§28.29 Definitions and Basic Properties

§28.29(i) Hill’s Equation

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

 28.29.1 $w^{\prime\prime}(z)+\left(\lambda+Q(z)\right)w=0,$ ⓘ Symbols: $z$: complex variable, $w(z)$: Mathieu’s equation solution, $\lambda$: parameter and $Q(z)$: function Referenced by: §28.29(ii), §28.29(ii), §28.29(ii), §28.29(iii), §28.29(iii), §28.29(iii) Permalink: http://dlmf.nist.gov/28.29.E1 Encodings: TeX, pMML, png See also: Annotations for §28.29(i), §28.29 and Ch.28

with

 28.29.2 $Q(z+\pi)=Q(z),$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $Q(z)$: function Permalink: http://dlmf.nist.gov/28.29.E2 Encodings: TeX, pMML, png See also: Annotations for §28.29(i), §28.29 and Ch.28

and

 28.29.3 $\int_{0}^{\pi}Q(z)\mathrm{d}z=0.$

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

§28.29(ii) Floquet’s Theorem and the Characteristic Exponent

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

 28.29.4 $\displaystyle w_{\mbox{\tiny I}}(z+\pi,\lambda)$ $\displaystyle=w_{\mbox{\tiny I}}(\pi,\lambda)w_{\mbox{\tiny I}}(z,\lambda)+w^{% \prime}_{\mbox{\tiny I}}(\pi,\lambda)w_{\mbox{\tiny II}}(z,\lambda),$ 28.29.5 $\displaystyle w_{\mbox{\tiny II}}(z+\pi,\lambda)$ $\displaystyle=w_{\mbox{\tiny II}}(\pi,\lambda)w_{\mbox{\tiny I}}(z,\lambda)+w^% {\prime}_{\mbox{\tiny II}}(\pi,\lambda)w_{\mbox{\tiny II}}(z,\lambda).$

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

 28.29.6 $-1<\Re\nu\leq 1$ ⓘ Symbols: $\Re$: real part and $\nu$: complex parameter Referenced by: §28.29(ii) Permalink: http://dlmf.nist.gov/28.29.E6 Encodings: TeX, pMML, png See also: Annotations for §28.29(ii), §28.29 and Ch.28

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

 28.29.7 $w(z+\pi)=e^{\pi\mathrm{i}\nu}w(z),$

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

 28.29.8 $\begin{bmatrix}w_{\mbox{\tiny I}}(\pi,\lambda)&w_{\mbox{\tiny II}}(\pi,\lambda% )\\ w^{\prime}_{\mbox{\tiny I}}(\pi,\lambda)&w^{\prime}_{\mbox{\tiny II}}(\pi,% \lambda)\end{bmatrix}.$

Equivalently,

 28.29.9 $2\cos\left(\pi\nu\right)=w_{\mbox{\tiny I}}(\pi,\lambda)+w_{\mbox{\tiny II}}^{% \prime}(\pi,\lambda).$

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

 28.29.10 $F_{\nu}(z)=e^{\mathrm{i}\nu z}P_{\nu}(z),$ ⓘ Defines: $F_{\nu}(z)$: Floquet solution (locally) Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/28.29.E10 Encodings: TeX, pMML, png See also: Annotations for §28.29(ii), §28.29 and Ch.28

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

If $\nu$ $(\neq 0,1)$ 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

 28.29.11 $w(z+\pi)=(-1)^{\nu}w(z)+cP(z),$

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

 28.29.12 $\begin{bmatrix}w_{\mbox{\tiny I}}(\pi,\lambda)&w_{\mbox{\tiny II}}(\pi,\lambda% )\\ w^{\prime}_{\mbox{\tiny I}}(\pi,\lambda)&w^{\prime}_{\mbox{\tiny II}}(\pi,% \lambda)\end{bmatrix}=\begin{bmatrix}(-1)^{\nu}&0\\ 0&(-1)^{\nu}\end{bmatrix}.$

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 $n\pi$, $n=3,4,\dots$, then all solutions are periodic with period $n\pi$.

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

 28.29.13 $w(z+\pi)+w(z-\pi)=2\cos\left(\pi\nu\right)w(z).$

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

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

 28.29.14 $\cos\left(\pi\nu\right)=w_{\mbox{\tiny I}}(\pi,\lambda).$

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 $w_{\mbox{\tiny I}}(z,\lambda),w_{\mbox{\tiny II}}(z,\lambda)$, respectively.

