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28 Mathieu Functions and Hill’s EquationHill’s Equation

§28.29 Definitions and Basic Properties

Contents

§28.29(i) Hill’s Equation

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

28.29.1 w′′(z)+(λ+Q(z))w=0,

with

28.29.2 Q(z+π)=Q(z),

and

28.29.3 0πQ(z)z=0.

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

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

The basic solutions wI(z,λ), wII(z,λ) are defined in the same way as in §28.2(ii) (compare (28.2.5), (28.2.6)). Then

28.29.4 wI(z+π,λ) =wI(π,λ)wI(z,λ)+wI(π,λ)wII(z,λ),
28.29.5 wII(z+π,λ) =wII(π,λ)wI(z,λ)+wII(π,λ)wII(z,λ).

Let ν be a real or complex constant satisfying (without loss of generality)

28.29.6 -1<ν1

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

28.29.7 w(z+π)=πνw(z),

iff πν is an eigenvalue of the matrix

28.29.8 [wI(π,λ)wII(π,λ)wI(π,λ)wII(π,λ)].

Equivalently,

28.29.9 2cos(πν)=wI(π,λ)+wII(π,λ).

This is the characteristic equation of (28.29.1), and cos(πν) is an entire function of λ. Given λ together with the condition (28.29.6), the solutions ±ν 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 ν (or Floquet solution). It has the form

28.29.10 Fν(z)=νzPν(z),

where the function Pν(z) is π-periodic.

If ν (0,1) is a solution of (28.29.9), then Fν(z), F-ν(z) comprise a fundamental pair of solutions of Hill’s equation.

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

28.29.11 w(z+π)=(-1)νw(z)+cP(z),

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

28.29.12 [wI(π,λ)wII(π,λ)wI(π,λ)wII(π,λ)]=[(-1)ν00(-1)ν].

The solutions of period π or 2π are exceptional in the following sense. If (28.29.1) has a periodic solution with minimum period nπ, n=3,4,, then all solutions are periodic with period nπ.

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

28.29.13 w(z+π)+w(z-π)=2cos(πν)w(z).

A nontrivial solution w(z) is either a Floquet solution with respect to ν, or w(z+π)-νπw(z) is a Floquet solution with respect to -ν.

In the symmetric case Q(z)=Q(-z), wI(z,λ) is an even solution and wII(z,λ) is an odd solution; compare §28.2(ii). (28.29.9) reduces to

28.29.14 cos(πν)=wI(π,λ).

The cases ν=0 and ν=1 split into four subcases as in (28.2.21) and (28.2.22). The π-periodic or π-antiperiodic solutions are multiples of wI(z,λ),wII(z,λ), 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 (λ)=wI(π,λ)+wII(π,λ)

is called the discriminant of (28.29.1). It is an entire function of λ. Its order of growth for |λ| is exactly 12; see Magnus and Winkler (1966, Chapter II, pp. 19–28).

For a given ν, the characteristic equation (λ)-2cos(πν)=0 has infinitely many roots λ. Conversely, for a given λ, the value of (λ) is needed for the computation of ν. 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 λn, n =0,1,2,, with (λn)=2,
28.29.17 μn, n =1,2,3,, with (μn)=-2.

In consequence, (28.29.1) has a solution of period π iff λ=λn, and a solution of period 2π iff λ=μn. Both λn and μn as n, and interlace according to the inequalities

28.29.18 λ0<μ1μ2<λ1λ2<μ3μ4<λ3λ4<.

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

28.29.19 N=1π0π(Q(x))2x,

we have for m

28.29.20 μ2m-1-(2m-1)2-N(4m)2 =o(m-2),
μ2m-(2m-1)2-N(4m)2 =o(m-2),
28.29.21 λ2m-1-(2m)2-N(4m)2 =o(m-2),
λ2m-(2m)2-N(4m)2 =o(m-2).

If Q(x) has k continuous derivatives, then as m

28.29.22 λ2m-λ2m-1 =o(1/mk),
μ2m-μ2m-1 =o(1/mk);

see Hochstadt (1963).

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