Abstract:
The overarching theme of my presentation is the quantitative description
of the spectrum of two-dimensional canonical systems of differential equations.
My fundamental results are two-sided estimates for the values along the imaginary
axis of the go-to functions encoding the spectrum – the imaginary part of the Weyl
coefficient (in limit point case) and entries of the monodromy matrix (in limit circle
case). In each case, the lower and upper estimates coincide up to a universal
multiplicative constant and are explicit in terms of the entries of the Hamiltonian.
These results can be used to answer several questions related to the growth
of the spectral measure and, for discrete spectrum, to the density of eigenvalues.
For example, I will present a criterion for a canonical system to have resolvents
belonging to a Schatten–von Neumann class with small index, e.g., trace class.
The density of eigenvalues of a canonical system in limit circle case can be described
in a particularly elegant way. Namely, one counts the number of points in
certain partitions of the base interval, corresponding to the rotation of the Hamiltonian,
and repeats this for decreasing “grain sizes”. The function mapping the
grain size to the number of partitioning points is then closely related to the eigenvalue
counting function.
In the final part of the talk I will consider canonical systems associated with
indeterminate Hamburger moment problems, making use of the algorithmic method
described above. Under mild well-behavedness assumptions, the growth of the
monodromy matrix can be computed explicitly. As an application, I determine the
order of a Jacobi matrix with power asymptotics whenever limit circle case takes
place.