33 research outputs found

### Existence of the D0-D4 Bound State: a detailed Proof

We consider the supersymmetric quantum mechanical system which is obtained by
dimensionally reducing d=6, N=1 supersymmetric gauge theory with gauge group
U(1) and a single charged hypermultiplet. Using the deformation method and
ideas introduced by Porrati and Rozenberg, we present a detailed proof of the
existence of a normalizable ground state for this system

### Microscopic Derivation of the Ginzburg-Landau Model

We present a summary of our recent rigorous derivation of the celebrated Ginzburgâ€“Landau (GL) theory, starting from the microscopic Bardeenâ€“Cooperâ€“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

### Onsager's Inequality, the Landau-Feynman Ansatz and Superfluidity

We revisit an inequality due to Onsager, which states that the (quantum)
liquid structure factor has an upper bound of the form (const.) x |k|, for not
too large modulus of the wave vector k. This inequality implies the validity of
the Landau criterion in the theory of superfluidity with a definite, nonzero
critical velocity. We prove an auxiliary proposition for general Bose systems,
together with which we arrive at a rigorous proof of the inequality for one of
the very few soluble examples of an interacting Bose fluid, Girardeau's model.
The latter proof demonstrates the importance of the thermodynamic limit of the
structure factor, which must be taken initially at k different from 0. It also
substantiates very well the heuristic density functional arguments, which are
also shown to hold exactly in the limit of large wave-lengths. We also briefly
discuss which features of the proof may be present in higher dimensions, as
well as some open problems related to superfluidity of trapped gases.Comment: 28 pages, 2 figure, uses revtex

### The BCS Functional for General Pair Interactions

The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed
attention as a description of fermionic gases interacting with local pairwise
interactions. We present here a rigorous analysis of the BCS functional for
general pair interaction potentials. For both zero and positive temperature, we
show that the existence of a non-trivial solution of the nonlinear BCS gap
equation is equivalent to the existence of a negative eigenvalue of a certain
linear operator. From this we conclude the existence of a critical temperature
below which the BCS pairing wave function does not vanish identically. For
attractive potentials, we prove that the critical temperature is non-zero and
exponentially small in the strength of the potential.Comment: Revised Version. To appear in Commun. Math. Phys

### A Minimization Method for Relativistic Electrons in a Mean-Field Approximation of Quantum Electrodynamics

We study a mean-field relativistic model which is able to describe both the
behavior of finitely many spin-1/2 particles like electrons and of the Dirac
sea which is self-consistently polarized in the presence of the real particles.
The model is derived from the QED Hamiltonian in Coulomb gauge neglecting the
photon field. All our results are non-perturbative and mathematically rigorous.Comment: 18 pages, 3 figure

### Renormalization and asymptotic expansion of Dirac's polarized vacuum

We perform rigorously the charge renormalization of the so-called reduced
Bogoliubov-Dirac-Fock (rBDF) model. This nonlinear theory, based on the Dirac
operator, describes atoms and molecules while taking into account vacuum
polarization effects. We consider the total physical density including both the
external density of a nucleus and the self-consistent polarization of the Dirac
sea, but no `real' electron. We show that it admits an asymptotic expansion to
any order in powers of the physical coupling constant \alphaph, provided that
the ultraviolet cut-off behaves as \Lambda\sim e^{3\pi(1-Z_3)/2\alphaph}\gg1.
The renormalization parameter $

### Stability and Instability of Relativistic Electrons in Classical Electro magnetic Fields

The stability of matter composed of electrons and static nuclei is
investigated for a relativistic dynamics for the electrons given by a suitably
projected Dirac operator and with Coulomb interactions. In addition there is an
arbitrary classical magnetic field of finite energy. Despite the previously
known facts that ordinary nonrelativistic matter with magnetic fields, or
relativistic matter without magnetic fields is already unstable when the fine
structure constant, is too large it is noteworthy that the combination of the
two is still stable provided the projection onto the positive energy states of
the Dirac operator, which defines the electron, is chosen properly. A good
choice is to include the magnetic field in the definition. A bad choice, which
always leads to instability, is the usual one in which the positive energy
states are defined by the free Dirac operator. Both assertions are proved here.Comment: LaTeX fil

### Unique Solutions to Hartree-Fock Equations for Closed Shell Atoms

In this paper we study the problem of uniqueness of solutions to the Hartree
and Hartree-Fock equations of atoms. We show, for example, that the
Hartree-Fock ground state of a closed shell atom is unique provided the atomic
number $Z$ is sufficiently large compared to the number $N$ of electrons. More
specifically, a two-electron atom with atomic number $Z\geq 35$ has a unique
Hartree-Fock ground state given by two orbitals with opposite spins and
identical spatial wave functions. This statement is wrong for some $Z>1$, which
exhibits a phase segregation.Comment: 18 page