ABSTRACT. A Thomas-Fermi-Weizs\"acker type theory is constructed, by means of which we are able to give a relatively simple proof of the stability of relativistic matter. Our procedure has the advantage over previous ones in that the lower bound on the critical value of the fine structure constant, $\alpha$, is raised from 0.016 to 0.77 (the critical value is known to be less than 2.72). When $\alpha =1/137$, the largest nuclear charge is 59 (compared to the known optimum value 87). Apart from this, our method is simple, for it parallels the original Lieb-Thirring proof of stability of nonrelativistic matter, and it adds another perspective on the subject.

PS file (146K) from Texas Math-Phys archive.

ABSTRACT. 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 $\alpha$, 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.

LaTeX file (60K) from Texas Math-Phys archive.

ABSTRACT. We prove that the spectrum of a suitably projected Dirac operator - sometimes called no-pair operator -- introduced by Brown and Ravenhall is bounded from below by some positive constant when the nuclear charge is below some critical charge. This bound improves a result obtained recently by Evans et al and disproves a conjecture made by Hardekopf and Sucher.

LaTeX2e file (24K) from Texas Math-Phys archive.

ABSTRACT. This paper is devoted to problems coming from statistical mechanics. The transfer matrix (or transfer operator) approach consists in reducing the analysis of asymptotic properties of statistical systems to the analysis of the spectral properties of their transfer operator. Sometimes the new problem appears to have a semi-classical nature. Our results concern the semi-classical analysis of the ground state for this operator with control with respect to the dimension. One basic technique is Sjöstrand's formalism of the 0-standard functions.

LaTeX2e file (96K) from Texas Math-Phys archive.

ABSTRACT. We give the first Lieb-Thirring type estimate on the sum of the negative eigenvalues of the Pauli operator that behaves as the corresponding semiclassical expression even in the case of strong non-homogeneous magnetic fields. This enables us, in the companion paper \cite{ES-II}, to obtain the leading order semiclassical eigenvalue asymptotic, which, in turn, leads to the proof of the validity of the magnetic Thomas-Fermi theory of \cite{LSY-II}. Our work generalizes the results of \cite{LSY-II} to non-homogeneous magnetic fields.

LaTeX2e file (114K) from Texas Math-Phys archive.

ABSTRACT. We give the leading order semiclassical asymptotics for the sum of the negative eigenvalues of the Pauli operator (in dimension two and three) with a strong non-homogeneous magnetic field. As in \cite{LSY-II} for homogeneous field, this result can be used to prove that the magnetic Thomas-Fermi theory gives the leading order ground state energy of large atoms. We develop a new localization scheme well suited to the anisotropic character of the strong magnetic field. We also use the basic Lieb-Thirring estimate obtained in our companion paper \cite{ES-I}.

LaTeX2e file (175K) from Texas Math-Phys archive.

ABSTRACT. This is the continuation of notes written for the NATO-ASI conference in Il Ciocco (Sept. 96) consisting in the analysis of the links between estimating the splitting between the two first eigenvalues for the Schr\"odinger operator $H$ and the proof of infrared estimates for quantities attached to Gaussian type measures. These notes were mainly reporting on the ``old'' contributions of Dyson, Fr\"ohlich, Glimm, Jaffe, Lieb, Simon, Spencer (in theseventies) in connection with more recent contributions of Pastur, Khoruzhenko, Barbulyak, Kondratev which treat in general more sophisticated models. Here we concentrate on the simplest model related to field theory and extend the results of Barbulyak-Kondratev by mixing ideas coming from Pastur-Khozurenko related to the use of Bogolyubov's inequality with classical inequalities due to Ginibre, Lebowitz, Sokal.... or in the case when the temperature $T$ is zero by applying rather elementary estimates on Schr\"odinger operators, in order to find lower bounds for second order moments attached to the measure $\phi \mapsto \Tr \phi \exp - \beta H/\tr \exp - \beta H$ with $\beta=\frac 1T$. This question was ``left to the reader'' in lectures given by J. Fr\"ohlich in 1976 \cite{Fr}, but we think that it is worthwhile to do this ``home work'' carefully.

LaTeX2e file (63K) from Texas Math-Phys archive.

ABSTRACT. The norm of an integral operator occurring in the partial wave decomposition of an operator $B$ introduced by Brown and Ravenhall in a model for relativistic one-electron atoms is determined. The result implies that $B$ is non-negative and has no eigenvalue at $0$ when the nuclear charge does not exceed a specified critical value.

LaTeX file (26K) from Texas Math-Phys archive.

ABSTRACT. We give the semi-classical asymptotics for the quantum current in 2 cases: First for $T>0$ ($T$ is the absolute temperature). Here we get a complete asymptotics. Then for $T=0$. Here it vanishes to the accessible orders.

LaTeX file (69K) from Texas Math-Phys archive.

ABSTRACT. We consider the Schr\"odinger operator $P_V(h)=-h^2\Delta +V$ where $V\in C^0(\r^n)$ such that $\lim _{|x|\to+\infty}V(x)=+\infty $. For every $\phi$ continuous convex with a support in $\r^+$, we state the following inequality $${\rm Tr}\big(\phi (E-P_V(h))\big)\leq {h^{-n}\over (2\pi )^n}\int_{\r ^n}\int_{\r ^n}\phi (E-\xi^2-V(x))\d x\d \xi $$ for all $E$ real and $h\in\r^+$ when $V$ is strictly convex and quadratic. When $\phi_\gamma=\max\{ 0,t\}^\gamma$ $\gamma\geq 1$ and $n\geq 3$, the inequality is the Lieb--Thirring's conjecture.

PS file (68K) from Texas Math-Phys archive.

ABSTRACT. Leading order semi-classical asymptotics are given for the distribution of the eigenvalues of Dirac and Pauli operators describing an electron in an electro-magnetic field. Minimal conditions are assumed on the electric and magnetic potentials to ensure the existence of only a finite number of eigenvalues outside the essential epectra. The method is based on coherent state analysis.

