arXiv subjects

Rupert L. Frank

Publications and source records attributed to Rupert L. Frank.

At least 163 records · Page 9Linked to original sources

arXiv2014

Trace class conditions for functions of Schrödinger operators

We consider the difference $f(-Δ+V)-f(-Δ)$ of functions of Schrödinger operators in $L^2(\mathbb R^d)$ and provide conditions under which this difference is trace class. We are particularly interested in non-smooth functions $f$ and in $V$ belonging only to some $L^p$ space. This is motivated by applications in mathematical physics related to Lieb--Thirring inequalities. We show that in the particular case of Schrödinger operators the well-known sufficient conditions on $f$, based on a general operator theoretic result due to V. Peller, can be considerably relaxed. We prove similar theorems for $f(-Δ+V)-f(-Δ)-\frac{d}{dα} f(-Δ+αV)|_{α=0}$. Our key idea is the use of the limiting absorption principle.

math.SP↗

arXiv2013

An extension problem for the CR fractional Laplacian

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.

math.DG↗

arXiv2013

Dynamics of a strongly coupled polaron

We study the dynamics of large polarons described by the Froehlich Hamiltonian in the limit of strong coupling. The initial conditions are (perturbations of) product states of an electron wave function and a phonon coherent state, as suggested by Pekar. We show that, to leading order on the natural time scale of the problem, the phonon field is stationary and the electron moves according to an effective linear Schroedinger equation.

math-ph↗

arXiv2013

Monotonicity of a relative Rényi entropy

We show that a recent definition of relative Rényi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of Müller-Lennert et al.

math-ph↗

arXiv2013

Strichartz inequality for orthonormal functions

We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schrödinger equation in a time-dependent potential and we show the existence of the wave operator in Schatten spaces.

math.AP↗

arXiv2013

Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems

In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo-Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Lieb's Translation Lemma and a Riesz energy version of the Brézis--Lieb lemma.

math.FA↗

arXiv2013

Refined Semiclassical Asymptotics for Fractional Powers of the Laplace Operator

We consider the fractional Laplacian on a domain and investigate the asymptotic behavior of its eigenvalues. Extending methods from semi-classical analysis we are able to prove a two-term formula for the sum of eigenvalues with the leading (Weyl) term given by the volume and the subleading term by the surface area. Our result is valid under very weak assumptions on the regularity of the boundary.

math.SP↗

arXiv2013

Stability estimates for the lowest eigenvalue of a Schrödinger operator

There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger eigenvalue under the constraint of a given L^p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for Hölder's inequality, which we believe to be new.

math.AP↗

arXiv2013

The ground state energy of a polaron in a strong magnetic field

We show that the ground state of a polaron in a homogeneous magnetic field $B$ and its energy are described by an effective one-dimensional minimization problem in the limit $B\to\infty$. This holds both in the linear Fröhlich and in the non-linear Pekar model and makes rigorous an argument of Kochetov, Leschke and Smondyrev.

math-ph↗

arXiv2013

Existence of ground states for negative ions at the binding threshold

As the nuclear charge Z is continuously decreased an N-electron atom undergoes a binding-unbinding transition at some critical Z_c. We investigate whether the electrons remain bound when Z=Z_c and whether the radius of the system stays finite as Z_c is approached. Existence of a ground state at Z_c is shown under the condition Z_c<N-K, where K is the maximal number of electrons that can be removed at Z_c without changing the ground state energy.

math-ph↗

arXiv2012

Ground state properties of multi-polaron systems

We summarize our recent results on the ground state energy of multi-polaron systems. In particular, we discuss stability and existence of the thermodynamic limit, and we discuss the absence of binding in the case of large Coulomb repulsion and the corresponding binding-unbinding transition. We also consider the Pekar-Tomasevich approximation to the ground state energy and we study radial symmetry of the ground state density.

math-ph↗

arXiv2012

Semi-classical analysis of the Laplace operator with Robin boundary conditions

We prove a two-term asymptotic expansion of eigenvalue sums of the Laplacian on a bounded domain with Neumann, or more generally, Robin boundary conditions. We formulate and prove the asymptotics in terms of semi-classical analysis. In this reformulation it is natural to allow the function describing the boundary conditions to depend on the semi-classical parameter and we identify and analyze three different regimes for this dependence.

math.SP↗

arXiv2012

Cwikel's theorem and the CLR inequality

We give a short proof of the Cwikel-Lieb-Rozenblum (CLR) bound on the number of negative eigenvalues of Schrödinger operators. The argument, which is based on work of Rumin, leads to remarkably good constants and applies to the case of operator-valued potentials as well. Moreover, we obtain the general form of Cwikel's estimate about the singular values of operators of the form $f(X) g(-i\nabla)$.

math.SP↗

arXiv2012

Remainder terms in the fractional Sobolev inequality

We show that the fractional Sobolev inequality for the embedding $\H \hookrightarrow L^{\frac{2N}{N-s}}(\R^N)$, $s \in (0,N)$ can be sharpened by adding a remainder term proportional to the distance to the set of optimizers. As a corollary, we derive the existence of a remainder term in the weak $L^{\frac{N}{N-s}}$-norm for functions supported in a domain of finite measure. Our results generalize earlier work for the non-fractional case where $s$ is an even integer.

math.AP↗

arXiv2012

A positive density analogue of the Lieb-Thirring inequality

The Lieb-Thirring inequalities give a bound on the negative eigenvalues of a Schrödinger operator in terms of an $L^p$ norm of the potential. This is dual to a bound on the $H^1$-norms of a system of orthonormal functions. Here we extend these to analogous inequalities for perturbations of the Fermi sea of non-interacting particles, i.e., for perturbations of the continuous spectrum of the Laplacian by local potentials.

math-ph↗

arXiv2012

A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality

We show that the sharp constant in the Hardy-Littlewood-Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the first one to do so for the whole parameter range.

math.FA↗

arXiv2012

Extended quantum conditional entropy and quantum uncertainty inequalities

Quantum states can be subjected to classical measurements, whose incompatibility, or uncertainty, can be quantified by a comparison of certain entropies. There is a long history of such entropy inequalities between position and momentum. Recently these inequalities have been generalized to the tensor product of several Hilbert spaces and we show here how their derivations can be shortened to a few lines and how they can be generalized. All the recently derived uncertainty relations utilize the strong subadditivity (SSA) theorem; our contribution relies on directly utilizing the proof technique of the original derivation of SSA.

quant-ph↗

arXiv2012

Symmetry of bipolaron bound states for small Coulomb repulsion

We consider the bipolaron in the Pekar--Tomasevich approximation and address the question whether the ground state is spherically symmetric or not. Numerical analysis has, so far, not completely settled the question. Our contribution is to prove rigorously that the ground state remains spherical for small values of the electron-electron Coulomb repulsion.

math-ph↗