COMPLEX METHODS IN HIGHER DIMENSIONS : RECENT TRENDS FOR SOLVING BOUNDARY VALUE AND INITIAL VALUE PROBLEMS

Joint Research on Environmental Science and Technology for the Eart

The Casimir Effect in Spheroidal Geometries

We study the zero point energy of massless scalar and vector fields subject to spheroidal boundary conditions. For massless scalar fields and small ellipticity the zero-point energy can be found using both zeta function and Green's function methods. The result agrees with the conjecture that the zero point energy for a boundary remains constant under small deformations of the boundary that preserve volume (the boundary deformation conjecture), formulated in the case of an elliptic-cylindrical boundary. In the case of massless vector fields, an exact solution is not possible. We show that a zonal approximation disagrees with the result of the boundary deformation conjecture. Applying our results to the MIT bag model, we find that the zero point energy of the bag should stabilize the bag against deformations from a spherical shape.Comment: 24 pages, 3 figures. To appear in Phys. Rev.

On the reduction of the multidimensional Schroedinger equation to a first order equation and its relation to the pseudoanalytic function theory

Given a particular solution of a one-dimensional stationary Schroedinger equation (SE) this equation of second order can be reduced to a first order linear differential equation. This is done with the aid of an auxiliary Riccati equation. We show that a similar fact is true in a multidimensional situation also. We consider the case of two or three independent variables. One particular solution of (SE) allows us to reduce this second order equation to a linear first order quaternionic differential equation. As in one-dimensional case this is done with the aid of an auxiliary Riccati equation. The resulting first order quaternionic equation is equivalent to the static Maxwell system. In the case of two independent variables it is the Vekua equation from theory of generalized analytic functions. We show that even in this case it is necessary to consider not complex valued functions only, solutions of the Vekua equation but complete quaternionic functions. Then the first order quaternionic equation represents two separate Vekua equations, one of which gives us solutions of (SE) and the other can be considered as an auxiliary equation of a simpler structure. For the auxiliary equation we always have the corresponding Bers generating pair, the base of the Bers theory of pseudoanalytic functions, and what is very important, the Bers derivatives of solutions of the auxiliary equation give us solutions of the main Vekua equation and as a consequence of (SE). We obtain an analogue of the Cauchy integral theorem for solutions of (SE). For an ample class of potentials (which includes for instance all radial potentials), this new approach gives us a simple procedure allowing to obtain an infinite sequence of solutions of (SE) from one known particular solution

On a complex differential Riccati equation

We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation as, e.g., the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical "one-dimensional" results we discuss new features of the considered equation like, e.g., an analogue of the Cauchy integral theorem