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Dirk Hundertmark

Publications and source records attributed to Dirk Hundertmark.

At least 37 records · Page 2Linked to original sources
arXiv2018

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 Schrödinger 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 Schrödinger-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.

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arXiv2018

The Lieb-Thirring inequality revisited

We provide new estimates on the best constant of the Lieb-Thirring inequality for the sum of the negative eigenvalues of Schrödinger operators, which significantly improve the so far existing bounds.

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arXiv2018

Nonlinear Schrödinger equation, differentiation by parts and modulation spaces

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in the modulation space $M_{p,q}^{s}(\mathbb R)$ where $1\leq q\leq2$, $2\leq p<\frac{10q'}{q'+6}$ and $s\geq0$. Moreover, for either $1\leq q\leq\frac32, s\geq0$ and $2\leq p\leq 3$ or $\frac32 \frac23-\frac1{q}$ and $2\leq p\leq 3$ or $\frac{18}{11} \frac23-\frac1{q}$ and $2\leq p<\frac{10q'}{q'+6}$ we show that the Cauchy problem is unconditionally wellposed in $M_{p,q}^{s}(\mathbb R).$ This improves \cite{NP}, where the case $p=2$ was considered and the differentiation by parts technique was introduced to a problem with continuous Fourier variable. Here the same technique is used, but more delicate estimates are necessary for $p\neq2$.

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arXiv2017

Solitary waves in nonlocal NLS with dispersion averaged saturated nonlinearities

A nonlinear Schrödinger equation (NLS) with dispersion averaged nonlinearity of saturated type is considered. Such a nonlocal NLS is of integro-differential type and it arises naturally in modeling fiber-optics communication systems with periodically varying dispersion profile (dispersion management). The associated constrained variational principle is shown to posses a ground state solution by constructing a convergent minimizing sequence through the application of a method similar to the classical concentration compactness principle of Lions. One of the obstacles in applying this variational approach is that a saturated nonlocal nonlinearity does not satisfy uniformly the so-called strict sub-additivity condition. This is overcome by applying a special version of Ekeland's variational principle.

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arXiv2017

Quantitative bounds versus existence of weakly coupled bound states for Schrödinger type operators

It is well-known that for usual Schroedinger operators weakly coupled bound states exist in dimensions one and two, whereas in higher dimensions the famous Cwikel-Lieb-Rozenblum bound holds. We show for a large class of Schrödinger-type operators with general kinetic energies that these two phenomena are complementary. In particular, we explicitly get a natural semi-classical type bound on the number of bound states precisely in the situation when weakly coupled bound states exist not.

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arXiv2017

Derivation of the Hartree equation for compound Bose gases in the mean field limit

We consider mixtures of Bose gases of different species. We prove that in the mean field limit and under suitable conditions on the initial condition a system composed of two Bose species can be effectively described by a system of coupled Hartree equations. Moreover, we derive quantitative bounds on the rates of convergence of the reduced density matrices in Sobolev trace norms. We treat both the non-relativistic case in the presence of an external magnetic field $A\in L^2_{\text{loc}}(\R^3;\R^3)$ and the semi-relativistic case.

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arXiv2016

On existence of global solutions of the one-dimensional cubic NLS for initial data in the modulation space $M_{p,q}(\mathbb R)$

We prove global existence for the one-dimensional cubic non-linear Schrödinger equation in modulation spaces $M_{p,p'}$ for $p$ sufficiently close to $2$. In contrast to known results, our result requires no smallness condition on initial data. The proof adapts a splitting method inspired by work of Vargas-Vega and Hyakuna-Tsutsumi to the modulation space setting and exploits polynomial growth of the free Schrödinger group on modulation spaces.

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arXiv2016

Thresholds for Existence of dispersion management solitons for general nonlinearities

We prove a threshold phenomenon for the existence of solitary solutions of the dispersion management equation for positive and zero average dispersion for a large class of nonlinearities. These solutions are found as minimizers of nonlinear and nonlocal variational problems which are invariant under a large non-compact group. There exists a threshold such that minimizers exist when the power of the solitons is bigger than the threshold. Our proof of existence of minimizers is rather direct and avoids the use of Lions' concentration compactness argument. The existence of dispersion managed solitons is shown under very mild conditions on the dispersion profile and the nonlinear polarization of optical active medium, which cover all physically relevant cases for the dispersion profile and a large class of nonlinear polarizations, for example, they are allowed to change sign.

