The authors study the phenomenon of avoided crossings of eigenvalues curves for boundary value problems related to differential equations of Heun’s class. The eigenvalues are given explicitly in asymptotic form taking into account power-type as well as exponentially small terms. In the research of the above problem the authors outgo from two following conjectures.
In one-dimensional models there is no single avoided crossing of only two eigenvalue curves. If there exists one avoided crossing of curves then there exists a large number of them (in the sense of the large parameter). They occur at certain values of an additional parameter, which controls the phenomenon.
These values of the controlling parameter constitute a periodic sequence.