A growth estimate for the monodromy matrix of a canonical system

We investigate the spectrum of 2-dimensional canonical systems in limit circle case. It is discrete and, by Krein–de Branges formula, cannot be more dense than integers. But many cases it will sparse. The a particular selfadjoint realisation coincides with zeroes one entry monodromy matrix system. Classical function theory thus establishes an immediate connection between growth and distribution spectrum. prove general flexible upper estimate for matrix, use to bound case continuous Hamiltonian, construct examples which show that this sharp. first two results run along lines earlier work R. Romanov, but significantly improve upon these results. This seen even on rough scale exponential order.