arXiv subjects

Dirk Hundertmark

Publications and source records attributed to Dirk Hundertmark.

At least 19 recordsLinked to original sources

arXiv2026

On Analyticity of Solitons for the Periodic Dispersion-Managed Nonlinear Schr\"odinger Equation

We explore the regularity of energy maximizers for the Lagrangian of a periodic dispersion managed fiber optic at fixed intensity, on a torus of length $L$ and with vanishing average dispersion. We show that the Fourier coefficients decay at a polynomial rate, and then upgrade this to exponential decay, so that the maximizers are analytic in space for large enough $L$. In addition, by an asymptotic comparison to the optimizers on the real line, we prove that the solutions are non-trivial. We also consider a conjecture that the maximizer necessarily has an underlying symmetry inherent to both the energy functional and the resulting Euler-Lagrange equation. All of our results are supported with illustrative numerical experiments.

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arXiv2026

Intrinsic Ultracontractivity for a class of Schroedinger Semigroups in $L^{2}(\mathbb{R}^{n})$ by Logarithmic Sobolev inequalities

In the first part of this article we present a growth condition on the potential $q$ in the Schr\"odinger operator $H=-\Delta + q(x)$ in $\mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ that implies Rosen inequalities for the ground state $\varphi$ of $H$, i.e. $\forall \varepsilon > 0 \exists \gamma(\varepsilon) > 0 \ : \ - \ln\left( \varphi(x) \right) \leq \varepsilon q(x) + \gamma(\varepsilon)$. While these inequalities are not particularly interesting in themselves, they offer Logarithmic Sobolev inequalities which are absolutely essential to prove an intrinsic ultracontractivity of the associated Schr\"odinger semigroup $\mathrm{e}^{-tH}$, i.e. $\forall t>0 \exists C_{t} > 0 \ : \ \left| \mathrm{e}^{-tH} u (x) \right| \ \leq \ C_{t} \varphi(x) \| u \|_{2}$ holds for every $u \in \mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ almost everywhere in $\mathbb{R}^{n}$ which we prove in the second part of this article. For proving Rosen inequalities we focus on solving a radial Schr\"odinger inequality and use Agmon's version of the comparison principle and Young's inequality for increasing functions. We follow the classic method proving intrinsic ultracontractivity of $\mathrm{e}^{-tH}$ by using weighted Sobolev function spaces, weighted Schr\"odinger semigroups and Logarithmic Sobolev inequalities.

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arXiv2025

On the Excess Charge Problem of Atoms

This paper establishes new bounds on the maximum number of electrons $ N_c(Z) $ that an atom with nuclear charge $Z$ can bind. Specifically, we show that \begin{equation*} N_c(Z) < 1.1185Z + O(Z^{1/3}) \end{equation*} with an explicit bound on the lower order term $O(Z^{1/3})$. This result improves long--standing bounds by Lieb and Nam obtained in 1984, respectively 2012. Our bounds show the fundamental difference between fermionic and bosonic atoms for finite $Z$ since for bosonic atoms it is known that $\lim N_c(Z)/Z = t_c \approx 1.21$ in the limit of large nuclear charges $Z$.

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arXiv2024

Why a System of Three Bosons on Separate Lines Can Not Exhibit the Confinement Induced Efimov Effect

We study a system of three bosons interacting with short-range potentials which can move along three different lines. Two of these lines are parallel to each other within one plane. The third line is constrained to a plane perpendicular to the first one. Recently it was predicted in physics literature that such a system exhibits the so-called confinement induced Efimov effect. We prove that this prediction is not correct by showing that this system has at most finitely many bound-states.

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arXiv2023

Absence of embedded eigenvalues of Pauli and Dirac operators

We consider eigenvalues of the Pauli operator in $\mathbb R^3$ embedded in the continuous spectrum. In our main result we prove the absence of such eigenvalues above a threshold which depends on the asymptotic behavior of the magnetic and electric field at infinity. We show moreover that the decay conditions on the magnetic and electric field are sharp. Analogous results are obtained for purely magnetic Dirac operators.

