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Michael Loss

Publications and source records attributed to Michael Loss.

At least 55 records · Page 3Linked to original sources
arXiv2011

Localization for the random displacement model

We prove spectral and dynamical localization for the multi-dimensional random displacement model near the bottom of its spectrum by showing that the approach through multiscale analysis is applicable. In particular, we show that a previously known Lifshitz tail bound can be extended to our setting and prove a new Wegner estimate. A key tool is given by a quantitative form of a property of a related single-site Neumann problem which can be described as "bubbles tend to the corners".

math-ph↗
arXiv2011

Sharp trace inequalities for fractional Laplacians

The sharp trace inequality of Jose Escobar is extended to traces for the fractional Laplacian on R^n and a complete characterization of cases of equality is discussed. The proof proceeds via Fourier transform and uses Lieb's sharp form of the Hardy-Littlewood-Sobolev inequality.

math.AP↗
arXiv2011

A new estimate on the indirect Coulomb Energy

We prove a new lower bound on the indirect Coulomb energy in quantum mechanics in terms of the single particle density of the system. The new universal lower bound is an alternative to the classical Lieb--Oxford bound (with a smaller constant, 1.45 < 1.68) but involving an additive kinetic energy term of the single particle density.

math-ph↗
arXiv2011

Hardy-Sobolev-Maz'ya inequalities for arbitrary domains

We prove a Hardy-Sobolev-Maz'ya inequality for arbitrary domains Ω\subset\R^N with a constant depending only on the dimension N\geq 3. In particular, for convex domains this settles a conjecture by Filippas, Maz'ya and Tertikas. As an application we derive Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators on domains.

math.AP↗
arXiv2010

On the Markov sequence problem for Jacobi polynomials

We give a simple and entirely elementary proof of Gasper's theorem on the Markov sequence problem for Jacobi polynomials. It is based on the spectral analysis of an operator that arises in the study of a probabilistic model of colliding molecules introduced by Marc Kac. In the process, we obtain some new integral formulas for ratios of Jacobi polynomials that generalize Gasper's product formula and a well known formula of Koornwinder.

math.CA↗
arXiv2008

Kramers degeneracy theorem in nonrelativistic QED

Degeneracy of the eigenvalues of the Pauli-Fierz Hamiltonian with spin 1/2 is proven by the Kramers degeneracy theorem. The Pauli-Fierz Hamiltonian at fixed total momentum is also investigated.

math-ph↗
arXiv2008

Low energy properties of the random displacement model

We study low-energy properties of the random displacement model, a random Schrödinger operator describing an electron in a randomly deformed lattice. All periodic displacement configurations which minimize the bottom of the spectrum are characterized. While this configuration is essentially unique for dimension greater than one, there are infinitely many different minimizing configurations in the one-dimensional case. The latter leads to unusual low energy asymptotics for the integrated density of states of the one-dimensional random displacement model. For symmetric Bernoulli-distributed displacements it has a $1/\log^2$-singularity at the bottom of the spectrum. In particular, it is not Hölder-continuous.

math-ph↗
arXiv2008

Self-adjointness via partial Hardy-like inequalities

Distinguished selfadjoint extensions of operators which are not semibounded can be deduced from the positivity of the Schur Complement (as a quadratic form). In practical applications this amounts to proving a Hardy-like inequality. Particular cases are Dirac-Coulomb operators where distinguished selfadjoint extensions are obtained for the optimal range of coupling constants.

math.AP↗
arXiv2007

Characterization of the critical magnetic field in the Dirac-Coulomb equation

We consider a relativistic hydrogenic atom in a strong magnetic field. The ground state level depends on the strength of the magnetic field and reaches the lower end of the spectral gap of the Dirac-Coulomb operator for a certain critical value, the critical magnetic field. We also define a critical magnetic field in a Landau level ansatz. In both cases, when the charge Z of the nucleus is not too small, these critical magnetic fields are huge when measured in Tesla, but not so big when the equation is written in dimensionless form. When computed in the Landau level ansatz, orders of magnitude of the critical field are correct, as well as the dependence in Z. The computed value is however significantly too big for a large Z, and the wave function is not well approximated. Hence, accurate numerical computations involving the Dirac equation cannot systematically rely on the Landau level ansatz. Our approach is based on a scaling property. The critical magnetic field is characterized in terms of an equivalent eigenvalue problem. This is our main analytical result, and also the starting point of our numerical scheme.

math.AP↗
arXiv2007

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

We consider the Dirichlet Laplacian with a constant magnetic field in a two-dimensional domain of finite measure. We determine the sharp constants in semi-classical eigenvalue estimates and show, in particular, that Polya's conjecture is not true in the presence of a magnetic field.

math-ph↗
arXiv2007

Lieb-Thirring inequalities with improved constants

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

math.AP↗
arXiv2007

Minimizing the ground state energy of an electron in a randomly deformed lattice

We provide a characterization of the spectral minimum for a random Schrödinger operator of the form $H=-Δ+ \sum_{i \in \Z^d}q(x-i-ω_i)$ in $L^2(\R^d)$, where the single site potential $q$ is reflection symmetric, compactly supported in the unit cube centered at 0, and the displacement parameters $ω_i$ are restricted so that adjacent single site potentials do not overlap. In particular, we show that a minimizing configuration of the displacements is given by a periodic pattern of densest possible $2^d$-clusters of single site potentials. The main tool to prove this is a quite general phenomenon in the spectral theory of Neumann problems, which we dub ``bubbles tend to the boundary.'' How should a given compactly supported potential be placed into a bounded domain so as to minimize or maximize the first Neumann eigenvalue of the Schrödinger operator on this domain? For square or rectangular domains and reflection symmetric potentials, we show that the first Neumann eigenvalue is minimized when the potential sits in one of the corners of the domain and is maximized when it sits in the center of the domain. With different methods we also show a corresponding result for smooth strictly convex domains.

math-ph↗