equivalent to a generalized Gagliardo-Nirenberg ineq
uality for systems. As a special case, we prove a logarithmic Sobolev inequality for infinite systems of mixed states. Optimal constants are determined and free energy estimates in connection with mixed states representations are also investigated.
We consider a Schrodinger operator on the half-line with a Dirichlet boundary condition at the origin and show that moments of its negative eigenvalues can be estimated by the part of the potential that is larger than the critical Hardy weight. The e
stimate is valid for the critical value of the moment parameter.
This paper considers Lieb-Thirring inequalities for higher order differential operators. A result for general fourth-order operators on the half-line is developed, and the trace inequality tr((-Delta)^2 - C^{HR}_{d,2} / (|x|^4) - V(x))^{-gamma} < C_g
amma int_{R^d} V(x)_+^{gamma + d/4} dx for gamma geq 1 - d/4, where C^{HR}_{d,2} is the sharp constant in the Hardy-Rellich inequality and where C_gamma > 0 is independent of V, is proved for dimensions d = 1,3. As a corollary of this inequality a Sobolev-type inequality is obtained.
Using a correspondence between the spectrum of the damped wave equation and non-self-adjoint Schroedinger operators, we derive various bounds on complex eigenvalues of the former. In particular, we establish a sharp result that the one-dimensional da
mped wave operator is similar to the undamped one provided that the L^1 norm of the (possibly complex-valued) damping is less than 2. It follows that these small dampings are spectrally undetectable.
We develop the differential aspects of a noncommutative geometry for the Quantum Hall Effect in the continuous, with the ambition of proving Kubos formula. Taking inspiration from the ideas developed by Bellissard during the 80s we build a Fredholm m
odule for the $C^*$-algebra of continuous magnetic operators, based on a Dirac operator closely related to the quantum harmonic oscillator. An important piece of Bellissards theory (the so-called second Connes formula) is proved. This work provides the continuation of the recent article [DS].
Philippe Briet
,Jean-Claude Cuenin
,Leonid Golinskii
.
(2019)
.
"Lieb-Thirring inequalities for an effective Hamiltonian of bilayer graphene"
.
Stanislav Kupin
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