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arXiv2022

Quantum Systems at The Brink

We present a method to calculate the asymptotic behavior of eigenfunctions of Schr\"odinger operators that also works at the threshold of the essential spectrum. It can be viewed as a higher order correction to the well-known WKB method which does need a safety distance to the essential spectrum. We illustrate its usefulness on examples of quantum particles in a potential well with a long-range repulsive term outside the well.

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arXiv2021

Quantum Systems at the Brink: Existence of Bound States, Critical Potentials and Dimensionality

One of the crucial properties of a quantum system is the existence of bound states. While the existence of eigenvalues below zero, i.e., below the essential spectrum, is well understood, the situation of zero energy bound states at the edge of the essential spectrum is far less understood. We present necessary and sufficient conditions for Schr\"odinger operators to have a zero energy bound state. Our sharp criteria show that the existence and non-existence of zero energy ground states depends strongly on the dimension and the asymptotic behavior of the potential. There is a spectral phase transition with dimension four being critical.

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arXiv2020

Quantum Systems at The Brink: Properties of Atomic Bound States at The Ionization Threshold

We give a rigorous argument that long--range repulsion stabilizes quantum systems; ground states of such quantum systems exist even when the ground state energy is precisely at the ionization threshold. For atomic systems at the critical nuclear charge, our bounds show that the ground state falls off like $\exp(-c\sqrt{|x|})$ for large $|x|$. This is much slower than what the WKB method predicts for bound states with energies strictly below the ionization threshold. For helium type systems at critical nuclear charge, we show that our upper bounds are sharp. This rigorously confirms predictions by quantum chemists.

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arXiv2020

On the asymptotic dynamics of 2-D magnetic quantum systems

In this work we provide results on the long time localisation in space (dynamical localisation) of certain two-dimensional magnetic quantum systems. The underlying Hamiltonian may have the form $H=H_0+W$, where $H_0$ is rotationally symmetric, has dense point spectrum, and $W$ is a perturbation that breaks the rotational symmetry. In the latter case, we also give estimates for the growth of the angular momentum operator in time.

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arXiv2020

On the van der Waals interaction between a molecule and a half-infinite plate

We consider a molecule in the Born-Oppenheimer approximation interacting with a plate of infinite thickness, i.e, a half--space, which is perfectly conducting or dielectric. It is well--known in the physics literature that in this case the atom or molecule is attracted by the plate at sufficiently large distances. This effect is analogous to the well--known van der Waals interaction between neutral atoms or molecules. We prove that the interaction energy $W$ of the system is given by $W(r,v)= -C(v)r^{-3} + \mathcal{O}(r^{-4})$, where $r$ is the distance between the molecule and the plate and $v$ indicates their relative orientation. Moreover, $C(v)$ is positive and continuous, thus the atom or molecule is always pulled towards the plate at sufficiently large distances, for all relative orientations $v$. For some specific systems we provide sharper estimates of $W(r,v)$. This asymptotic behavior is well--known in the physics literature, however, we are not aware of any previous rigorous results, even on existence of a ground state of the system. For pedagogical reasons, we often start with the case of a hydrogen atom and then we generalize the arguments to deal with a general molecule.

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arXiv2020

Well-posedness of dispersion managed nonlinear Schr\"odinger equations

We prove local and global well-posedness results for the Gabitov-Turitsyn or dispersion managed nonlinear Schr\"odinger equation with a large class of nonlinearities and arbitrary average dispersion on $L^2(\mathbb{R})$ and $H^1(\mathbb{R})$. Moreover, when the average dispersion is non-negative, we show that the set of ground states is orbitally stable. This covers the case of non-saturated and saturated nonlinear polarizations and yields, for saturated nonlinearities, the first proof of orbital stability.