TeX file (96K) from Texas Math-Phys archive.

ABSTRACT. Stability of matter with Coulomb forces has been proved for non-relativistic dynamics, including arbitrarily large magnetic fields, and for relativistic dynamics without magnetic fields. In both cases stability requires that the fine structure constant $\alpha$ be not too large. It was unclear what would happen for {\it both} relativistic dynamics {\it and} magnetic fields, or even how to formulate the problem clearly. We show that the use of the Dirac operator allows both effects, provided the filled negative energy `sea' is defined properly. The use of the free Dirac operator to define the negative levels leads to catastrophe for any $\alpha$, but the use of the Dirac operator {\it with} magnetic field leads to stability.

TeX file (18K) from Texas Math-Phys archive.

ABSTRACT. A minimax principle is derived for the eigenvalues in the spectral gap of a possibly non-semibounded self adjoint operator. It allows us to bound the $n$-th eigenvalue of the Dirac operator with Coulomb potential from below by the $n$-th eigenvalue of a semibounded Hamiltonian which is of interest in the context of stability of matter. As a second application we show that the Dirac operator with suitable non-positive potential has at least as many discrete eigenvalues as the Schr\"odinger operator with the same potential.

PS file (379K) from Texas Math-Phys archive.

ABSTRACT. We give a criterion on when a form perturbation of the free Dirac operator defines a form perturbation of the second quantized free Dirac field. Moreover, we discuss the links between the various criteria in the literature for the existence of the second quantization of the Dirac operator inspired mainly by Klaus, Nenciu, and Scharf.

PS file from the Mathematical Physics Electronic Journal

ABSTRACT. A lower bound for the ground state energy of a one particle relativistic Hamiltonian - sometimes called no-pair operator - is provided.

PS file from Los Alamos archive.

ABSTRACT. Some properties of a pseudo-relativistic Hamiltonian describing a one electron atom - an appropriately projected Dirac operator with Coulomb potential - proposed by Brown and Ravenhall are proven. Self-adjointness is investigated and the explicit form of the Friedrichs extension is given. The behavior of the operator near the critical coupling constant $Z_c$ is studied and the essential spectrum is determined in the case of $Z=Z_c$ .

PS file from the Texas archive.

ABSTRACT. In this paper we consider the Hamiltonian of the standard model of non-relativistic QED. In this model non-relativistic quantum particles interact with quantized electro-magnetic field and their interaction is subjected to an ultra-violet cut-off. We prove absence of excited states and absolute continuity of the spectrum for sufficiently small charges under conditions on the coupling functions which milder than those of \cite{BachFroehlichSigal1996a}. We use the method of positive commutators with the ``conjugate'' operator deformed appropriately in order to accomodate the interaction term.

PS file (290 K) from the Texas archive.

ABSTRACT. We consider a system of finitely many nonrelativistic, quantum mechanical electrons bound to static nuclei. The electrons are minimally coupled to the quantized electromagnetic field; but we impose an ultraviolet cutoff on the electromagnetic vector potential appearing in covariant derivatives, and the interactions between the radiation field and electrons localized very far from the nuclei are turned off. For a class of Hamiltonians we prove exponential localization of bound states, establish the existence of a ground state, and derive sufficient conditions for its uniqueness. Furthermore, we show that excited bound states of the unperturbed system become unstable and turn into resonances when the electrons are coupled to the radiation field. To this end we develop a novel renormalization transformation which acts directly on the space of Hamiltonians.

PS file (946 K) from the Texas archive.

ABSTRACT. In this paper we present a self-contained and detailed exposition of the new renormalization group technique proposed in \cite{BachFroehlichSigal1995,BachFroehlichSigal1997a}. Its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator, the partition function, or correlation functions. We apply this renormalization transformation to a Hamiltonian describing the physics of an atom interacting with the quantized electromagnetic field, and we prove that excited atomic states turn into resonances when the coupling between electrons and field is nonvanishing.

PS file (974 K) from the Texas archive.

ABSTRACT. We consider Dirichlet eigenfunctions of membrane problems. A counterexample to Payne's nodal line conjecture is given, i.e. a domain in $\mathbb R^2$ (not simply connected) whose second eigenfunction has a nodal set disjoint from the boundary. Also a domain in $\mathbb R^2$ is given whose second eigenvalue has multiplicity three. Furthermore, some sufficient conditions are given which imply that an eigenfunction of a Dirichlet membrane problem in $\mathbb R^n$ has a zero set which hits the boundary.

PS file (522K) from the Texas archive.

ABSTRACT. ABSTRACT. Let $u\not\equiv\operatorname{const}$ satisfy an elliptic equation $L_0u\equiv\sum a_{ij}D_{ij}u+\sum b_jD_ju=0$ with smooth coefficients in a domain in $\mathbf R^n$. It is shown that the critical set $|\nabla u|^{-1}\{0\}$ has locally finite $n-2$ dimensional Hausdorff measure. This implies in particular that for a solution $u\not\equiv0$ of $(L_0+c)u=0$, with $c\in C^\infty$, the critical zero set $u^{-1}\{0\}\cap|\nabla u|^{-1}\{0\}$ has locally finite $n-2$ dimensional Hausdorff measure.

ABSTRACT. A virial theorem is established for the operator proposed by Brown and Ravenhall as a model for relativistic one-electron atoms. As a consequence, it is proved that the operator has no eigenvalues greater than $\max(m c^2, 2 \alpha Z - \frac{1}{2})$, where $\alpha$ is the fine structure constant, for all values of the nuclear charge $Z$ below the critical value $Z_c$: in particular there are no eigenvalues embedded in the essential spectrum when $Z \leq 3/4 \alpha$. Implications for the operators in the partial wave decomposition are also described.

LaTeX file (42 K) from the Texas archive.

