Abstract
This paper deals with the eigenvalue problem of Hamiltonian operator matrices with at least one invertible off-diagonal entry. The ascent and the algebraic multiplicity of their eigenvalues are determined by using the properties of the eigenvalues and associated eigenvectors. The necessary and sufficient condition is further given for the eigenvector (root vector) system to be complete in the Cauchy principal value sense.
Highlights
1 Introduction The method of separation of variables, known as the Fourier method, is one of the most effective tools in analytically solving the problems from mathematical physics. This method will lead to the eigenvalue problem of self-adjoint operators
As the orthogonality and the completeness of eigenvector systems can not be guaranteed for these problems, the traditional method of separation of variables fails to work
The orthogonality and the completeness of eigenvector systems are, respectively, replaced by the symplectic orthogonality and the completeness in the Cauchy principal value (CPV) sense; the eigenvalue problem of self-adjoint operators becomes that of Hamiltonian operator matrices admitting the representation
Summary
The method of separation of variables, known as the Fourier method, is one of the most effective tools in analytically solving the problems from mathematical physics. [ ] Let T be a linear operator in a Banach space, and let u be an eigenvector of T associated with the eigenvalue λ. Let λ and μ be the eigenvalues of the Hamiltonian operator matrix H, and let the associated eigenvectors be u = (x y )T and v = (f g )T , respectively.
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