Problems About Symmetric Extension And Self-adjoint Extension Of Differential Operators

Linear Hamiltonian operator, as its widespread use in common life, has become one of the most important differential operators, while the Friedrichs extension is also the most important part of researches. Friedrichs extension, whose name after Friedrichs, is an special adjoint extension. That is to say, for a semi-bounded densely defined operator, there exists a self-adjoint extension which has the same bound. With the appearance of complex differential operators, the J-symmetric operators have won peo-ple’s attention. In the third and fourth chapters, we major in the J-adjoint operators and obtain some conclusions.The thesis is divided into four sections according to contents.Chapter1Preference, we introduce the main contents of this thesis.Chapter2We consider the following2n-order linear Hamiltonian system Jy’(t)=(λW(t)+P(t))y(t),(2.2.1) t∈I,I=[a1,b1]∪[a2, b2].Where W(t) and P(t) are real integrable functions on interval I, and W=W*(?)0, P=P*, λ is a complex spectral parameter, J is2n x2n-order matrix and satisfies J*=J-1=-J, that isIn is n-order unit matrix. The Hamiltonian operator (2.2.2) are defined on I and is bouded. Suppose that TF is an extension operator of the Hamiltonian Operator and its domain is T0, let LF=D(TF)/D(T0), then TF is T0’s Friedrichs extension (?) LF=Span｛en+1, en+2,…,e2n,e3n+1,e3n+2,…,e4n,fn+1,fn+2,…,f2n,f3n+1,f3n+2,…,f4n｝.Chapter3In this chapter, symplectic spaces are defined by the domains of the maximal and the minimal operator in linear Hamiltonian system. A natural bi-unique correspondence is built between the J-symmetric extension and J-Lagrange sub-manifold,thus we obtain the theory of J-symmetric extensions for linear Hamil-tonian systems.That is to say, L is L ’s k(0≤κ≤2n)dimensional J-Lagrangian complete manifold (?)(?)αsj,bsj∈C,s=1,2,...,κ,s.t. L=span｛a11e11+...+a1,2ne2n+b11f1+...+b1,2nf2n,...,ακ1e1+...+ακ,2ne2n+bκ1f1+...+bκ,2nf2n｝, and also satisfies (1) ranκ(A|B)=2n,(2)αiOαjT=0,whereA=(αsj)κ×n,B (bsj)κ×n.Chapter4In this chapter,we will use the theory in chapter2to obtain some J-symmetric extensions of Hamiltonian system.