§28.29 Definitions and Basic Properties

Referenced by:: Ch.28
Permalink:: http://dlmf.nist.gov/28.29

§28.29(i) Hill’s Equation
§28.29(ii) Floquet’s Theorem and the Characteristic Exponent
§28.29(iii) Discriminant and Eigenvalues in the Real Case

§28.29(i) Hill’s Equation

Notes:: See Magnus and Winkler (1966, p. 8), Arscott (1964b, §7.1), and McLachlan (1947, §6.10).
Keywords:: Hill’s equation
Referenced by:: §29.7(ii)
Permalink:: http://dlmf.nist.gov/28.29.i

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)
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with

28.29.2 $Q(z+\pi)=Q(z),$

Symbols:: $z$ : complex variable and $Q(z)$ : function
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and

28.29.3 $\int _{0}^{\pi}Q(z)dz=0.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $Q(z)$ : function
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$Q(z)$ is either a continuous and real-valued function for $z\in\Real$ 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

Notes:: See Magnus and Winkler (1966, §§1.2 and 1.3).
Keywords:: Floquet solutions, Floquet’s theorem, Hill’s equation, characteristic equation, pseudoperiodic solutions
Permalink:: http://dlmf.nist.gov/28.29.ii

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

Symbols:: $z$ : complex variable, $w(z)$ : Mathieu’s equation solution, $w_{{\mbox{\tiny I}}}(z;a,q)$ : fundamental solution, $w_{{\mbox{\tiny II}}}(z;a,q)$ : fundamental solution and $\lambda$ : parameter
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28.29.5 $w_{{\mbox{\tiny II}}}(z+\pi,\lambda)=w_{{\mbox{\tiny II}}}(\pi,\lambda)w_{{\mbox{\tiny I}}}(z,\lambda)+w^{{\prime}}_{{\mbox{\tiny II}}}(\pi,\lambda)w_{{\mbox{\tiny II}}}(z,\lambda).$

Symbols:: $z$ : complex variable, $w(z)$ : Mathieu’s equation solution, $w_{{\mbox{\tiny I}}}(z;a,q)$ : fundamental solution, $w_{{\mbox{\tiny II}}}(z;a,q)$ : fundamental solution and $\lambda$ : parameter
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Let $\nu$ be a real or complex constant satisfying (without loss of generality)

28.29.6 $-1<\realpart{\nu}\leq 1$

Symbols:: $\realpart{}$ : real part and $\nu$ : complex parameter
Referenced by:: §28.29(ii)
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throughout this section. Then (28.29.1) has a nontrivial solution $w(z)$ with the pseudoperiodic property

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

Symbols:: $e$ : base of exponential function, $z$ : complex variable, $\nu$ : complex parameter and $w(z)$ : Mathieu’s equation solution
Referenced by:: §28.29(ii)
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iff $e^{{\pi i\nu}}$ is an eigenvalue of the matrix

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

Symbols:: $w(z)$ : Mathieu’s equation solution, $w_{{\mbox{\tiny I}}}(z;a,q)$ : fundamental solution, $w_{{\mbox{\tiny II}}}(z;a,q)$ : fundamental solution and $\lambda$ : parameter
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Equivalently,

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

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\nu$ : complex parameter, $w_{{\mbox{\tiny I}}}(z;a,q)$ : fundamental solution, $w_{{\mbox{\tiny II}}}(z;a,q)$ : fundamental solution and $\lambda$ : parameter
Referenced by:: §28.29(ii), §28.29(ii), §28.29(ii)
Permalink:: http://dlmf.nist.gov/28.29.E9
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This is the characteristic equation of (28.29.1), and $\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)$ 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 $F_{{\nu}}(z)=e^{{i\nu z}}P_{{\nu}}(z),$

Symbols:: $e$ : base of exponential function, $z$ : complex variable, $\nu$ : complex parameter and $F_{{\nu}}(z)$ : Floquet solution
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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 $w(z+\pi)=(-1)^{{\nu}}w(z)+cP(z),$

Symbols:: $z$ : complex variable, $\nu$ : complex parameter, $w(z)$ : Mathieu’s equation solution and $c$ : constant
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where $c$ is a constant. The case $c=0$ is equivalent to

