Spectral estimations for canonical systems

Two-dimensional canonical systems are boundary value problems of the form Jy'(x) = -zH(x)y(x), x epsilon (0, L), L less than or equal to infinity, z epsilon C, with y(1) (0) = 0 and Weyl's limit point case at L. The 2 x 2 matrix valued function H is real, symmetric and nonnegative, J = ((0)(1) (-1)(0)). The c orrespondence between canonical systems and their 0 Titchmarsh-Weyl coeffic ients Q is a bijection between the class of all matrix functions H with tr H(x) = 1 a. e. on (0, L) and the class of the Nevanlinna functions N augmen ted by the function Q = infinity. Each Titchmarsh-Weyl coefficient Q epsilo n N can be represented by means of a measure a, the so-called spectral meas ure of the canonical system. In this note matrix functions H are specified whose corresponding spectral measures a satisfy conditions of the form S--i nfinity(+infinity) d sigma(lambda)/1+\lambda\gamma < <infinity> or integral (1)(-1) d sigma(lambda)/\lambda\(gamma) < +<infinity>, gamma epsilon [0,2] . Herewith we generalize corresponding results of M. G. KREIN and I.S. KAC for so-called vibrating strings.