The dirac second-order equation and an improved quasirelativistic theory of atoms

A transformation decoupling the Dirac second-order equations for the large and small components of the radial atomic wavefunction has been derived. The decoupling procedure is exact for one-electron atoms and approximate, though very accurate, for a general case of an atomic spherical potential. Its connection with the quasirelativistic theory of atoms has been discussed. The transformation has been applied to deriving an optimized quasirelativistic equation and to establishing relations between the quasirelativistic and the Dirac wavefunctions. Formulas resulting from these considerations allow improvement of quasirelativistic expectation values of operators so that the error, relative to the Dirac values, is reduced by a factor of 10.     

Capacity and quantum mechanical tunneling

We connect the notion of capacity of sets in the theory of symmetric Markov process and Dirichlet forms with the notion of tunneling through the boundary of sets in quantum mechanics. In particular we show that for diffusion processes the notion appropriate to a boundary without tunneling is more refined than simply capacity zero. We also discuss several examples in ^d .     

Shallow arch model. An asymptotic approach

We examine an asymptotic expansion of the time dependent displacement field of a solid, three-dimensional elastic arch. We show that it is possible to derive from the principles of virtual work four different models of arches. We also show that the approximate equations of motion are universal within a class of homogeneous, hyperelastic materials provided that the deformations of the solid arch are not too large.     

Generalized one-electron spin functions and self-similarity measures

A simple method for construction of eigenfunctions of one-electron spin angular momentum operators from products of the primitive one-electron spin functions is presented. Properties of these functions and their applications to the evaluation of some integrals met in theory of quantum similarity are briefly discussed.     

Schrödinger Operator Perturbed by Operators Related to Null Sets

We discuss the Schrödinger operator with positive singular perturbations given by operators which act in the space constructed by a positive measure supported by a null set. We construct examples when perturbations are given by the one-dimensional Laplacian on a segment.     

Asymptotic models for a long,elastic cylinder

We study asymptotic expansions for the displacement field of a long elastic cylinder under various constitutive assumptions. We show that under simple hypotheses it is possible to derive from the equations of continuum mechanics two known beam equations and several different string models. Some of the string models correspond to those studied by S. Antman and R. Dickey. We also show that under our assumptions the problem of asymptotic expansion can be reduced to that of algebraic geometry.Research partially supported by NSF Grant DMR-8612369.     

Matrix elements of one- and two-electron operators

A graphical representation of matrix elements of spin-free one- and two-electron operators is used for deriving a simple algorithm for the evaluation of their values. The method covers all the cases which may occur when wave functions are taken as mutually orthogonal antisymmetrized products of spinorbitals (which are assumed to form an orthonormal set) and are eigenfunctions of L ² and L _z operators. The resulting formulas are suitable as well for computer programming as for hand calculations.

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Zusammenfassung Mit Hilfe einer graphischen Darstellung wird ein einfacher Algorithmus zur Bestimmung der Werte der Matrix-Elemente von spinfreien Ein- und Zweielektronenoperatoren abgeleitet. Durch diese Methode werden alle Fälle erfaßt, die auftreten, wenn die Wellenfunktionen wechselweise orthogonale, antisymmetrisierte Produkte von Spinorbitalen (von denen angenommen wird, daß sie einen orthogonalen Satz bilden) darstellen und Eigenfunktion der L ² und L _z Operatoren sind. Die erhaltenen Formeln eignen sich für Rechnungen mit und ohne Verwendung eines Computers.

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Sponsored by the Mathematical Institute, Polish Academy of Sciences.