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

Symbols:: $\nu$ : complex parameter, $w(z)$ : Mathieu’s equation solution, $w_{{\mbox{\tiny I}}}(z;a,q)$ : fundamental solution, $w_{{\mbox{\tiny II}}}(z;a,q)$ : fundamental solution and $\lambda$ : parameter
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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 $w(z+\pi)+w(z-\pi)=2\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)w(z).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $z$ : complex variable, $\nu$ : complex parameter and $w(z)$ : Mathieu’s equation solution
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A nontrivial solution $w(z)$ is either a Floquet solution with respect to $\nu$ , or $w(z+\pi)-e^{{i\nu\pi}}w(z)$ 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 $\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)=w_{{\mbox{\tiny I}}}(\pi,\lambda).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\nu$ : complex parameter, $w_{{\mbox{\tiny I}}}(z;a,q)$ : fundamental solution and $\lambda$ : parameter
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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

Notes:: See Magnus and Winkler (1966, pp. 11–36 and §2.4).
Keywords:: Hill’s equation, discriminant
Permalink:: http://dlmf.nist.gov/28.29.iii

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

The function

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

Symbols:: $w_{{\mbox{\tiny I}}}(z;a,q)$ : fundamental solution, $w_{{\mbox{\tiny II}}}(z;a,q)$ : fundamental solution and $\lambda$ : parameter
Permalink:: http://dlmf.nist.gov/28.29.E15
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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 $\bigtriangleup(\lambda)-2\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)=0$ 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 $\lambda _{n},\; n=0,1,2,\dots,\mbox{ with $\bigtriangleup(\lambda _{n})=2$},$

Symbols:: $n$ : integer and $\lambda _{n}$ : eigenvalues
Referenced by:: §28.30(i)
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28.29.17 $\mu _{n},\; n=1,2,3,\dots,\mbox{ with $\bigtriangleup(\mu _{n})=-2$}.$

Symbols:: $n$ : integer and $\mu _{n}$ : eigenvalues
Referenced by:: §28.30(i)
Permalink:: http://dlmf.nist.gov/28.29.E17
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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)
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Assume that the second derivative of $Q(x)$ in (28.29.1) exists and is continuous. Then with

28.29.19 $N=\frac{1}{\pi}\int _{0}^{\pi}\left(Q(x)\right)^{2}dx,$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $Q(z)$ : function and $N$
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we have for $m\to\infty$

28.29.20

$\mu _{{2m-1}}-(2m-1)^{2}-\dfrac{N}{(4m)^{2}}=\mathop{o\/}\nolimits\!\left(m^{{-2}}\right),$

$\mu _{{2m}}-(2m-1)^{2}-\dfrac{N}{(4m)^{2}}=\mathop{o\/}\nolimits\!\left(m^{{-2}}\right),$

Symbols:: $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $m$ : integer, $\mu _{n}$ : eigenvalues and $N$
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28.29.21

$\lambda _{{2m-1}}-(2m)^{2}-\dfrac{N}{(4m)^{2}}=\mathop{o\/}\nolimits\!\left(m^{{-2}}\right),$

$\lambda _{{2m}}-(2m)^{2}-\dfrac{N}{(4m)^{2}}=\mathop{o\/}\nolimits\!\left(m^{{-2}}\right).$

Symbols:: $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $m$ : integer, $\lambda _{n}$ : eigenvalues and $N$
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If $Q(x)$ has $k$ continuous derivatives, then as $m\to\infty$

28.29.22

$\lambda _{{2m}}-\lambda _{{2m-1}}=\mathop{o\/}\nolimits\!\left(\ifrac{1}{m^{k}}\right),$

$\mu _{{2m}}-\mu _{{2m-1}}=\mathop{o\/}\nolimits\!\left(\ifrac{1}{m^{k}}\right);$

Symbols:: $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $m$ : integer, $\lambda _{n}$ : eigenvalues, $\mu _{n}$ : eigenvalues and $k$ : number
Permalink:: http://dlmf.nist.gov/28.29.E22
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see Hochstadt (1963).

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

§28.29 Definitions and Basic Properties

Contents

§28.29(i) Hill’s Equation

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

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