Robert Seiringer

Publications (69)42.23 Total impact

• Article: Translation-invariant quasi-free states for fermionic systems and the BCS approximation

  Gerhard Bräunlich, Christian Hainzl, Robert Seiringer
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  ABSTRACT: We study translation-invariant quasi-free states for a system of fermions with two-particle interactions. The associated energy functional is similar to the BCS functional but includes also direct and exchange energies. We show that for suitable short-range interactions, these latter terms only lead to a renormalization of the chemical potential, with the usual properties of the BCS functional left unchanged. Our analysis thus represents a rigorous justification of part of the BCS approximation. We give bounds on the critical temperature below which the system displays superfluidity.
  05/2013;
• Article: On contact interactions as limits of short-range potentials

  Gerhard Bräunlich, Christian Hainzl, Robert Seiringer
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  ABSTRACT: We reconsider the norm resolvent limit of $-\Delta + V_\ell$ with $V_\ell$ tending to a point interaction in three dimensions. We are mainly interested in potentials $V_\ell$ modelling short range interactions of cold atomic gases. In order to ensure stability the interaction $V_\ell$ is required to have a strong repulsive core, such that $\lim_{\ell \to 0} \int V_\ell >0$. This situation is not covered in the previous literature.
  05/2013;
• Article: Equivalence of two definitions of the effective mass of a polaron

  Elliott H. Lieb, Robert Seiringer
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  ABSTRACT: Two definitions of the effective mass of a particle interacting with a quantum field, such as a polaron, are considered and shown to be equal in models similar to the Froehlich polaron model. These are: 1. the mass defined by the low momentum energy $E(P) \approx E(0)+P^2/2M$ of the translation invariant system constrained to have momentum $P$ and 2. the mass $M$ of a simple particle in an arbitrary slowly varying external potential, $V$, described by the nonrelativistic Schroedinger equation, whose ground state energy equals that of the combined particle/field system in a bound state in the same $V$.
  04/2013;
• Article: Existence of ground states for negative ions at the binding threshold

  Jacopo Bellazzini, Rupert L. Frank, Elliott H. Lieb, Robert Seiringer
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  ABSTRACT: As the nuclear charge Z is continuously decreased an N-electron atom undergoes a binding-unbinding transition at some critical Z_c. We investigate whether the electrons remain bound when Z=Z_c and whether the radius of the system stays finite as Z_c is approached. Existence of a ground state at Z_c is shown under the condition Z_c<N-K, where K is the maximal number of electrons that can be removed at Z_c without changing the ground state energy.
  01/2013;
• Article: Disordered Bose Einstein Condensates with Interaction

  Robert Seiringer, Jakob Yngvason, Valentin A. Zagrebnov
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  ABSTRACT: We study the effects of random scatterers on the ground state of the one-dimensional Lieb-Liniger model of interacting bosons on the unit interval in the Gross-Pitaevskii regime. We prove that Bose Einstein condensation survives even a strong random potential with a high density of scatterers. The character of the wave function of the condensate, however, depends in an essential way on the interplay between randomness and the strength of the two-body interaction. For low density of scatterers or strong interactions the wave function extends over the whole interval. High density of scatterers and weak interaction, on the other hand, leads to localization of the wave function in a fragmented subset of the interval.
  09/2012;
• Article: Ground state properties of multi-polaron systems

  Rupert L. Frank, Elliott H. Lieb, Robert Seiringer, Lawrence E. Thomas
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  ABSTRACT: We summarize our recent results on the ground state energy of multi-polaron systems. In particular, we discuss stability and existence of the thermodynamic limit, and we discuss the absence of binding in the case of large Coulomb repulsion and the corresponding binding-unbinding transition. We also consider the Pekar-Tomasevich approximation to the ground state energy and we study radial symmetry of the ground state density.
  09/2012;
• Article: Microscopic Derivation of the Ginzburg-Landau Model

  Rupert L. Frank, Christian Hainzl, Robert Seiringer, Jan Philip Solovej
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  ABSTRACT: We present a summary of our recent 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.
  09/2012;
• Article: Further implications of the Bessis-Moussa-Villani conjecture