ABSTRACT. The purpose of this paper is to study the transition from the classical to the quantum asymptotics for the integrated density of states of an unbounded random Jacobi matrix. Therefore, we give precise results on the behavior of the tail of the integrated density of states near infinity. We study the evolution of these asymptotics when the decay of the tail of the distribution of the random potential increases.

256K, Uuencoded Gzipped Postscript from the Texas archive.

ABSTRACT. As it appears in recent articles by Helffer or Sj\"ostrand and Naddaf-Spencer, the analysis, in the context of the statistical mechanics, of measures of the type $\exp - \Phi (x) \; dx$ is connected with the analysis of suitable Witten Laplacians on $1$-forms. For illustrating this point of view, we present here remarks about the Brascamp-Lieb inequalities and its extensions and prove the decay of the correlation in some cases when $\Phi$ is weakly non convex.

39K, LATEX file from the Texas archive.

ABSTRACT. This is the continuation of a previous article
``Remarks on decay of correlations and Witten Laplacians
-- Brascamp-Lieb inequalities and semiclassical limit --''
and devoted to the analysis of Laplace integrals attached to the measure $\exp - \Phi(X)\;dX$ for suitable families of phase $\Phi$ appearing naturally in the context of statistical mechanics. The main application treated in Part I was a semi-classical one ($\Phi=\Psi/h$ and $h\ar 0$) and the assumptions on the phase were related to weak non convexity.
We first analyze in the same spirit the case when the coefficient of the interaction $\Jg$ is possibly large and give rather explicit lower bounds for the lowest eigenvalue of the Witten Laplacian on $1$-forms. We also analyze the case $\Jg$ small by discussing first an unpublished proof of Bach-Jecko-Sj\"ostrand and then an alternative approach based on the analysis of a family of $1$-dimensional Witten Laplacians. We also compare with the results given by Sokal's approach. In part III of this serie, we shall analyze, in a less explicit way but in a more general context, applications to the logarithmic Sobolev inequality.

44K LATEX file from the Texas archive.

ABSTRACT. This is the continuation of our two previous articles devoted to the use of Witten Laplacians for analyzing Laplace integrals in statistical mechanics. The main application treated in Part I was a semi-classical one. The second application was more perturbative in spirit and gave very explicit estimates for the lower bound of the Witten Laplacian in the case of a quartic model. We shall relate in this third part our studies of the Witten Laplacian with the existence of uniform logarithmic Sobolev inequalities through a criterion of B. Zegarlinski. More precisely, our main contribution is to show how to control the decay of correlations uniformly with respect to various parameters, under a natural condition of strict convexity at infinity of the single-spin phase and when the nearest neighbor interaction is small enough.

39K LATEX file from the Texas archive.

ABSTRACT. We establish the phase diagram of the local mean field model, called Kac model. This enables us to solve a conjecture made by Kac on the spectral gap of the transfer operator associated to this model.

70K LATEX file from the Texas archive.

ABSTRACT. We show that the multiplicity of the eigenvalues of the Laplace Beltrami operator on compact Riemannian surfaces with genus zero is bounded by $m(\lambda_k) \le 2k-3 $ for $k\ge3$. Here we label the eigenvalues in the following way: $0=\lambda_1<\lambda_2\le \lambda_3\dots$.

LaTeX (17K) file from the Texas archive.

ABSTRACT. For a membrane in the plane the multiplicity of the k-th eigenvalue is known to be not greater than 2k-1. Here we prove that it is actually not greater than 2k-3, for k\geq3.

PS (582K) file from the Texas archive.

ABSTRACT. We study the energy of relativistic electrons and positrons interacting through the second quantized Coulomb interaction and a self-generated magnetic field. As states we allow generalized Hartree-Fock states in the Fock space. Our main result is the assertion of positivity of the energy, if the atomic numbers and the fine structure constant are not too big. We also discuss the dependence of the result on the dressing of the electrons (choice of subspaces defining the electrons).

PS (483K) file from the Texas archive.

ABSTRACT: A study is made of an integral identity introduced recently by B.~Helffer and J.~Sjöstrand; in comparison with the Brascamp--Lieb inequality it is a more flexible and in some contexts stronger means for the analysis of correlation asymptotics in statistical mechanics. Using functional analysis, viz. a detailed version of the Closed Range Theorem, the identity's validity is shown to depend upon and be implied by explicitly given spectral properties of Witten--Laplacians on Euclidean space, and the formula is moreover deduced from the obtained abstract expression for the range projection. As a corollary, a generalised version of Brascamp--Lieb's inequality is obtained. Explicit criteria for the Witten-Laplacians are found from compactness of embeddings and from the Weyl calculus, which give results for closed range, strict positivity, essential self-adjointness and domain characterisations.

gzipped PS file from the Texas archive

ABSTRACT. For bounded potentials which behave like \(-cx^{-\gamma}\) at infinity we investigate whether discrete eigenvalues of the radial Dirac operator $H_{\kappa}$ accumulate at +1 or not. It is well known that $\gamma=2$ is the critical exponent. We show that \(c=1/8+\kappa(\kappa+1)/2\) is the critical coupling constant in the case $\gamma=2$. Our approach is to transform the radial Dirac equation into a Sturm-Liouville equation nonlinear in the spectral parameter and to apply a new, general result on accumulation of eigenvalues of such equations.

LaTeX file (41 K) from the Texas archive

ABSTRACT. In appropriate units, the Brown-Ravenhall Hamiltonian for a system of $1$ electron relativistic molecules with $K$ fixed nuclei having charge and position $Z_k, R_k$, $k=1,2, \ldots,K$, is of the form $\bB_{1,K}= \Lambda_+ \bigl( D_0 + \alpha V_c\bigr) \Lambda_+ $, where $\Lambda_+ $ is the projection onto the positive spectral subspace of the free Dirac operator $D_0$ and $V_c= - \sum_{k=1}^K \frac{\alpha Z_k}{\lmod \bx-R_k \rmod} + \sum_{k<l, \ k,l=1}^K \frac{\alpha Z_k Z_l}{\lmod R_k-R_l \rmod} $, with $\alpha$ Sommerfeld's fine structure constant. It is proved that for $\alpha Z_k \leq \alpha Z_c = \frac{2}{\pi /2 + 2/ \pi}$ , $k=1,2, \ldots,K$, \ $\bB_{1,K} \geq \operatorname{const} \cdotp K$.

