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We derive out a complete series expression of Hamiltonian eigenvalues withoutany approximation and cut in the general quantum systems based on Wang's formalframework \cite{wang1}. In particular, we then propose a calculating approachof eigenvalues of arbitrary Hamiltonian via solving an algebra equationsatisfied by a kernal function, which involves the contributions from all orderperturbations. In order to verify the validity of our expressions and revealthe power of our approach, we calculate the ground state energy of a quarticanharmonic oscillator and have obtained good enough results comparing with theknown one.

Quantum mechanics in the general quantum systems (V): Hamiltonian eigenvalues