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arXiv2020

Absence of positive eigenvalues of magnetic Schr\"odinger operators

We study sufficient conditions for the absence of positive eigenvalues of magnetic Schr\"odinger operators in $\mathbb{R}^d,\, d\geq 2$. In our main result we prove the absence of eigenvalues above certain threshold energy which depends explicitly on the magnetic and electric field. A comparison with the examples of Miller--Simon shows that our result is sharp as far as the decay of the magnetic field is concerned. As applications, we describe several consequences of the main result for two-dimensional Pauli and Dirac operators, and two and three dimensional Aharonov--Bohm operators.

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arXiv2019

Quantum Systems at The Brink. Existence and Decay Rates of Bound States at Thresholds; Atoms

It is well known that $N$-electron atoms undergoes unbinding for a critical charge of the nucleus $Z_c$, i.e. the atom has eigenstates for the case $Z> Z_c$ and it has no bound states for $Z<Z_c$. In the present paper we derive upper bound for the bound state for the case $Z=Z_c$ under the assumption $Z_c<N-K$ where $K$ is the number of electrons to be removed for atom to be stable for $Z=Z_c$ without any change in the ground state energy. We show that the eigenvector decays faster as $\exp\left(-C\sum\sqrt{|x|_{k}}\right)$ where we sum K largest values of $|x_j|$, $j\in\{1,\ldots,N\}$. Our method do not require Born-Oppenheimer approximation.

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arXiv2019

Quantum Systems at The Brink: Helium-type systems

In the present paper we study two challenging problems for helium-type systems. Existence of eigenvalues at thresholds and the asymptotic behavior of the corresponding eigenfunctions. Since the usual methods for addressing these problems need a safety distance to the essential spectrum, they cannot be applied in critical cases, when an eigenvalue enters the continuum. We develop a method to address both problems and derive sharp upper and lower bounds for the asymptotic behavior of the ground state of critical helium-type systems at the threshold of the essential spectrum. This is the first proof of the precise asymptotic behavior of the ground state for this benchmark problem in quantum chemistry. Moreover, our bounds describe precisely how the asymptotic decay of the ground state changes, when the system becomes critical. In addition, we show the existence of a ground state of this quantum critical system with a finite nuclear mass. Previously this had been known only in the Born-Oppenheimer approximation of infinite nuclear mass.

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arXiv2019

On the global well-posedness of the quadratic NLS on $L^2(\mathbb{R}) + H^1(\mathbb{T})$

We study the one dimensional nonlinear Schr\"odinger equation with power nonlinearity $|u|^{\alpha - 1} u$ for $\alpha \in [1,5]$ and initial data $u_0 \in L^2(\mathbb{R}) + H^1(\mathbb{T})$. We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity ($\alpha = 2$) we obtain global well-posedness in the space $C(\mathbb{R}, L^2(\mathbb R) + H^1(\mathbb T))$ via Gronwall's inequality.

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arXiv2019

Knocking out teeth in one-dimensional periodic NLS

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in one dimension with initial data $u_{0}$ in $H^{s_{1}}(\mathbb R)+H^{s_{2}}(\mathbb T), 0\leq s_{1}\leq s_{2}.$ In addition, we show that if $u_{0}\in H^{s}(\mathbb R)+H^{\frac12+ε}(\mathbb T)$ where $ε>0$ and $\frac16\leq s\leq\frac12$ the solution is unique in $H^{s}(\mathbb R)+H^{\frac12+ε}(\mathbb T).$ Our main tool is a normal form type reduction via the use of the differentiation by parts technique.

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arXiv2019

Van der Waals-London interaction of atoms with pseudo-relativistic kinetic energy

We consider a multiatomic system where the nuclei are assumed to be point charges at fixed positions. Particles interact via Coulomb potential and electrons have pseudo-relativistic kinetic energy. We prove the van der Waals-London law, which states that the interaction energy between neutral atoms decays as the sixth power of the distance $|D|$ between the atoms. We rigorously compute all the terms in the binding energy up to the order $|D|^{-9}$ with error term of order $\mathcal{O}(|D|^{-10})$ . As intermediate steps we prove exponential decay of eigenfunctions of multiparticle Schr\"odinger operators with permutation symmetry imposed by the Pauli principle and new estimates of the localization error.

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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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