LaTeX File (61K) from the Texas archive

ABSTRACT. We investigate nodal sets of magnetic Schroedinger operators with zero magnetic field, acting on a non simply connected domain in $\r^2$. For the case of circulation $1/2$ of the magnetic vector potential around each hole in the region, we obtain a characterisation of the nodal set, and use this to obtain bounds on the multiplicity of the groundstate. For the case of one hole and a fixed electric potential, we show that the first eigenvalue takes its highest value for circulation $1/2$.

LaTeX file (129K) from the Texas archive

ABSTRACT. We study a class of holomorphic complex measures, which is close in an appropriate sense to a complex Gaussian. we show that these measures can be reduced to a product of real Gaussians with the aid of a maximum principle in the complex domain. The formulation of this problem has its origin in the study of a certain class of random Schroedinger operators, for which we show that the expectation value of the Green's function decays exponentially.

TeX file from the Texas archive

ABSTRACT. Under suitable analyticity conditions on the probability distribution, we study the expectation of the Green function. W@e give precise results about domains of holomorphic extensions in energy and exponential decay. The key ingredients (as in the previous paper) is the construction of a probability measure in the complex domain after contour deformation. This permits us to avoid the use of perturbation series. Compared to the method in the previous paper, the variant here seems limited to the random Schroedinger equation, in which case it permits to treat more general probability distributions.

TeX file from the Texas archive

ABSTRACT. The present paper continues Sjöstrand's study of correlation functions of lattice field theories by means of Witten's deformed Laplacian. Under the assumptions specified in the paper and for sufficiently low temperature, we derive an estimate for the spectral gap of a certain Witten Laplacian which enables us to prove the exponential decay of the two-point correlation function and, further, to derive its asymptotics, as the distance between the spin sites becomes large. Typically, our assumptions do not require uniform strict convexity and apply to Hamiltonian functions which have a single, nondegenerate minimum and no other extremal point.

PS (534K) file from the Texas archive

ABSTRACT. In this work, we give an improvement of the adiabatic approximation of the resolvent and wave operators for diatomic molecules obtained by Klein-Martinez-Wang. Following Mourre's commutator method, we use a new kind of escape function. In the same way, we deal with matricial Schroedinger operators and the Dirac operator with scalar electric potential.

LaTeX file (89 K) from the Texas archive

ABSTRACT. The aim of this paper is to justify the physical intuition that the quantum scattering of a diatomic molecule is well approached by a classical 2-body system when the nuclei's mass are large. To this end, we compute the action of an elactic scattering operator on quantum observable, microlocalized by coherent states. An adiabatic operator makes the connection between the quantum scattering and the classical one.

LaTeX file (117K) from the Texas archive

ABSTRACT. We point out the basic bound on the singular values of integral operators of the type a(x)b(D), from which Cwikel's inequality and its generalizations can easily be derived.

PS file (104 K) from the Texas archive

ABSTRACT. In this paper we obtain a new version of stability of matter, which in particular shows that Thomas-Fermi theory is asymptotically correct in the limit of large nuclear charges uniformly in the number of nuclei. As a consequence we give a new lower bound on the volume of matter with an improved dependence on the nuclear charges.

LaTeX file (71 K) from the Texas archive

ABSTRACT. For a pseudo-relativistic model of matter, based on the no-pair Hamiltonian, we prove that the inclusion of the interaction with the self-generated magnetic field leads to instability for all positive values of the fine structure constant. This is true no matter whether this interaction is accounted for by the Breit potential, by an external magnetic field which is chosen to minimize the energy, or by the quantized radiation field.

LATeX 2e file (46K) from the Texas archive

ABSTRACT. We investigate stability of matter of the Hartree-Fock functional of the relativistic electron-positron field -- in suitable second quantization -- interacting via a second quantized Coulomb field and a classical magnetic field. We are able to show that stability holds for a range of nuclear charges $Z_1,..,Z_K\leq Z$ and fine structure constants $\alpha$ that include the physical value of $\alpha$ and elements up to holmium ($Z=67$).

gzipped PS file from DOCUMENTA MATHEMATICA.

ABSTRACT.
We show how a matrix version of the Buslaev-Faddeev-Zakharov trace formulae for a one-dimensional Schr\"odinger operator leads to Lieb-Thirring inequalities with sharp constants $L^{\mbox{\footnotesize\upshape cl}}_{\gamma,d}$ with $\gamma\ge 3/2$ and arbitrary $d\ge 1$.

PS file (201K) from the Texas archive

ABSTRACT. Improved estimates on the constants $L_{\gamma,d}$, for $1/2<\gamma<3/2$, $d\in N$ in the inequalities for the eigenvalue moments of Schr\"{o}dinger operators are established.

PS file from the Texas archive

ABSTRACT. In this note, we give a proof of Log-Sobolev inequality for unbounded spins systems with weak assumptions on the potentials.

TeX file from the Texas archive

ABSTRACT.

We prove the minimax principle for eigenvalues in spectral gaps introduced in \cite{GriesemerSiedentop1999} based on an alternative set of hypotheses. In the case of the Dirac operator these new assumptions allow for potentials with Coulomb singularites. gzipped PS file from DOCUMENTA MATHEMATICA

ABSTRACT. In continuation with \cite{He1} and an unpublished paper by F.~Daumer \cite{Da}, we would like to explore the variation of the splitting for a Schr\"odinger operator coming from field theory when the potential changes as a function of the temperature from a one-well situation to a double well situation. We are for the moment only able to treat the fixed dimension case. But, because the most interesting thing would be to analyze the dependence on the dimension (in the perspective of applications to statistical mechanics) we have tried to follow some of the steps in the proof with respect to the dimension. All this analysis was strongly motivated by a course by M.~Kac where some heuristical discussion based on some Born-Oppenheimer analysis is presented. So our work can be considered as a first attempt for justifying this discussion.