  Elliott H. Lieb, Robert Seiringer
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  ABSTRACT: We find further implications of the BMV conjecture, which states that for hermitian matrices A and B, the function Tr exp(A - t B) is the Laplace transform of a positive measure.
  06/2012;
• Source

  Available from: ArXiv

  Article: Symmetry of bipolaron bound states for small Coulomb repulsion

  Rupert L. Frank, Elliott H. Lieb, Robert Seiringer
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  ABSTRACT: We consider the bipolaron in the Pekar--Tomasevich approximation and address the question whether the ground state is spherically symmetric or not. Numerical analysis has, so far, not completely settled the question. Our contribution is to prove rigorously that the ground state remains spherical for small values of the electron-electron Coulomb repulsion.
  01/2012;
• Source

  Available from: ArXiv

  Article: Lieb-Thirring Inequality for a Model of Particles with Point Interactions

  Rupert L. Frank, Robert Seiringer
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  ABSTRACT: We consider a model of quantum-mechanical particles interacting via point interactions of infinite scattering length. In the case of fermions we prove a Lieb-Thirring inequality for the energy, i.e., we show that the energy is bounded from below by a constant times the integral of the particle density to the power 5/3.
  12/2011;
• Source

  Available from: Mathieu Lewin

  Article: A positive density analogue of the Lieb-Thirring inequality

  Rupert L. Frank, Mathieu Lewin, Elliott H. Lieb, Robert Seiringer
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  ABSTRACT: The Lieb-Thirring inequalities give a bound on the negative eigenvalues of a Schr\"odinger operator in terms of an $L^p$ norm of the potential. This is dual to a bound on the $H^1$-norms of a system of orthonormal functions. Here we extend these to analogous inequalities for perturbations of the Fermi sea of non-interacting particles, i.e., for perturbations of the continuous spectrum of the Laplacian by local potentials.
  08/2011;
• Source

  Available from: ArXiv

  Article: The BCS gap equation for spin-polarized fermions

  Abraham Freiji, Christian Hainzl, Robert Seiringer
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  ABSTRACT: We study the BCS gap equation for a Fermi gas with unequal population of spin-up and spin-down states. For $\cosh(\delta_\mu/T) \leq 2$, with $T$ the temperature and $\delta_\mu$ the chemical potential difference, the question of existence of non-trivial solutions can be reduced to spectral properties of a linear operator, similar to the unpolarized case studied previously in \cite{FHNS,HHSS,HS}. For $\cosh(\delta_\mu/T) > 2$ the phase diagram is more complicated, however. We derive upper and lower bounds for the critical temperature, and study their behavior in the small coupling limit.
  07/2011;
• Source

  Available from: ArXiv

  Article: Binding of Polarons and Atoms at Threshold

  Rupert L. Frank, Elliott H. Lieb, Robert Seiringer
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  ABSTRACT: If the polaron coupling constant $\alpha$ is large enough, bipolarons or multi-polarons will form. When passing through the critical $\alpha_c$ from above, does the radius of the system simply get arbitrarily large or does it reach a maximum and then explodes? We prove that it is always the latter. We also prove the analogous statement for the Pekar-Tomasevich (PT) approximation to the energy, in which case there is a solution to the PT equation at $\alpha_c$. Similarly, we show that the same phenomenon occurs for atoms, e.g., helium, at the critical value of the nuclear charge. Our proofs rely only on energy estimates, not on a detailed analysis of the Schr\"odinger equation, and are very general. They use the fact that the Coulomb repulsion decays like $1/r$, while `uncertainty principle' localization energies decay more rapidly, as $1/r^2$.
  06/2011;
• Source

  Available from: ArXiv

  Article: Low Density Limit of BCS Theory and Bose-Einstein Condensation of Fermion Pairs

  Christian Hainzl, Robert Seiringer
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  ABSTRACT: We consider the low density limit of a Fermi gas in the BCS approximation. We show that if the interaction potential allows for a two-particle bound state, the system at zero temperature is well approximated by the Gross-Pitaevskii functional, describing a Bose-Einstein condensate of fermion pairs.
  05/2011;
• Source

  Available from: ArXiv

  Article: Energy cost to make a hole in the Fermi sea.

