Luis A. Seco

On the energy of a large atom

We announce a proof of an asymptotic formula for the groundstate energy of a large atom. The early work of Thomas-Fermi, Hartree-Fock, Dirac, and Scott predicted that for an atomic number Z , the energy is E(Z) « -cQZ ^ + c{Z 2 c2Z 513 for known c0, cx, and c2 (see [5]). Schwinger [7] observed an additional effect and set down the modified formula E(Z) « -cQZ +cxZ •yC2Z . Our proof shows that Schwingers formula is correct. We give the precise formulation of the problem. For a fixed nucleus of charge Z and quantized electrons x{, . . . , xN e R , the Hamiltonian HN z is the self-adjoint operator

Asymptotic neutrality of large ions

It is proved that a nucleus of chargeZ can bind at mostZ+O(Za) electrons, witha=47/56.

Bound on the ionization energy of large atoms

We present a simple argument which gives a bound on the ionization energy of large atoms that implies the bound on the excess charge of Fefferman and Seco [2].

Aperiodicity of the Hamiltonian flow in the Thomas-Fermi potential.

In [FS1] we announced a precise asymptotic formula for the ground-state energy of a non-relativistic atom. The purpose of this paper is to establish an elementary inequality that plays a crucial role in our proof of that formula. The inequality concerns the Thomas-Fermi potentialnVTF = -y(ar) / r, a > 0, where y(r) is defined as the solution ofnni y(x) = x-1/2y3/2(x),ni y(0) = 1,ni y(8) = 0.

Computer Assisted Lower Bounds for Atomic Energies

For an atom of nuclear charge Z, the ground state energy is defined to be the lowest possible value of the energy Hamiltonian. We describe an algorithm to produce rigorous lower bounds for the ground state energy of atoms as well as its implementation.

Harmonic analysis in value at risk calculations

Value at Risk is a measure of risk exposure of a portfolio and is defined as the maximum possible loss in a certain time frame, typically 1-20 days, and within a certain confidence, typically 95%. Full valuation of a portfolio under a large number of scenarios is a lengthy process. To speed it up, one can make use of the total delta vector and the total gamma matrix of a portfolio and compute a Gaussian integral over a region bounded by a quadric. We use methods from harmonic analysis to find approximate analytic formulas for the Value at Risk as a function of time and of the confidence level. In this framework, the calculation is reduced to the problem of evaluating linear algebra invariants such as traces of products of matrices, which arise from a Feynmann expansion. The use of Fourier transforms is crucial to resum the expansions and to obtain formulas that smoothly interpolate between low and large confidence levels, as well as between short and long time horizons.

Weyl sums and atomic energy oscillations

We extend Van der Corputs method for exponential sums to study an oscillating term appearing in the quantum theory of large atoms. We obtain an interpretation in terms of classical dynamics and we produce sharp asymptotic upper and lower bounds for the oscillations.