TeX File (58K, LATeX2e) from the Texas archive.

ABSTRACT.

ABSTRACT. In this paper we give two different variational characterizations for the eigenvalues of of $H+V$ where $H$ denotes the free Dirac operator and $V$ is a scalar potential. The first one is a min-max involving a Rayleigh quotient. The second one consists in minimizing an appropriate nonlinear functional. Both methods can be applied to potentials which have singularities as strong as the Coulomb potential.

PS file from the Texas archive.

ABSTRACT. This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the essential spectrum of a self-adjoint unbounded operator. We prove an abstract theorem, then we apply it to the case of Dirac operators with a Coulomb-like potential. The result is optimal for the Coulomb potential.

PS file from the Texas archive

ABSTRACT. Abstract. In this paper we prove the existence of infinitely many solutions of the Dirac-Fock equations with $N$ electrons turning around a nucleus of atomic charge $Z$, satisfying $N < Z+1$ and $\alpha \max(Z,N) < \frac{2}{\frac{\pi}{2}+\frac{2}{\pi}}$, where $\alpha$ is the fundamental constant of the electromagnetic interaction. This work is an improvement of an article of Esteban-S\'er\'e, where the same result was proved under more restrictive assumptions on $N$.

PS file from the Texas archive

ABSTRACT. We apply a method developed by J.Howland and K.Yajima in conjunction with ideas from the analysis of Hamiltonians with constant electric fields to obtain absence of bound states and asymptotic completeness for 2-body short-range systems in an external time-periodic electric field.

PS file from the Texas archive

ABSTRACT. By adapting methods developed in a book by Yu. Safarov and D. Vassiliev to the semi-classical situation we obtain a new two-term asymptotic formula for the counting function of eigenvalues of $ h $ pseudodifferential operators in the limit as $ h \to 0 $. Recent results on clustering of eigenvalues obtained by V. Petkov and G. Popov follow in Corollary and, as an application, we consider the semi-classical behaviour of the quantum current studied recently by S. Fournais.

PS file from the Texas archive

ABSTRACT. We consider systems of static nuclei and electrons --atoms and molecules-- coupled to the quantized radiation field. The interactions between electrons and the soft modes of the quantized electromagnetic field are described by minimal coupling, p --> p - e A(x), where A(x) is the electromagnetic vector potential with an ultraviolet cutoff. If the interactions between the electrons and the quantized radiation field are turned off, the atom or molecule is assumed to have at least one bound state. We prove that, for sufficiently small values of the feinstructure constant alpha, the interacting system has a ground state corresponding to the bottom of its energy spectrum and that the excited states of the atom or molecule above the ground state turn into metastable states whose life-times we estimate. Furthermore the energy spectrum is absolutely continuous, except, perhaps, in a small interval above the ground state energy and around the threshold energies of the atom or molecule.

PS file (570 K) from the Texas archive

ABSTRACT. We study the current of the Pauli operator in a strong constant magnetic field. We prove that in the semi-classical limit the persistent current and the current from the interaction of the spin with the magnetic field cancel, in the case where the magnetic field is very strong. Furthermore we calculate the next term in the asymptotics and estimate the error. Finally, we discuss the connection between this work and the semi-classical estimate of the energy in strong magnetic fields proved by Lieb, Solovej and Yngvason.

PS file (468 K) from the Texas archive

ABSTRACT. We construct a set in ${\mathbf R}^D$ with the property that the nodal surface of the second eigenfunction of the Dirichlet Laplacian is closed, i.e. does not touch the boundary of the domain. The construction is explicit in all dimensions $D \geq 2$ and we obtain explicit control of the connectivity of the domain.

PS file (198 K) from the Texas archive

ABSTRACT. In appropriate units, the no-pair Hamiltonian for a system of $1$ electron relativistic molecules with $K$ fixed nuclei, having charge and position $Z_k, \bR_k$, $k=1,2, \ldots, K$, is of the form $\bB_{1,K}= \Lambda_+ \bigl( D_0 + \alpha V_c\bigr) \Lambda_+ $, where $\Lambda_+ $ is the projection onto the positive spectral subspace of the free Dirac operator $D_0$ and $V_c= - \sum\limits_{k=1}^K \frac{\alpha Z_k}{\lmod \bx-\bR_k \rmod} + \sum\limits_{k≤l, \ k,l=1}^K \frac{\alpha Z_k Z_l}{\lmod \bR_k-\bR_l \rmod} $, with $\alpha$ Sommerfeld's fine structure constant. We discuss the background and significance of our result that for $\alpha Z_k \leq \alpha Z_c = \frac{2}{\pi /2 + 2/ \pi}$ , $k=1,2, \ldots,K$, \ and $\alpha \leq \frac{2 \pi}{(\pi^2 +4)(2+ \sqrt{1+ \pi /2})}$, \ \ \ $\bB_{1,K} \geq \mathop{\mathrm const} \cdotp K$, and give an outline of the main features of our proof.

This paper has not been archived but has appeared in J. Phys. A, vol.32(11), L129--L132 (1999)

ABSTRACT. A rigorous derivation of macroscopic spin-wave equations is demonstrated. We introduce a macroscopic mean-field limit and derive the so-called Landau-Lifshitz equations for spin waves. We first discuss the ferromagnetic Heisenberg model at $T=0$ and finally extend our analysis to general spin hamiltonians for the same class of ferromagnetic ground states.

