A generalization of Mathieu’s equation (28.2.1) is Hill’s equation
is either a continuous and real-valued function for or an analytic function of in a doubly-infinite open strip that contains the real axis. is the minimum period of .
The basic solutions , are defined in the same way as in §28.2(ii) (compare (28.2.5), (28.2.6)). Then
Let be a real or complex constant satisfying (without loss of generality)
throughout this section. Then (28.29.1) has a nontrivial solution with the pseudoperiodic property
iff is an eigenvalue of the matrix
This is the characteristic equation of (28.29.1), and is an entire function of . Given together with the condition (28.29.6), the solutions of (28.29.9) are the characteristic exponents of (28.29.1). A solution satisfying (28.29.7) is called a Floquet solution with respect to (or Floquet solution). It has the form
where the function is -periodic.
If is a solution of (28.29.9), then , comprise a fundamental pair of solutions of Hill’s equation.
If or , then (28.29.1) has a nontrivial solution which is periodic with period (when ) or (when ). Let be a solution linearly independent of . Then
where is a constant. The case is equivalent to
The solutions of period or are exceptional in the following sense. If (28.29.1) has a periodic solution with minimum period , , then all solutions are periodic with period .
Furthermore, for each solution of (28.29.1)
A nontrivial solution is either a Floquet solution with respect to , or is a Floquet solution with respect to .
In the symmetric case , is an even solution and is an odd solution; compare §28.2(ii). (28.29.9) reduces to
The cases and split into four subcases as in (28.2.21) and (28.2.22). The -periodic or -antiperiodic solutions are multiples of , respectively.
For details and proofs see Magnus and Winkler (1966, §1.3).
is assumed to be real-valued throughout this subsection.
is called the discriminant of (28.29.1). It is an entire function of . Its order of growth for is exactly ; see Magnus and Winkler (1966, Chapter II, pp. 19–28).
For a given , the characteristic equation has infinitely many roots . Conversely, for a given , the value of is needed for the computation of . For this purpose the discriminant can be expressed as an infinite determinant involving the Fourier coefficients of ; see Magnus and Winkler (1966, §2.3, pp. 28–36).
To every equation (28.29.1), there belong two increasing infinite sequences of real eigenvalues:
In consequence, (28.29.1) has a solution of period iff , and a solution of period iff . Both and as , and interlace according to the inequalities
Assume that the second derivative of in (28.29.1) exists and is continuous. Then with
we have for
If has continuous derivatives, then as
see Hochstadt (1963).
For further results, especially when is analytic in a strip, see Weinstein and Keller (1987).