Abstract
We investigate the Hill differential equation where A(t), B(t), and D(t) are trigonometric polynomials. We are interested in solutions that are even or odd, and have period π or semi-period π. The above equation with one of the above conditions constitutes a regular Sturm-Liouville eigenvalue problem. We investigate the representation of the four Sturm-Liouville operators by infinite banded matrices.
Highlights
The first known appearance of the Ince equation,(1+ a cos(2t )) y′′(t ) + (bsin (2t )) y′(t ) + (λ + d cos(2t )) y (t ) = 0, is in Whittaker’s paper ([1], Equation (5)) on integral equations
Whittaker emphasized the special case a = 0, and this special case was later investigated in more detail by Ince [2] [3]
The coexistence problem which has no simple solution for the general Hill equation has a complete solution for the Ince equation
Summary
One of the important features of the Ince equation is that the corresponding Ince differential operator when applied to Fourier series can be represented by an infinite tridiagonal matrix. It is this part of the theory that makes the Ince equation interesting. When studying the Ince equation, it became apparent that many of its properties carry over to a more general class of equations “the generalized Ince equation”. These linear second order differential equations describe. Moussa important physical phenomena which exhibit a pronounced oscillatory character; behavior of pendulum-like systems, vibrations, resonances and wave propagation are all phenomena of this type in classical mechanics, (see for example [7]), while the same is true for the typical behavior of quantum particles (Schrödinger’s equation with periodic potential [8])
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