TeX file from the Texas archive

ABSTRACT. We continue with this paper our study of the thermodynamic limit for various models of Quantum Chemistry, this time focusing on the Hartree and the restricted Hartree model. For the restricted Hartree model, we prove the existence of the thermodynamic limit for the energy per unit volume. We also define a periodic problem associated to the hartree model, and we prove that it is well-posed.

PS File (192 K) from the Texas archive

ABSTRACT. The Brown-Ravenhall Hamiltonian is a model for the behavior of $N$ electrons in a field of $K$ fixed nuclei having the atomic numbers ${\bf Z}=(Z_1,\ldots,Z_K)$, which is written, in appropriate units, as $$B=\Lambda_{+,N}\left(\sum_{n=1}^N D_0^{(n)} +\alpha V_c\right)\Lambda_{+,N}$$ acting on the $N$-fold antisymmetric tensor product $\mathfrak H_N$ of $\Lambda_+(L^2({\mathbb R}^3)\otimes {\mathbb C}^4)$, where $D_0^{(n)}$ denotes the free Dirac operator $D_0$ acting on the $n$-th particle, $\Lambda_+$ denotes the projection onto the positive spectral subspace of $D_0$, $\Lambda_{+,N}$ the projection onto $\mathfrak H_N$ and the potential $V_c$ is the usual Coulomb interaction of the particles, coupled by the constant $\alpha$. It is proved in the massless case that for any $\gamma <2/(2/\pi+\pi/2)$ there exists an $\alpha_0$ such that for all $\alpha<\alpha_0$ and $\alpha Z_k\leq\gamma$ $(k=1,\ldots K)$ we have stability, i.e., $B\geq 0$. Using numerical calculations we get stability for the physical value $\alpha\approx 1/137$ up to $Z_k\leq 88$ $(k=1,\ldots K)$.

PS file from MPEJ.

ABSTRACT. Let $E(B,Z,N)$ denote the ground state energy of an atom with $N$ electrons and nuclear charge $Z$ in a homogeneous magnetic field $B$. We study the asymptotics of $E(B,Z,N)$ as $B\to \infty$ with $N$ and $Z$ fixed but arbitrary. It is shown that the leading term has the form $(\ln B)^2 e(Z,N)$, where $e(Z,N)$ is the ground state energy of a system of $N$ {\em bosons} with delta interactions in {\em one} dimension. This extends and refines previously known results for $N=1$ on the one hand, and $N,Z\to\infty$ with $B/Z^3\to\infty$ on the other hand.

LaTeX file from the Texas archive

ABSTRACT. We study the asymptotic behavior, as Planck's constant $\hbar \to 0$, of the number of discrete eigenvalues and the Riesz means of Pauli and Dirac operators with a magnetic field $\mu\mathbf{B}(x)$ and an electric field. The magnetic field strength $\mu$ is allowed to tend to infinity as $\hbar\to 0$. Two main types of results are established: in the first $\mu\hbar\le constant$ as $\hbar\to 0$, with magnetic fields of arbitrary direction; the second results are uniform with respect to $\mu\ge 0$ but the magnetic fields have constant direction. The results on the Pauli operator complement recent work of Sobolev.

PS file (154 K) from the Texas archive

ABSTRACT. In this paper we describe an intrinsically geometric way of producing magnetic fields on $\S^3$ and $\R^3$ for which the corresponding Dirac operators have a non-trivial kernel. In many cases we are able to compute the dimension of the kernel. In particular we can give examples where the kernel has any given dimension. This generalizes the examples of Loss and Yau (Commun. Math. Phys. 104 (1986) 283-290).

PS file (133K) from the Los Alamos archive

ABSTRACT. We continue here our study of the thermodynamic limit for various models of Quantum Chemistry, this time focusing on the Hartree-Fock type models. For the reduced Hartree-Fock model, we prove the existence of the thermodynamic limit for the ground-state energy per unit volume. We also suggest a periodic problem associated to the Hartree-Fock model, and prove that it is well-posed.

PS file (675 K) from the Texas archive

ABSTRACT. Two results are proved for $\mathrm{nul} \ \mathbb{P}_A$, the dimension of the kernel of the Pauli operator $\mathbb{P}_A = \bigl\{ \bbf{\sigma} \cdotp \bigl(\frac{1}{i} \bbf{\nabla} + \vec{A} \bigr) \bigr\} ^2 $ in $[L^2 (\mathbb{R}^3)]^2$: (i) for $|\vec{B}| \in L^{3/2} (\mathbb{R}^3),$ where $\vec{B} = \mathrm{curl} \vec{A}$ is the magnetic field, $\mathrm{nul} \ \mathbb{P}_{tA} = 0$ except for a finite number of values of $t$ in any compact subset of $(0, \infty)$; (ii) \ $\bigl\{ \ \vec{B}: \ \mathrm{nul} \ \mathbb{P}_{ A} = 0, \ \ | \vec{B} | \in L^{3/2}(\mathbb{R}^3) \ \bigr\} $ contains an open dense subset of $[L^{3/2}(\mathbb{R}^3)]^3$.

PS file (209K) from the Texas archive

ABSTRACT. We construct the zero'th order low-temperature WKB-phase for the first eigenfunction of a transfer operator in a large domain around a non-degenerate critical point for the potential. The zero'th order low-temperature phase is shown to solve the eikonal equation in the strong-coupling limit and we obtain non-local estimates on the zero'th order phase, which are preserved in the limit. We furthermore use the IMS localization technique to study the two highest eigenvalues of the transfer operator in the case where V is allowed to have many non-degenerate global minima.

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ABSTRACT. With a special `Ansatz' we analyse the regularity properties of atomic electron wavefunctions and electron densities. In particular we prove an a priori estimate, $\sup_{y\in B(x,R)}|\nabla\psi(y)| \leq C(R)\,\sup_{y\in B(x,2R)}|\psi(y)|$ and obtain for the spherically averaged electron density, $\widetilde\rho(r)$, that $\widetilde\rho''(0)$ exists and is non-negative.

LaTeX file (64K) from the Texas archive

ABSTRACT. 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/. 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].

