The Completeness Of The Eigenfunction Systems Of A Class Of Infinite Dimensional Hamiltonian Operators

The method of separation of variables is effective to solve partial differential equations. For instance, it can solve equation like the wave equation, heat equation and harmonic equation. In applying this method to a given differential equation, the equation is usually transformed into the Sturm-Liouville problem, which has developed into systematic theory. However, the Sturm-Liouville problem has its own limitations; i.e., it is essentially self-adjoint operators' eigen-problem. Due to this restriction, it is limited in terms of area of application. A question arises is there an alternative? Namely, can the method of separation be based on the eigen-problem of the operators with some special properties? The present paper examines the equation, which can be transformed into the Sturm-Liouville problem after separating variables, in the Hamiltonian systems, from the perspective of infinite dimensional Hamiltonian operators' eigen-problem. By investigating the completeness of the eigenfunction system, the paper aims to provide a theoretical basis for employing the method of separation of variables based on Hamiltonian systems for equations of this specific type.