Estimates for the Weyl coefficient of a two-dimensional canonical system

Abstract

For a two-dimensional canonical system y⁰.t/ D zJH.t/y.t/ on some interval .a; b/ whose Hamiltonian H is a.e. positive semi-definite and which is regular at a and in the limit point case at b, denote by q<inf>H</inf> its Weyl coefficient. De Branges’ inverse spectral theorem states that the assignment H 7! q<inf>H</inf> is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions. We give upper and lower bounds for jq<inf>H</inf> .z/j and Im q<inf>H</inf> .z/ when z tends to i1 non-tangentially. These bounds depend on the Hamiltonian H near the left endpoint a and determine jq<inf>H</inf> .z/j up to universal multiplicative constants. We obtain that the growth of jq<inf>H</inf> .z/j is independent of the off-diagonal entries of H and depends monotonically on the diagonal entries in a natural way. The imaginary part is, in general, not fully determined by our bounds (in a forthcoming work we shall prove that for “most” Hamiltonians also Im q<inf>H</inf> .z/ is fully determined). We translate the asymptotic behaviour of q<inf>H</inf> to the behaviour of the spectral measure 0<inf>H</inf> of H by means of Abelian-Tauberian results and obtain conditions for membership of growth classes defined by weighted integrability condition (Kac classes) or by boundedness of the tails at ±1 with respect to a weight function. Moreover, we apply our results to Krein strings and Sturm-Liouville equations.