The paper from the journal.

ABSTRACT. We calculate the asymptotic form of the quantum current/magnetisation of a non-interacting electron gas at zero temperature. The calculation uses coherent states and a novel commutator identity for the current operator.

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ABSTRACT. In this paper we study the asymptotic form of the magnetisation and current of large atoms in strong constant magnetic fields. We prove that the Magnetic Thomas-Fermi theory gives the right magnetisation/current for magnetic field strengths which satisfy $B \leq Z^{4/3}$.

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ABSTRACT. The model considered here is the `jellium' model in which there is a uniform, fixed background with charge density $-e\rho$ in a large volume $V$ and in which $N=\rho V$ particles of electric charge $+e$ and mass $m$ move --- the whole system being neutral. In 1961 Foldy used Bogolubov's 1947 method to investigate the ground state energy of this system for bosonic particles in the large $\rho$ limit. He found that the energy per particle is $-0.402 \, r_s^{-3/4}{me^4}/{\hbar^2}$ in this limit, where $r_s=(3/4\pi \rho)^{1/3}e^2m/\hbar^2$. Here we prove that this formula is correct, thereby validating, for the first time, at least one aspect of Bogolubov's pairing theory of the Bose gas.

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ABSTRACT. Taking into account relativistic effects in quantum chemistry is crucial for accurate computations involving heavy atoms. Standard numerical methods can deal with the problem of {\rm variational collapse} and the appearance of {\rm spurious roots} only in special cases. The goal of this letter is to provide a general and robust method to compute particle bound states of the Dirac equation.

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ABSTRACT. We study the exponential decay asymptotics of correlations at large distance, associated to a measure of Laplace type, in the semi-classical limit. The new feature compared to earlier works by V. Bach, T. Jecko and the author, is that we get full asymptotics of the decay rate and the prefactor, instead of just the leading terms, and that we treat the thermodynamical limit. As before, we study the Witten Laplacian via a Grushin (Feshbach) problem, but we now have to use higher order problems, involving multiparticle states.

ABSTRACT. In this article, we would like to review recent results concerning the links between the decay of correlations, the spectral gap and the Log-Sobolev inequalities. This was motivated by various papers by Antoniouk&Antoniouk, Zegarlinski and Yoshida. We are mainly reporting on contributions by Helffer-Sj\"ostrand, Helffer, Yoshida and Bodineau-Helffer but also present some new results.

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ABSTRACT. This is a survey on a recent paper of the present authors (CMP 1999). We explain in detail the origin of the problem in superconductivity as first presented by Berger and Rubinstein (CMP 1999), recall our results and explain the extension to the Dirichlet case. As illustration of the theory, we detail some semi-classical aspects and give examples where our results are sharp.

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ABSTRACT. In a previous article, the first two authors have proved that the existence of zero modes of Pauli operators is a rare phenomenon; inter alia, it is proved that for $|\vec{B}| \in L^{3/2}(\mathbb{R}^3)$, the set of magnetic fields $\vec{B}$ which do not yield zero modes contains an open dense subset of $[L^{3/2}(\mathbb{R}^3)]^3$. Here the analysis is taken further, and it is shown that Sobolev, Hardy and Cwikel-Lieb-Rosenbljum (CLR) inequalities hold for Pauli operators for all magnetic fields in the aforementioned open dense set.

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ABSTRACT. It is proved that the existence of zero modes of Weyl-Dirac operators is a rare phenomenon. An estimate of the multiplicity is given in term of the magnetic potential.

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ABSTRACT. It is proved that for $V_+ = \max(V,0)$ in the subspace $ L^1 ( \mathbb{R}^+ , \ L^{\infty}(\mathbb{S}^1), \ rdr)$ of $L^1 (\mathbb{R}^2)$, there is a Cwikel-Lieb-Rosenblum type inequality for the number of negative eigenvalues of the operator $\biggl( \frac{1}{i} \vec{\nabla} + \vec{A} \biggr)^2 - V$ in $L^2 (\mathbb{R}^2)$ when $\vec{A}$ is an Aharonov-Bohm magnetic potential with non-integer flux. It is shown that $ L^1 ( \mathbb{R}^+ , \ L^{\infty}(\mathbb{S}^1), \ rdr)$ can not be replaced by $L^1 (\mathbb{R}^2)$ in the inequality.

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ABSTRACT. We consider the relativistic electron-positron field interacting with itself via the Coulomb potential defined with the physically motivated, positive, density-density quartic interaction. The more usual normal-ordered Hamiltonian differs from the bare Hamiltonian by a quadratic term and, by choosing the normal ordering in a suitable, self-consistent manner, the quadratic term can be seen to be equivalent to a renormalization of the Dirac operator. Formally, this amounts to a Bogolubov-Valatin transformation, but in reality it is non-perturbative, for it leads to an inequivalent, fine-structure dependent representation of the canonical anticommutation relations. This non-perturbative redefinition of the electron/positron states can be interpreted as a mass, wave-function and charge renormalization, among other possibilities, but the main point is that a non-perturbative definition of normal ordering might be a useful starting point for developing a consistent quantum electrodynamics.

Paper in PDF format from the journal

ABSTRACT. In this paper, the Hartree-Fock equations are proved to be the non relativistic limit of the Dirac-Fock equations as far as convergence of ``stationary states" is concerned. This property is used to derive a meaningful definition of ``ground state" energy and ``ground state" solutions for the Dirac-Fock model.

PS file (523K) from the Texas archive

ABSTRACT. This is a survey of the authors' recent results on the spectral asymptotics for the Schroedinger, Pauli, and Dirac operators in strong magnetic fields.

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ABSTRACT. In this note we consider an Hamiltonian with cutoffs describing the interaction of relativistic electrons and positrons in a Coulomb potential with transversal photons in Coulomb gauge. We prove that the Hamiltonian is self-adjoint in the Fock space and has a ground state for a sufficiently small coupling constant.