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

§28.29(iii) Discriminant and Eigenvalues in the Real Case

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

The function

 28.29.15 $\bigtriangleup(\lambda)=w_{\mbox{\tiny I}}(\pi,\lambda)+w_{\mbox{\tiny II}}^{% \prime}(\pi,\lambda)$

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

For a given $\nu$, the characteristic equation $\bigtriangleup(\lambda)-2\cos\left(\pi\nu\right)=0$ has infinitely many roots $\lambda$. Conversely, for a given $\lambda$, the value of $\bigtriangleup(\lambda)$ 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:

 28.29.16 $\displaystyle\lambda_{n},\;n$ $\displaystyle=0,1,2,\dots,\mbox{ with \bigtriangleup(\lambda_{n})=2},$ ⓘ Defines: $\lambda_{n}$: eigenvalues (locally) Symbols: $n$: integer Referenced by: §28.30(i) Permalink: http://dlmf.nist.gov/28.29.E16 Encodings: TeX, pMML, png See also: Annotations for §28.29(iii), §28.29 and Ch.28 28.29.17 $\displaystyle\mu_{n},\;n$ $\displaystyle=1,2,3,\dots,\mbox{ with \bigtriangleup(\mu_{n})=-2}.$ ⓘ Defines: $\mu_{n}$: eigenvalues (locally) Symbols: $n$: integer Referenced by: §28.30(i) Permalink: http://dlmf.nist.gov/28.29.E17 Encodings: TeX, pMML, png See also: Annotations for §28.29(iii), §28.29 and Ch.28

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 $\mu_{n}\to\infty$ as $n\to\infty$, and interlace according to the inequalities

 28.29.18 $\lambda_{0}<\mu_{1}\leq\mu_{2}<\lambda_{1}\leq\lambda_{2}<\mu_{3}\leq\mu_{4}<% \lambda_{3}\leq\lambda_{4}<\cdots.$ ⓘ Symbols: $\lambda_{n}$: eigenvalues and $\mu_{n}$: eigenvalues Referenced by: §28.30(i) Permalink: http://dlmf.nist.gov/28.29.E18 Encodings: TeX, pMML, png See also: Annotations for §28.29(iii), §28.29 and Ch.28

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

 28.29.19 $N=\frac{1}{\pi}\int_{0}^{\pi}\left(Q(x)\right)^{2}\mathrm{d}x,$ ⓘ Defines: $N$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\int$: integral, $x$: real variable and $Q(z)$: function Permalink: http://dlmf.nist.gov/28.29.E19 Encodings: TeX, pMML, png See also: Annotations for §28.29(iii), §28.29 and Ch.28

we have for $m\to\infty$

 28.29.20 $\displaystyle\mu_{2m-1}-(2m-1)^{2}-\dfrac{N}{(4m)^{2}}$ $\displaystyle=o\left(m^{-2}\right),$ $\displaystyle\mu_{2m}-(2m-1)^{2}-\dfrac{N}{(4m)^{2}}$ $\displaystyle=o\left(m^{-2}\right),$ ⓘ Symbols: $o\left(\NVar{x}\right)$: order less than, $m$: integer, $\mu_{n}$: eigenvalues and $N$ Permalink: http://dlmf.nist.gov/28.29.E20 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.29(iii), §28.29 and Ch.28
 28.29.21 $\displaystyle\lambda_{2m-1}-(2m)^{2}-\dfrac{N}{(4m)^{2}}$ $\displaystyle=o\left(m^{-2}\right),$ $\displaystyle\lambda_{2m}-(2m)^{2}-\dfrac{N}{(4m)^{2}}$ $\displaystyle=o\left(m^{-2}\right).$ ⓘ Symbols: $o\left(\NVar{x}\right)$: order less than, $m$: integer, $\lambda_{n}$: eigenvalues and $N$ Permalink: http://dlmf.nist.gov/28.29.E21 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.29(iii), §28.29 and Ch.28

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

 28.29.22 $\displaystyle\lambda_{2m}-\lambda_{2m-1}$ $\displaystyle=o\left(\ifrac{1}{m^{k}}\right),$ $\displaystyle\mu_{2m}-\mu_{2m-1}$ $\displaystyle=o\left(\ifrac{1}{m^{k}}\right);$ ⓘ Symbols: $o\left(\NVar{x}\right)$: order less than, $m$: integer, $\lambda_{n}$: eigenvalues, $\mu_{n}$: eigenvalues and $k$: number Permalink: http://dlmf.nist.gov/28.29.E22 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.29(iii), §28.29 and Ch.28

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