Intersection of Spectrum for Two Sturm-Liouville Problems with Periodic Boundary Conditions

In order to study the intersection of the spectra for two Sturm-Liouville (SL) problems with periodic boundary conditions (BCs), a two-dimensional vectorial SL problem is constructed. In this question the spectral sets of two one-dimensional SL problems are equal to the spectral sets of the two-dimensional vectorial SL problem. Then an upper bound M_Q of the double eigenvalues of the two-dimensional SL problem is calculated. It is concluded that the eigenvalues greater than M_Q of two-dimensional SL problem are single eigenvalues, and there are only a limited number of non single eigenvalues. Using the relationship between the spectral set of one-dimensional SL problem and two-dimensional vectorial SL problem, it is obtained that the number of cross spectra (same eigenvalues) of two one-dimensional SL problems with periodic BCs is limited, and the number of double eigen‐ values of one-dimensional SL problem with periodic BCs is also limited. At the same time, the upper bound esti‐ mation of the maximum double eigenvalues is calculated.