Analyticity of extremisers to the Airy Strichartz inequality

We prove that there exists an extremal function to the Airy Strichartz inequality, $e^{-t\partial_x^3}: L^2(\mathbb{R})\to L^8_{t,x}(\mathbb{R}^2)$ by using the linear profile decomposition. Furthermore we show that, if $f$ is an extremiser, then $f$ is extremely fast decaying in Fourier space and so $f$ can be extended to be an entire function on the whole complex domain. The rapid decay of the Fourier transform of extremisers is established with a bootstrap argument which relies on a refined bilinear Airy Strichartz estimate and a weighted Strichartz inequality.Comment: 18 page

Exponential decay of eigenfunctions and generalized eigenfunctions of a non self-adjoint matrix Schr\"odinger operator related to NLS

We study the decay of eigenfunctions of the non self-adjoint matrix operator \calH = (\begin{smallmatrix} -\Delta +\mu+U & W \W & \Delta -\mu -U \end{smallmatrix}), for $\mu>0$, corresponding to eigenvalues in the strip -\mu<\re E <\mu.Comment: 16 page

Classical magnetic Lifshits tails in three space dimensions: impurity potentials with slow anisotropic decay

We determine the leading low-energy fall-off of the integrated density of states of a magnetic Schroedinger operator with repulsive Poissonian random potential in case its single impurity potential has a slow anisotropic decay at infinity. This so-called magnetic Lifshits tail is shown to coincide with the one of the corresponding classical integrated density of states

The Fate of Lifshitz Tails in Magnetic Fields

We investigate the integrated density of states of the Schr\"odinger operator in the Euclidean plane with a perpendicular constant magnetic field and a random potential. For a Poisson random potential with a non-negative algebraically decaying single-impurity potential we prove that the leading asymptotic behaviour for small energies is always given by the corresponding classical result in contrast to the case of vanishing magnetic field. We also show that the integrated density of states of the operator restricted to the eigenspace of any Landau level exhibits the same behaviour. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a delta-function impurity potential.Comment: 19 pages LaTe

Variational Estimates for Discrete Schr\"odinger Operators with Potentials of Indefinite Sign

Let $H$ be a one-dimensional discrete Schr\"odinger operator. We prove that if \sigma_{\ess} (H)\subset [-2,2], then $H-H_0$ is compact and \sigma_{\ess}(H)=[-2,2]. We also prove that if $H_0 + \frac14 V^2$ has at least one bound state, then the same is true for $H_0 +V$. Further, if $H_0 + \frac14 V^2$ has infinitely many bound states, then so does $H_0 +V$. Consequences include the fact that for decaying potential $V$ with $\liminf_{|n|\to\infty} |nV(n)| > 1$, $H_0 +V$ has infinitely many bound states; the signs of $V$ are irrelevant. Higher-dimensional analogues are also discussed.Comment: 17 page

Cwikel's bound reloaded

There are a couple of proofs by now for the famous Cwikel--Lieb--Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schr\"odinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel's proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years. It turns out that this common belief, i.e, Cwikel's approach yields bad constants, is not set in stone: We give a drastic simplification of Cwikel's original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible and leads to rather precise bounds for a large class of Schr\"odinger-type operators with generalized kinetic energies. Moreover, it highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis.Comment: 30 page