§28.29 Definitions and Basic Properties

A generalization of Mathieu’s equation (28.2.1) is Hill’s equation

with

and

$Q(z)$ is either a continuous and real-valued function for $z\in ℝ$ or an analytic function of $z$ in a doubly-infinite open strip that contains the real axis. $\pi$ is the minimum period of $Q$.

The basic solutions $w_{\text{I}}(z,\lambda )$, $w_{\text{II}}(z,\lambda )$ are defined in the same way as in §28.2(ii) (compare (28.2.5), (28.2.6)). Then

Let $\nu$ be a real or complex constant satisfying (without loss of generality)

throughout this section. Then (28.29.1) has a nontrivial solution $w(z)$ with the pseudoperiodic property

iff ${\mathrm{e}}^{\pi \mathrm{i}\nu }$ is an eigenvalue of the matrix

Equivalently,

This is the characteristic equation of (28.29.1), and $\cos \left(\pi \nu \right)$ is an entire function of $\lambda$. Given $\lambda$ together with the condition (28.29.6), the solutions $\pm \nu$ 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 $\nu$ (or Floquet solution). It has the form

where the function $P_{\nu }(z)$ is $\pi$-periodic.

If $\nu$ $(\ne 0,1)$ is a solution of (28.29.9), then $F_{\nu }(z)$, $F_{-\nu }(z)$ comprise a fundamental pair of solutions of Hill’s equation.

If $\nu =0$ or $1$, then (28.29.1) has a nontrivial solution $P(z)$ which is periodic with period $\pi$ (when $\nu =0$) or $2\pi$ (when $\nu =1$). Let $w(z)$ be a solution linearly independent of $P(z)$. Then

where $c$ is a constant. The case $c=0$ is equivalent to

The solutions of period $\pi$ or $2\pi$ are exceptional in the following sense. If (28.29.1) has a periodic solution with minimum period $n\pi$, $n=3,4,\mathrm{\dots }$, then all solutions are periodic with period $n\pi$.

Furthermore, for each solution $w(z)$ of (28.29.1)

A nontrivial solution $w(z)$ is either a Floquet solution with respect to $\nu$, or $w(z+\pi )-{\mathrm{e}}^{\mathrm{i}\nu \pi }w(z)$ is a Floquet solution with respect to $-\nu$.

In the symmetric case $Q(z)=Q(-z)$, $w_{\text{I}}(z,\lambda )$ is an even solution and $w_{\text{II}}(z,\lambda )$ is an odd solution; compare §28.2(ii). (28.29.9) reduces to

The cases $\nu =0$ and $\nu =1$ split into four subcases as in (28.2.21) and (28.2.22). The $\pi$-periodic or $\pi$-antiperiodic solutions are multiples of $w_{\text{I}}(z,\lambda ),w_{\text{II}}(z,\lambda )$, respectively.

For details and proofs see Magnus and Winkler (1966, §1.3).

$Q(x)$ is assumed to be real-valued throughout this subsection.

The function

is called the discriminant of (28.29.1). It is an entire function of $\lambda$. Its order of growth for $|\lambda |\to \mathrm{\infty }$ is exactly $\frac{1}{2}$; see Magnus and Winkler (1966, Chapter II, pp. 19–28).

For a given $\nu$, the characteristic equation $△(\lambda )-2\cos \left(\pi \nu \right)=0$ has infinitely many roots $\lambda$. Conversely, for a given $\lambda$, the value of $△(\lambda )$ is needed for the computation of $\nu$. For this purpose the discriminant can be expressed as an infinite determinant involving the Fourier coefficients of $Q(x)$; 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 $\pi$ iff $\lambda ={\lambda }_n$, and a solution of period $2\pi$ iff $\lambda ={\mu }_n$. Both ${\lambda }_n$ and ${\mu }_n\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$, and interlace according to the inequalities

Assume that the second derivative of $Q(x)$ in (28.29.1) exists and is continuous. Then with

we have for $m\to \mathrm{\infty }$

If $Q(x)$ has $k$ continuous derivatives, then as $m\to \mathrm{\infty }$

see Hochstadt (1963).

For further results, especially when $Q(z)$ is analytic in a strip, see Weinstein and Keller (1987).