SPACE, TIME AND FUNDAMENTAL INTERACTIONS

Darboux transformation for the simplest phenomenological brane model

Yurov A.V., Maksyutov A.R., Trunin A.S.

The method of isospectral symmetries is used to integrate the Einstein equations in the presence of a family of parallel 3-branes immersed in an ambient five-dimensional space filled with gravity and a scalar field. In the presence of orbifold symmetry, for a given stationary brane, part of the resulting equations can be reduced to the Schrodinger equation, the discrete non-negative spectrum of which determines a set of particles on the brane. The Schrodinger equation admits a Darboux transformation, which, when using a regular support function, changes the spectrum to a finite number of levels without changing the others. Thus, the described technique allows one to construct finite sets of particles with a given mass distribution. In the approximation of fine-tuning between the brane tension and the value of the vacuum energy density, exact solutions are constructed for a brane with zero curvature. An exact derivation of the generalized Crum formulas is given.