A Special Case of De Branges' Theorem on Monodromy Matrix: Associated Riemann Surface Is of Widom Type with Direct Cauchy Theorem

Let H 0 (t); H 1 (t) be real 2 2 matrix{functions with entries from L 1 (0; 1), H 1 (t) = H 1 (t); H 0 (t) 0. We associate with these data the solution of the Cauchy problem for the diierential system dA(t; z) dt = A(t;z)fzH 0 (t) + H 1 (t)gJ; A(0;z) = 1 2 ; where J = 0 1 ?1 0 : The matrix{function A(z) = A(1;z) is called the monodromy matrix of the corresponding system 5]. More generally, let 0 (t) be a continuous nondecreasing real 2 2 matrix{function of t 2 0; 1], spf 0 (1) ? 0 (0)g < 1; and 1 (t) be a real symmetric 2 2 matrix{functions, whose entries are absolutely continuous functions with respect to the measure spfd 0 (t)g. In this case A(t;z) is deened as the solution of the matrix integral equation A(t;z) = 1 2 + Z t 0 A(s;z)fz d 0 (s) + d 1 (s)gJ; (0.1) and as before A(z) = A(1;z). How to restore the system on the monodromy matrix? When it could be done? Do we have a uniqueness theorem? These problems were solved in the whole generality by L. de Branges 2]. His theorem states, that if A(z) is an entire 2 2 matrix{function, which satisses the following properties: A(z) = A(z) (0.2.1) det A(z) = 1 (0.2.2) 1 2 PETER YUDITSKII then A(z) is the monodromy matrix of a system (0.1). Normalization 1 (t) = 0 deenes 0 (t) in unique way up to a continuous monotonic change of variable t, 0; 1] ! 0; 1]. The main problem here is to prove that 1 2 and A(z) could be included in a monotonic continuous chain of entire matrix-functions A(;z): J z ? z A(;z)JA(;z) z ? z A(z)JA(z) z ? z and that this chain is complete i.e.: under some normalization any divisor A 1 (z) of A(z), is present there (9 1 : A 1 (z) = A(1 ; z)). Set R = f(z;) : dettA(z) ? ] = 0g: Except some very special cases, when A(z) is a linear polynomial, or it is the mon-odromy matrix of a system with constant coeecients, R is two{sheeted Riemann surface (see section 1) and we deene R + = f(z;) : dettA(z) ? ] = 0; jj < 1g: According to the deenition is an inner function …