多区间线性哈密顿系统的GKN理论-应用数学专业论文.docxVIP

i i GKN , , , , , , . , . Hilbert, Glazman,Krein,N Hilbert  GKN , GK N . G , , GKN . GKN . [19] , : J y′(t) = (λW (t) + Q(t))y(t), t ∈ I I = I1 ∪ I2 ∪ . . . ∪ Ii ∪ . . . = [a1, b1] ∪ [a2, b2] ∪ . . . ∪ [ai, bi] ∪ . . . , i ∈ Z+, a1 b1  a2  b2  . . . bi?1  ai  bi  . . . . l : ly(t) = J y′(t) ? Q(t)y(t). Di(l) = yi : I ?→ C2n|yi ∈ A D(l) = 与 Di(l), i∈z i ii ? H ilb er t (  ) , , GKN , GKN . ; ; ? ; GK N ; . PAGE PAGE iii GKN Theory for Linear Hamiltonian Systems with a Countable Number of I Abstract The spectrum of di?erential operators is related to mang disparate and deep topics in mathematics and physics,including quantum mechanics,stability of ?uid and biolog- ical engineering.The structures they present have the profound signi?cance of mathe- matical models, Hence,researching on the theory of linear Hamiltonian systems is of theoretical and practical signi?cance. In 1850s, 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 con?ned to real-valued quasi-di?erential expression of second order.Everitt and Zettl extended the GKN theory to the quasi-di?erential expressions of arbitrary order with a counta