Jakob Reiffenstein: Growth of Weyl coefficient and monodromy matrix of a canonical system

Abstract:

For two-dimensional canonical systems of differential equations, we investigate the growth of analytic functions encoding the spectrum – the Weyl coefficient (in limit point case) and the monodromy matrix (in limit circle case). We give a two-sided estimate for the imaginary part of the Weyl coefficient along the imaginary axis, in a similar way as the Jitomirskaya-Last inequality describes the absolute value of the Weyl coefficient. This way we can characterize integrability of a given function with respect to the spectral measure. In the case of discrete spectrum we also want to separate the eigenvalues from the weights put on them. To do this we describe the growth in the spectral parameter of the fundamental solution. This yields, e.g., an equivalent criterion for a canonical system to have resolvents belonging to a Schatten–von Neumann class with small index (most notably trace class).