Abstract

Polynomial Supersymmetry and Chains of Darboux Transformations for the One-Dimensional Dirac Equation

V.Bagrov, A.A.Pecheritsin, B.F.Samsonov (Tomsk State University, Tomsk, Russia)

e-mail: bagrov@phys.tsu.ru

Darboux transformation operator L for the Dirac equation is defined as a differential-matrix operator satisfying the intertwining relation LH=hL. Here the Dirac Hamiltonian H is supposed to be exactly solvable with a known system of eigenspinors f(x,E). The eigenspinors of the Hamiltonian h, g(x,E), can be found by the action of the operator L: g(x,E)=Lf(x,E). By iteration of this procedure one gets a chain of transformations. Compact formulae are obtained for resulting action of a chain. The components of the eigenspinors are expressed through the determinants similar to these which appear in the chain of transformations for the Schrodinger equation. It is shown that a polynomial superalgebra underlies the chain of transformations. A one-to-one correspondence between spaces of solutions of the initial and transformed Hamiltonians is established. The Jost function for the transformed Hamiltonian is analysed.

Section : 12