Papers by Dirk Hundertmark
Springer eBooks, 2002
We give a proof of the Lieb-Thirring inequality in the critical case d = 1, γ = 1/2, which yields... more We give a proof of the Lieb-Thirring inequality in the critical case d = 1, γ = 1/2, which yields the best possible constant.
Journal of Statistical Physics, Jul 1, 1995
We investigate the integrated density of states of the Schrödinger operator in the Euclidean plan... more We investigate the integrated density of states of the Schrödinger 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.
Documenta Mathematica, 2017
The Maxwell-Boltzmann functional equation has recently attraction renewed interest since besides ... more The Maxwell-Boltzmann functional equation has recently attraction renewed interest since besides its importance in Boltzmann's kinetic theory of gases it also characterizes maximizers of certain bilinear estimates for solutions of the free Schrödinger equation. In this note we give a short and simple proof that, under some mild growth restrictions, any measurable complex-valued solution of the Maxwell-Boltzmann equation is a Gaussian. This covers most, if not all, of the applications.
Journal of Applied Physics, Jul 16, 2021
The fundamental sensitivity limit of atomic force microscopy is strongly correlated to the therma... more The fundamental sensitivity limit of atomic force microscopy is strongly correlated to the thermal noise of the cantilever oscillation. A method to suppress this unwanted noise is to reduce the bandwidth of the measurement, but this approach is limited by the speed of the measurement and the width of the cantilever resonance, commonly defined through the quality factor Q. However, it has been shown that optomechanical resonances in interferometers might affect the cantilever oscillations resulting in an effective quality factor Q eff. When the laser power is sufficiently increased the cantilever oscillations might even reach the regime of self-oscillation. In this self-oscillation state, the noise of the system is partially determined by the interaction with the laser light far from equilibrium. Here, we show and discuss how tuning of the laser power leads to nonlinear optomechanical effects that can dramatically increase the effective quality factor of the cantilever leading to out-of-equilibrium noise. We model the effects using a fourth order nonlinearity of the damping coefficient.
Trends in mathematics, 2020
The original version of the chapter was inadvertently published with an error in the abstract. On... more The original version of the chapter was inadvertently published with an error in the abstract. On line 6 of the abstract, the name 'Okoudjou' was misspelled as 'Oboudjou'. The chapter has now been corrected and approved by the author.
Communications in Mathematical Physics, May 1, 2000
It was shown in [2] that the energy of the relativistic electron-positron field interacting via a... more It was shown in [2] that the energy of the relativistic electron-positron field interacting via a second quantized Coulomb potential in Hartree-Fock approximation is positive provided the fine structure constant is not bigger than 4/π. We show that it is instable if the fine structure constant is above this critical value. We do this partially by adapting an idea of Chaix, Iracane, and Lions [4] leading to an instability problem similar to the one of the no-pair hydrogen hamiltonian [5].
Journal of Functional Analysis, Dec 1, 2003
For Jacobi matrices with a n = 1 + (−1) n αn −γ , b n = (−1) n βn −γ , we study bound states and ... more For Jacobi matrices with a n = 1 + (−1) n αn −γ , b n = (−1) n βn −γ , we study bound states and the Szegő condition. We provide a new proof of Nevai's result that if γ > 1 2 , the Szegő condition holds, which works also if one replaces (−1) n by cos(µn). We show that if α = 0, β = 0, and γ < 1 2 , the Szegő condition fails. We also show that if γ = 1, α and β are small enough (β 2 + 8α 2 < 1 24 will do), then the Jacobi matrix has finitely many bound states (for α = 0, β large, it has infinitely many).
Oxford University Press eBooks, Jul 24, 2008
We give short introduction to some aspects of the theory of Anderson localization.
Physica D: Nonlinear Phenomena, Oct 1, 2017
The averaged dispersion managed nonlinear Schrödinger equation with saturated nonlinearity is con... more The averaged dispersion managed nonlinear Schrödinger equation with saturated nonlinearity is considered. It is shown that under rather general assumptions on the saturated nonlinearity, the ground state solution corresponding to the dispersion managed soliton can be found for both zero residual dispersion and positive residual dispersion. The same applies to diffraction management solitons, which are a discrete version describing certain waveguide arrays.
Siam Journal on Mathematical Analysis, 2017
We prove a threshold phenomenon for the existence of solitary solutions of the dispersion managem... more 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. Contents 1. Introduction 1 1.1. The variational problems 1 1.2. The connection with nonlinear optics 5 2. Nonlinear estimates 8 2.1. Fractional Bilinear Estimates 8 2.2. Splitting the nonlocal nonlinearity 3. Strict subadditivity of the ground state energy 4. The existence proof 5. Threshold phenomena 6. Nonexistence Appendix A. Tightness and strong convergence in L 2 Appendix B. Galilei transformations and space-time localization properties of Gaussian coherent states References
Letters in Mathematical Physics, Nov 27, 2015
We show that any weak solution of the dispersion management equation describing dispersion manage... more We show that any weak solution of the dispersion management 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.
Crelle's Journal, Jan 7, 2004
The diamagnetic inequality for the magnetic Schrö dinger semigroup is extended to the di¤erence o... more The diamagnetic inequality for the magnetic Schrö dinger semigroup is extended to the di¤erence of the semigroups of magnetic Schrö dinger operators with Neumann and Dirichlet boundary conditions on arbitrary open domains and rather general magnetic vector potentials A and potentials V. In particular, this bound renders moot all the technical issues in the recent proofs of the independence of the boundary conditions for the integrated density of states for magnetic Schrö dinger operators: Independence of the boundary conditions for the free case, that is, for vanishing potentials and vector potentials, immediately implies independence of the boundary conditions of the integrated density of states for a large class of magnetic Schrö dinger operators.
arXiv (Cornell University), Jul 29, 2021
One of the crucial properties of a quantum system is the existence of bound states. While the exi... more 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 Schrödinger 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. CONTENTS 1. Introduction 1 2. Definitions and preparations 7 3. Proof of the non-existence result 13 4. Proof of the existence result 16 Appendix A. An example in search of a theorem 22 References 27
Archive for Rational Mechanics and Analysis, Apr 12, 2017
It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properti... more 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 (R d) ∩ L log L(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.
arXiv (Cornell University), Aug 27, 2018
We provide new estimates on the best constant of the Lieb-Thirring inequality for the sum of the ... more 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.
Advances in Theoretical and Mathematical Physics, 1998
We give a proof of the Lieb-Thirring inequality in the critical case d = 1, 7 = 1/2, which yields... more We give a proof of the Lieb-Thirring inequality in the critical case d = 1, 7 = 1/2, which yields the best possible constant.
arXiv (Cornell University), Oct 31, 2016
It is well-known that for usual Schrödinger operators weakly coupled bound states exist in dimens... more It is well-known that for usual Schrödinger 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.
arXiv (Cornell University), Dec 23, 2020
We give a rigorous argument that long-range repulsion stabilizes quantum systems; ground states o... more 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 |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.
Kinetic and Related Models, 2017
We study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a log... more 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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Papers by Dirk Hundertmark