Darboux transformation of boundary conditions of regular Dirac Sturm--Liouville problem

发布时间：2011-07-17 14:00:51

a rXiv:0711.1675v1 [he p-th] 11 N ov 2 7Darboux transformation of boundary conditions of regular Dirac Sturm—Liouville problem Ekaterina Pozdeeva ∗and Alexander Tarasov Joint Institute for Nuclear Research,141980,Dubna,Moscow region,Russia Abstract It is shown that boundary conditions of the Darboux transformed Dirac Sturm—Liouville problem are always zero-valued independently on boundary conditions of initial problem.1.Introduction The study of the properties of the Darboux transformation [1,2,3,4,5](or SUSY)attracts more and more attention in the past two decades.Recent the developments in this area are reflected in the special issue of J.Phys.A 34where the references to the early paper can be found.Despite in depth studies,many problems in given field have yet to be solved.In particular relationship between Green functions of two Hamiltonians being SUSY partners have been studied only in few paper [6,7,8,9,10].All these paper deal with the Green functions of Schr¨o dinger Hamiltonians.The attempts to generalize some of these results to the case of Dirac problem have been only recently [11,12,13,14].They lead to the necessity of consideration of the Darboux transformation of boundary conditions of the initial Sturm—Liouville problem.In the paper [14]this problem have been considered for some particular cases,that admit the analytical solution of the Sturm—Liouville problem.Considered in[14]the boundary values of the Darboux transformed solutions of these problem are always equal zero.The last means that both components of the spinorial solution on the ends of an interval,on which leads to the natural assumption that this conclusion is also valid in general case,but not only in particular cases,considered in [14].The proving of this statement is the aim of the present paper.In Section 2we derive two different forms of the relation between the solutions of the initial and

Darboux transformation of boundary conditions of regular Dirac Sturm--Liouville problem

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