On semibounded canonical systems

We present two inverse spectral relations for canonical differential equations Jy′(x)=-zH(x)y(x)Jy′(x)=-zH(x)y(x), x∈[0,L)x∈[0,L): Denote by QHQH the Titchmarsh–Weyl coefficient associated with this equation. We show: If the Hamiltonian H   is on some interval [0,ϵ)[0,ϵ) of the formH(x)=v(x)2v(x)v(x)1with a nondecreasing function v  , then limx↘0v(x)=limy→+∞QH(iy)limx↘0v(x)=limy→+∞QH(iy). If H   is of the above form on some interval [l,L)[l,L), then limx↗Lv(x)=limz↗0QH(z)limx↗Lv(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.