GKN Theory For Linear Hamiltonian Systems With A Countable Number Of Intervals

The spectrum of differential operators is related to mang disparate and deep topics in mathematics and physics,including quantum mechanics,stability of fluid and biolog-ical engineering.The structures they present have the profound significance of mathe-matical models, Hence,researching on the theory of linear Hamiltonian systems is of theoretical and practical significance.In1850s, Glazman,Krein and Naimark founded a famous theory named GKN theory.This theory gives an algebraic isomorphism between the set of Lagrangian com-plex sympletic subspaces and the set of self-adjoint operators in the Hilbert space.The original GKN theory was confined to real-valued quasi-differential expression of second order.Everitt and Zettl extended the GKN theory to the quasi-differential expressions of arbitrary order with a countable number of intervals on the real line.In paper [19], the spectral theory of linear Hamilton systems are studied, the GKN theory for this systems is established. The research of this paper is the GKN theory for linear Hamiltonian systems with a countable number of intervals.The thesis is dividded into two sections according to contents.Chapter1Preference, we introduce the main contents of this paper with several intervalsChapter2In this chapter, we consider the following linear Hamiltonian systems. Jy’(t)=(λW(t)+Q(t))y(t), t∈I where I=I1U I2U... U Ii U...=[a1b1] U [a2, b2]U...U [ai, bi] U..., i∈Z+, and a1<b1<a2<b2<...<bi-1<ai<bi<The formal Hamiltonian operator is defined as l:ly(t)=Jy’(t)-Q(t)y(t) acting on Di(l)={yi:Iâ†'C2n|yi∈ACloc{Ii)} In the appropriate Hilber space by the form of Hamiltonian operator we give thedefnitions of the maximal and minimal operators and symplectic spaces,and we givesthe one-to-one correspondence between the set of self-adjoint extensions of the minimaloperator and the set of Lagrangian symplectic subspaces.