On the Hill Discriminant of Lam´e’s Differential Equation

. Lam´e’s differential equation is a linear differential equation of the second order with a periodic coefficient involving the Jacobian elliptic function sn depending on the modulus k , and two additional parameters h and ν . This differential equation appears in several applications, for example, the motion of coupled particles in a periodic potential. Stability and existence of periodic solutions of Lam´e’s equations is determined by the value of its Hill discriminant D ( h, ν, k ). The Hill discriminant is compared to an explicitly known quantity including explicit error bounds. This result is derived from the observation that Lam´e’s equation with k = 1 can be solved by hypergeometric functions because then the elliptic function sn reduces to the hyperbolic tangent function. A connection relation be-tween hypergeometric functions then allows the approximation of the Hill discriminant by a simple expression. In particular, one obtains an asymptotic approximation of D ( h, ν, k ) when the modulus k tends to 1.


Introduction
Kim, Levi and Zhou [4] consider two elastically coupled particles positioned at x(t), y(t) in a periodic potential V (x).The system is described by where γ denotes the coupling constant.Let x(t) = y(t) = p(t) be a synchronous solution.If we linearize the system around this synchronous solution, x = p + ξ, y = p + η, and set u = ξ + η, w = ξ − η, then we obtain ü + V ′′ (p)u = 0, ẅ + (2γ + V ′′ (p))w = 0. (1.1) These are Hill equations [5], that is, they are of the form ẅ + q(t)w = 0 (1.2) with a periodic coefficient function q(t), say of period σ > 0. In this and many other applications the Hill discriminant D associated with (1.2) plays an important role.The discriminant D is defined as the trace of the endomorphism w(t) → w(t + σ) of the two-dimensional solution space of (1.2) onto itself.It is well known [5] that equation (1.2) is stable if |D| < 2 and unstable if |D| > 2. The condition D = 2 is equivalent to the existence of a nontrivial solution with period σ while D = −2 is equivalent to the existence of a nontrivial solution with semi-period σ.
In this work, we are interested in the special case V (x) = − cos x.Then p(t) is a solution of the differential equation p+sin p = 0 of the mathematical pendulum.We get [7,Section 22.19 with parameters h = k 2 (2γ + 1) and ν = 1.There is no explicit formula for the corresponding Hill discriminant D = D(h, ν, k).However, in [4] a remarkable asymptotic formula for this Hill discriminant as E → 0 (or k → 1) is given.It is shown that where The main result of this paper is Theorem 5.1 which improves on (1.4) in three directions.
1. We allow any real ν in place of ν = 1.Since we may replace ν by −1 − ν we assume ν ≥ − 1 2 without loss of generality.2. We provide explicit values for the amplitude a and the phase shift ϕ in (1.4) 3. We give explicit error bounds.This makes it possible to prove stability of the Lamé equation in some cases.
The idea behind the proof of Theorem 5.1 is the observation that Lamé's equation (1.3) with k = 1 can be solved in terms of the hypergeometric function F (a, b; c, x).Then well-known connection relations between hypergeometric functions play a crucial role.
As a preparation, we present some elementary results on linear differential equations of the second order in Section 2. In Section 3, we give a quick review of the Lamé equation.In Section 4, we consider the special case of the Lamé equation when k = 1.In Section 5 we combine our results to obtain Theorem 5.1.

Lemmas on second order linear equations
Let u be the solution of the initial value problem where q, r : [a, b] → R are continuous functions.By the variation of constants formula [2, Section 2.6], where y(t) = L(t, s) is the solution of determined by the initial conditions y(s) = 0, y ′ (s) = 1.Let L 1 , L 2 be constants such that Then it follows that where Let y be a solution of (2.1) and w a solution of w ′′ + p(t)w = 0 with y(a) = w(a) and y ′ (a) = w ′ (a).Then Proof .For u = y − w, we have The desired result follows from (2.3).■ Lemma 2.2.Let q : [a, b] → (0, ∞) be continuously differentiable and monotone.Set Let y 1 , y 2 be the solutions of (2.1) determined by y and, if q is nonincreasing, Proof .Suppose first that q is nondecreasing.Set Then If q is nonincreasing, we argue similarly using v j (t) = y ′ j (t) 2 + q(t)y j (t) 2 in place of u j .■

Lamé's equation
For h ∈ R, ν ≥ − 1 2 , k ∈ (0, 1), we consider the Lamé equation [1, Section IX] and [3, Section XV] This is a Hill equation with period 2K(k), where K = K(k) is the complete elliptic integral of the first kind: Equation (3.1) also makes sense for k = 1 [7, Section 22.5 (ii)] when it becomes Of course, this is not a Hill equation anymore.Let be the solutions of (3.1) determined by the initial conditions If k = 1, this is true for all 0 ≤ s ≤ t.
Proof .This follows from Lemma 2.2.■ In the next theorem, we use the complete elliptic integral E = E(k) of the second kind: Theorem 3.2.Suppose that h > 0 and h − ν(ν + 1) > 0. Then where the constants C are formed with k = 1.
The estimate for y ′ 2 is proved similarly. where Also note that [7, formula (19.9.1)]

H. Volkmer
This can be confirmed by direct computation.In order to determine the behaviour of the functions w j (t) as R ∋ t → ∞, we use the connection formula [7, formula (15.8.4)] and find w j (t) = Re(v j (t)), where and .
Proof .Using (4.5) and (5.1), we have 3).Now we use Theorems 3.2 and 4.1 to estimate As an illustration, take h = 6, ν = 1 2 and k = 1 − e −τ .Figure 1 depicts the graphs of τ → D 6, 1  2 , k (red) and τ → 2 Re Be 2iωK (black).These graphs are hard to distinguish for τ > 2. The Hill discriminant D 6, 1 2 , k is computed using (5.1).The values of y 1 (K) and y ′ 2 (K) are found by numerical integration of Lamé's equation (1.3) using the software Maple.6 Discussion and further work In Theorem 5.1, we presented an asymptotic formula describing the behavior of the discriminant of the Lamé equation (1.3) as k → 1.The proof is based on the fact that the Lamé equation approaches the associated Legendre (a special case of the hypergeometric) differential equation, and the known behavior of the hypergeometric function as the independent variable tends to 1.As we know from [4] a less precise formula describing the asymptotic behavior as E → 0 also exists for more general potentials in (1.1).It is an interesting research question whether there exist other potentials that allow an explicit determination of the amplitude and phase shift in this asymptotic formula.
This gives the desired statements (a) and (b) substituting the values for C j and C ′ j .■ We may use K(k) = ln(4/k ′ ) + O k ′2 ln k ′ [7, formula (19.12.1)] and e is − e it ≤ |s − t| for s, t ∈ R to obtain