Use of symbolic methods in analyzing an integral operator
The “stability of matter” problem in theoretical quantum mechanics is to derive from basic theory a mathematically rigorous lower bound on the energy per nucleus of an arbitrary configuration of nuclei and electrons. Such a lower bound provides a theoretical explanation of why the electrons do not simply collapse into the nuclei. The existence of a lower bound for the energy was originally proved by Dyson and Lenard in . Lieb and Thirring [4,3] later established a much better bound, coming within a factor of about 5.4 of the value suggested by experimental data. Recently, C. Fefferman  has presented a method that promises to yield a further improvement. Roughly, the idea is to express the total energy as an integral over all balls (of all sizes) in R3 , and then for each ball that contains nuclei to assign its energy in equal shares to all nuclei in it, and for each ball that contains no nuclei to assign its energy to the nearest nucleus. The lowest energy assigned to any nucleus is obviously a lower bound for the average energy per nucleus. Arguments given in  provide bounds for the energy contributed to a nucleus by all balls except those that are contained within a sphere of radius 28 about the nucleus and have their center within a distance 5 of the nucleus, where 25 is the distance to the nearest other nucleus. It is also shown in  that a lower bound for the energy contributed by the remaining balls can be obtained in terms of the sum of the negative eigenvalues of the quadratic form Q = K – V described below, which is essentially the part contributed by the same family of balls to the energy for a single electron in the field of a stationary nucleus with charge Z. (In the limit as δ goes to infinity, K becomes the kinetic energy given by the Laplacian, and V the Coulomb energy (given by a “1/r” potential) for such an electron.) The present paper discusses the computational problem of getting a rigorous lower bound on the negative eigenvalues of Q . As stated above, Fefferman is entirely responsible for formulating the problem; in addition, he has been closely involved in the computational analysis and some of what is reported here is joint work with him.
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