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arXiv2016

Discrete diffraction managed solitons: Threshold phenomena and rapid decay for general nonlinearities

We prove a threshold phenomenon for the existence/non-existence of energy minimizing solitary solutions of the diffraction management equation for strictly positive and zero average diffraction. Our methods allow for a large class of nonlinearities, they are, for example, allowed to change sign, and the weakest possible condition, it only has to be locally integrable, on the local diffraction profile. The solutions are found as minimizers of a nonlinear and nonlocal variational problem which is translation invariant. There exists a critical threshold ?cr such that minimizers for this variational problem exist if their power is bigger than ?cr and no minimizers exist with power less than the critical threshold. We also give simple criteria for the finiteness and strict positivity of the critical threshold. Our proof of existence of minimizers is rather direct and avoids the use of Lions' concentration compactness argument. Furthermore, we give precise quantitative lower bounds on the exponential decay rate of the diffraction management solitons, which confirm the physical heuristic prediction for the asymptotic decay rate. Moreover, for ground state solutions, these bounds give a quantitative lower bound for the divergence of the exponential decay rate in the limit of vanishing average diffraction. For zero average diffraction, we prove quantitative bounds which show that the solitons decay much faster than exponentially. Our results considerably extend and strengthen the results of [15] and [16].

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arXiv2015

Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction

We study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a logarithmic singularity of the collision kernel for grazing collisions. Even though in this situation the Boltzmann operator enjoys only a very weak coercivity estimate, it still leads to strong smoothing of weak solutions in accordance to the smoothing expected by an analogy with a logarithmic heat equation.

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arXiv2015

Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules

It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian. In particular, the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in $L^1_2(\mathbb{R}^d)\cap L\log L(\mathbb{R}^d)$, i.e., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molecules.

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arXiv2013

Exponential Decay for dispersion managed solitons for general dispersion profiles

We show that any weak solution of the Gabitov-Turitsyn equation describing dispersion managed solitons together with its Fourier transform decay exponentially. This strong regularity result extends a recent result of Erdogan, Hundertmark and Lee in two directions, to arbitrary non-negative average dispersion and, more importantly, to rather general dispersion profiles, which cover most, if not all, physically relevant cases.

math.AP↗
arXiv2010

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.

math.AP↗
arXiv2010

On non-local variational problems with lack of compactness related to non-linear optics

We give a simple proof of existence of solutions of the dispersion manage- ment and diffraction management equations for zero average dispersion, respectively diffraction. These solutions are found as maximizers of non-linear and non-local vari- ational problems which are invariant under a large non-compact group. Our proof of existence of maximizer is rather direct and avoids the use of Lions' concentration compactness argument or Ekeland's variational principle.

math-ph↗
arXiv2008

Heat-flow monotonicity of Strichartz norms

Most notably we prove that for $d=1,2$ the classical Strichartz norm $$\|e^{i sΔ}f\|_{L^{2+4/d}_{s,x}(\mathbb{R}\times\mathbb{R}^d)}$$ associated to the free Schrödinger equation is nondecreasing as the initial datum $f$ evolves under a certain quadratic heat-flow.

math.CA↗
arXiv2008

Super-exponential decay of Diffraction Managed Solitons

This is the second part of a series of papers where we develop rigorous decay estimates for breather solutions of an averaged version of the non-linear Schrödinger equation. In this part we study the diffraction managed discrete non-linear Schrödinger equation, an equation which describes coupled waveguide arrays with periodic diffraction management geometries. We show that, for vanishing average diffraction, all solutions of the non-linear and non-local diffraction management equation decay super-exponentially. As a byproduct of our method, we also have a simple proof of existence of diffraction managed solitons in the case of vanishing average diffraction.

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