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ABSTRACT. For a Dirac operator in $\R ^3$ with an electric potential behaving at infinity like a power of $ | x | $, we prove the existence of resonances and we study, when $c\rightarrow + \infty $, the asymptotic expansion of their real part, and an estimation of their imaginary part, generalizing an old result of Titchmarsh.

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ABSTRACT. We present a generalization of the Fefferman-de la Llave decomposition of the Coulomb potential to quite arbitrary radial functions V on Rⁿ going to zero at infinity. As a byproduct, we obtain conditions for positive definiteness of V, thereby improving results of Askey.

PS file from the Los Alamos arXive

ABSTRACT. Scattering theory for the Nelson model is studied. We show Rosen estimates and we prove the existence of a ground state for the Nelson Hamiltonian. Also we prove that it has a locally finite pure point spectrum outside its thresholds. We study the asymptotic fields and the existence of the wave operators. Finally we show asymptotic completeness for the Nelson Hamiltonian.

PS file from the Texas archive

ABSTRACT. Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, a lot of papers devoted to the analysis in a semi-classical regime of the lowest eigenvalue of the Schr\"odinger operator with magnetic field have appeared recently. Here we would like to mention the works by Bernoff-Sternberg, Lu-Pan and Del Pino-Felmer-Sternberg. This recovers partially questions analyzed in a different context by the authors around the question of the so called magnetic bottles. Our aim is to analyze the former results, to treat them in a more systematic way and to improve them by giving sharper estimates of the remainder. In particular, we improve significatively the lower bounds and as a byproduct we solve a conjecture proposed by Bernoff-Sternberg concerning the localization of the ground state inside the boundary in the case with constant magnetic fields.

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ABSTRACT. We study the scattering theory for a class of non-relativistic quantum field theory models describing a confined non-relativistic atom interacting with a relativistic bosonic field. We construct invariant spaces $\cH_{\c}^{\pm}$ which are defined in terms of propagation properties for large times and which consist of states containing a finite number of bosons in the region $\{|x|\geq \c t\}$ for $t\to \pm \infty$. We show the existence of asymptotic fields and we prove that the associated asymptotic CCR representations preserve the spaces $\cH_{\c}^{\pm}$ and induce on these spaces representations of Fock type. For these induced representations, we prove the property of {\em geometric asymptotic completeness}, which gives a characterization of the vacuum states in terms of propagation properties. Finally we show that a positive commutator estimate imply the {\em asymptotic completeness} property, ie the fact that the vacuum states of the induced representations coincide with the bound states of the Hamiltonian.

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ABSTRACT. We continue the study started by the first author of the semiclassical Kac Operator. This kind of operator has been obtained for example by M. Kac as he was studying a 2D spin lattice by the so-called "transfer operator method". We are interested here in the thermodynamical limit $\Lambda(h)$ of the ground state energy of this operator. For Kac's spin model, $\Lambda(h)$ is the free energy per spin, and the semiclassical regime corresponds to the mean-field approximation. Under suitable assumptions, which are satisfied by many examples comming from statistical mechanics, we construct a formal asymptotic expansion for $\Lambda(h)$ in powers of $h$, from which we derive precise estimates. We work in the setting of \emph{standard functions} introduced by J. Sjöstrand for the study of similar questions in the case of Schröodinger operators.

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ABSTRACT. Let $H=-\Delta+V$ be a two-dimensional Schr\"odinger operator defined on a bounded domain $\Omega \subset \mathbb{R}^2$ with Dirichlet boundary conditions on $\partial \Omega$. Suppose that $H$ commutes with the actions of the dihedral group $\mathbb D_{2n}$, the group of the regular $n$-gone. We analyze completely the multiplicity of the groundstate eigenvalues associated to the different symmetry subspaces related to the irreducible representations of $\mathbb D_{2n}$. In particular we find that the multiplicities of these groundstate eigenvalues equal the degree of the corresponding irreducible representation. We also obtain an ordering of these eigenvalues. In addition we analyze the qualitative properties of the nodal sets of the corresponding ground state eigenfunctions.

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ABSTRACT. In this paper we improve the estimate obtained by Lu-Pan on the value of the upper critical field $H_{C_3}(\kappa)$ for a cylindrical superconductor with cross section $\Omega$ being an arbitrary $2$-dimensional smooth bounded domain. We also show that, when a homogeneous magnetic field is applied along the axis of the cylinder with magnitude of the field close to $H_{C_3}$, superconductivity nucleates first at the surface of the sample where the curvature of the boundary is maximal.

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ABSTRACT. Jansen and He{\ss} -- correcting an earlier paper of Douglas and Kroll -- have derived a (pseudo-)relativistic energy expression which is very successfull in describing heavy atoms. It is an approximate no-pair Hamiltonian in the Furry picture. We show that their energy in the one-particle Coulomb case, and thus the resulting self-adjoint Hamiltonian and its spectrum, is bounded from below for $\alpha Z\leq 1.006$.

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ABSTRACT. We study the photoelectric effect on the example of a simplified model of an atom with a single bound state, coupled to the quantized electromagnetic field. For this model, we show that Einstein's prediction for the photoelectric effect is qualitatively and quantitatively correct to leading order in the coupling parameter. More specifically, considering the ionization of the atom by an incident photon cloud consisting of $N$ photons, we prove that the total ionized charge is additive in the $N$ involved photons. Furthermore, if the photon cloud is approaching the atom from a large distance or is monochromatic, the kinetic energy of the ejected electron is shown to be given by the difference of the photon energy of each single photon in the photon cloud and the ionization energy.

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ABSTRACT. We study the free energy of continuous spin-systems on $\Bbb Z^d$, in the framework of Laplace integrals and transfer operators. Under a weak coupling condition, we show that the free energy in the low-temperature limit is determined, up to an exponentially small error, by the restriction to a neighbourhood of global minima of the energy. We have results for some single- and double-well problems.

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