On transformations of canonical systems

Abstract. We present two inverse spectral relations for canonical differential equations Jy′(x) = −zH(x)y(x), x ∈ [0, L): Denote by QH the Titchmarsh-Weyl coefficient associated with this equation. We show: If the Hamiltonian H is on some interval [0, ) of the form H(x) = v(x)2 v(x) v(x) 1 with a nondecreasing function v, then limx↘0 v(x) = limy→+∞QH(iy). If H is of the above form on some interval [l, L), then limx↗L v(x) = limz↗0QH(z). In particular, these results are applicable to semibounded canonical systems, or canonical systems with a finite number of negative eigenvalues, respectively