The spectrum of infinite dimensional Hamiltonian operators are considered and the sufficient condition for infinite dimensional Hamiltonian operators have only pure imaginary spectrum is obtained.

If A,B are given,one sees that∩C∈S(X)σi(HC)=σiπ(A),∪C∈S(X)σi(HC)=σi(A),∩F∈S(X)σi(HF)=σiπ(APR(B)⊥),∪F∈S(X)σi(HF)=σi(APR(B)⊥),where σi(T),σiπ(T),PM andS(X) denote the pure imaginary spectrum of T,the pure imaginary approximate point spectrum of T,the orthogonal projection(from X) onto M and the set of all self-adjoint operators(not necess.

In a base class,abstract class is defined by declaring pure virtual function,a derived class must define all inherited pure virtual functions of its base classes to be concrete.