  Rupert L Frank, Mathieu Lewin, Elliott H Lieb, Robert Seiringer
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  ABSTRACT: The change in energy of an ideal Fermi gas when a local one-body potential is inserted into the system, or when the density is changed locally, are important quantities in condensed matter physics. We show that they can be rigorously bounded from below by a universal constant times the value given by the semiclassical approximation.
  Physical Review Letters 04/2011; 106(15):150402. · 7.37 Impact Factor
• Source

  Available from: ArXiv

  Article: Derivation of Ginzburg-Landau theory for a one-dimensional system with contact interaction

  Rupert L. Frank, Christian Hainzl, Robert Seiringer, Jan Philip Solovej
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  ABSTRACT: In a recent paper we give the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Here we present our results in the simplified case of a one-dimensional system of particles interacting via a delta-potential.
  03/2011;
• Source

  Available from: ArXiv

  Article: Microscopic Derivation of Ginzburg-Landau Theory

  Rupert L. Frank, Christian Hainzl, Robert Seiringer, Jan Philip Solovej
  [show abstract] [hide abstract]
  ABSTRACT: 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.
  02/2011;
• Source

  Available from: ArXiv

  Article: Binding, Stability, and Non-binding of Multi-polaron Systems

  Rupert L. Frank, Elliott H. Lieb, Robert Seiringer, Lawrence E. Thomas
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  ABSTRACT: The binding of polarons, or its absence, is an old and subtle topic. After defining the model we state some recent theorems of ours. First, the transition from many-body collapse to the existence of a thermodynamic limit for N polarons occurs precisely at U=2\alpha, where U is the electronic Coulomb repulsion and \alpha is the polaron coupling constant. Second, if U is large enough, there is no multi-polaron binding of any kind. We also discuss the Pekar-Tomasevich approximation to the ground state energy, which is valid for large \alpha. Finally, we derive exact results, not reported before, about the one-dimensional toy model introduced by E. P. Gross. Comment: 12 pages; contribution to the proceedings of the conference QMath 11 (Hradec Kralove, September 2010); clarification added after Theorem 4.1
  10/2010;
• Article: Bipolaron and N-polaron binding energies.

  Rupert L Frank, Elliott H Lieb, Robert Seiringer, Lawrence E Thomas
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  ABSTRACT: The binding of polarons, or its absence, is an old and subtle topic. Here we prove two things rigorously. First, the transition from many-body collapse to the existence of a thermodynamic limit for N polarons occurs precisely at U=2α, where U is the electronic Coulomb repulsion and α is the polaron coupling constant. Second, if U is large enough, there is no multipolaron binding of any kind. Considering the known fact that there is binding for some U>2α, these conclusions are not obvious and their proof has been an open problem for some time.
  Physical Review Letters 05/2010; 104(21):210402. · 7.37 Impact Factor
• Source

  Available from: ArXiv

  Article: Stability and Absence of Binding for Multi-Polaron Systems

  Rupert L. Frank, Elliott H. Lieb, Robert Seiringer, Lawrence E. Thomas
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  ABSTRACT: We resolve several longstanding problems concerning the stability and the absence of multi-particle binding for N\geq 2 polarons. Fr\"ohlich's 1937 polaron model describes non-relativistic particles interacting with a scalar quantized field with coupling \sqrt\alpha, and with each other by Coulomb repulsion of strength U. We prove the following: (i) While there is a known thermodynamic instability for U<2\alpha, stability of matter does hold for U>2\alpha, that is, the ground state energy per particle has a finite limit as N\to\infty. (ii) There is no binding of any kind if U exceeds a critical value that depends on \alpha but not on N. The same results are shown to hold for the Pekar-Tomasevich model.
  04/2010;