Rupert L. Frank

Uniqueness of non-linear ground states for fractional Laplacians in {\mathbb{R}}

We prove uniqueness of ground state solutions Q = Q(|x|) > 0 for the nonlinear equation (−∆)Q+Q−Q = 0 in R, where 0 < s < 1 and 0 < α < 4s 1−2s for s < 1/2 and 0 < α < ∞ for s > 1/2. Here (−∆) denotes the fractional Laplacian in one dimension. In particular, we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for s = 1/2 and α = 1 in [Acta Math., 167 (1991), 107–126]. As a technical key result in this paper, we show that the associated linearized operator L+ = (−∆)+1− (α+1)Q is nondegenerate; i. e., its kernel satisfies kerL+ = span {Q}. This result about L+ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for nonlinear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.

Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators

We show that the Lieb-Thirring inequalities on moments of negative eigenvalues of Schrodinger-like operators remain true, with possibly different constants, when the critical Hardy-weight C │x│^(-2) is subtracted from the Laplace operator. We do so by first establishing a Sobolev inequality for such operators. Similar results are true for fractional powers of the Laplacian and the Hardy-weight and, in particular, for relativistic Schrodinger operators. We also allow for the inclusion of magnetic vector potentials. As an application, we extend, for the first time, the proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge Zɑ = 2/π, for ɑ less than some critical value.

Monotonicity of a relative Rényi entropy

We show that a recent definition of relative Renyi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of Muller-Lennert et al. [“On quantum Renyi entropies: A new definition, some properties,” J. Math. Phys. 54, 122203 (2013); e-print arXiv:1306.3142v1; see also e-print arXiv:1306.3142].

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.

Eigenvalue bounds for Schrödinger operators with complex potentials

We show that the absolute values of non-positive eigenvalues of Schrodinger operators with complex potentials can be bounded in terms of L_p-norms of the potential. This extends an inequality of Abramov, Aslanyan and Davies to higher dimensions and proves a conjecture by Laptev and Safronov. Our main ingredient are the uniform Sobolev inequalities of Kenig, Ruiz and Sogge.

The Critical Temperature for the BCS Equation at Weak Coupling

For the BCS equation with local two-body interaction λV(x), we give a rigorous analysis of the asymptotic behavior of the critical temperature as γ»0. We derive necessary and sufficient conditions onV(x) for the existence of a nontrivial solution for all values of γ>0.

Müller’s exchange-correlation energy in density-matrix-functional theory

The increasing interest in the Muller density-matrix-functional theory has led us to a systematic mathematical investigation of its properties. This functional is similar to the Hartree-Fock functional, but with a modified exchange term in which the square of the density matrix (x, x′) is replaced by the square of y^(1/2)(x,x′). After an extensive introductory discussion of densitymatrix-functional theory we show, among other things, that this functional is convex (unlike the HF functional) and that energy minimizing y’s have unique densities p(r), which is a physically desirable property often absent in HF theory. We show that minimizers exist if N ≤ Z, and derive various properties of the minimal energy and the corresponding minimizers. We also give a precise statement about the equation for the orbitals of y, which is more complex than for HF theory. We state some open mathematical questions about the theory together with conjectured solutions.

MICROSCOPIC DERIVATION OF GINZBURG-LANDAU THEORY

We give the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof.

A Simple Proof of Hardy-Lieb-Thirring Inequalities

We give a short and unified proof of Hardy-Lieb-Thirring inequalities for moments of eigenvalues of fractional Schrödinger operators. The proof covers the optimal parameter range. It is based on a recent inequality by Solovej, Sørensen, and Spitzer. Moreover, we prove that any non-magnetic Lieb-Thirring inequality implies a magnetic Lieb-Thirring inequality (with possibly a larger constant).

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.