ACTA issues

A growth condition for Hamiltonian systems related with Kre\u?n strings

Henrik Winkler, Harald Woracek

Acta Sci. Math. (Szeged) 80:1-2(2014), 31-94

Abstract. We study two-dimensional Hamiltonian systems of the form $(*)$ $y'(x)=zJH(x)y(x),\quad x\in[s_-,s_+)$, where the Hamiltonian $H$ is locally integrable on $[s_-,s_+)$ and nonnegative, and $J:=\big({0 -1\atop1 0}\big )$. The spectral theory of the equation changes depending on the growth of $H$ towards the endpoint $s_+$; the classical distinction into the Weyl alternatives `limit point' or `limit circle' case. A refined measure for the growth of a limit point Hamiltonian $H$ can be obtained by comparing with $H$-polynomials. This growth measure is concretised by a number $\Delta(H)\in\bb N_0\cup\{\infty\}$ and appeared first in connection with a Pontryagin space analogue of the equation $(*)$. It is known that the growth restriction `$\Delta(H)< \infty $' has some striking consequences on the spectral theory of the equation; in many respects, the case `limit point but still $\Delta(H)< \infty $' is similar to the limit circle case. In general, the number $\Delta(H)$ is given in a rather implicit way, difficult to handle and not suitable for concrete calculations. In the present paper we provide a more accessible way to compute $\Delta(H)$ for some particular classes of Hamiltonians which occur in connection with Sturm--Liouville equations and Kre\u?n strings.

DOI: 10.14232/actasm-012-028-8

AMS Subject Classification (1991): 34B20, 37J99, 34L40; 47B50, 47E05

Keyword(s): Hamiltonian system, Kre\u?n string, growth towards singular endpoint

Received May 8, 2012, and in revised form August 27, 2013. (Registered under 28/2012.)