Eigenvalue Approximations for Linear Periodic Differential Equations with a Singularity

where q is a real-valued, periodic function with period a. We further suppose that q has a single non-integrable singularity within [0, a] which is repeated by periodicity. We take the singularity to be at the point a 2 . More particularly, we suppose that q ∈ Lloc[0, a 2 ) ∪ ( a 2 , a]. It is well known, see for example [4] and [6], that for certain types of singularity (1.1) may be “regularized” in the sense that it may be transformed to a differential equation all of whose coefficients belong to L[0, a], and has spectral properties related to those of (1.1). This regularized form of (1.1) also gives rise to a Floquet theory similar to that described in [3] and [5]. Our object in this paper is to derive asymptotic estimates for the eigenvalues of (1.1) on [0, a] with periodic and semi-periodic boundary conditions. Our approach to regularizing (1.1) follows that used in [1], [4] and [6]. We illustrate our methods by calculating asymptotic estimates for the periodic and semi-periodic eigenvalues of (1.1) in the case where q(x) = 1/|x− 1| for x ∈ [0, 2] \ {1} and repeats by periodicity.


Introduction
We consider the second order, linear differential equation where q is a real-valued, periodic function with period a.We further suppose that q has a single non-integrable singularity within [0, a] which is repeated by periodicity.We take the singularity to be at the point a 2 .More particularly, we suppose that q ∈ L 1 loc [0, It is well known, see for example [4] and [6], that for certain types of singularity (1.1) may be "regularized" in the sense that it may be transformed to a differential equation all of whose coefficients belong to L 1 [0, a], and has spectral properties related to those of (1.1).This regularized form of (1.1) also gives rise to a Floquet theory similar to that described in [3] and [5].Our object in this paper is to derive asymptotic estimates for the eigenvalues of (1.1) on [0, a] with periodic and semi-periodic boundary conditions.Our approach to regularizing (1.1) follows that used in [1], [4] and [6].We illustrate our methods by calculating asymptotic estimates for the periodic and semi-periodic eigenvalues of (1.1) in the case where q(x) = 1/|x − 1| for x ∈ [0, 2] \ {1} and repeats by periodicity.
EJQTDE, 1999 No. 7, p.1 Our approach follows that of [4] and [6].The differences are necessitated by the fact that the singularity at a 2 is repeated by periodicity and when we consider (1.1) on intervals of the form [τ, τ + a] for 0 < τ < a we have to take account of the three possibilities: (c) τ = a 2 so there are singularities at both endpoints of [τ, τ + a].
Case a) and b) are similar so we will concentrate on case a) and describe the changes necessary for case b).Case c) is somewhat different and we consider this in §8 below.
We define the space where AC[c, d] denotes the set of functions absolutely continuous on [c, d].For α ∈ AC * [0, a] to be chosen later we define quasi-derivatives, y [i] as follows.
We set α := n r=1 f r where The choice of n depends on the singularity of q at a 2 .It is shown in [6] for example that if q(x) = (x − a 2 ) −k then an n may be found so that (2.5) regularizes (1.1) for 1 ≤ k < 2.
An example of a choice of α is given in §9 below.

Floquet Theory
It is well known that (2.3) gives rise to a number of eigenvalue problems over [0, a].We refer to [3], [5], and [7] for the details, but we summarize some of the results below.
The periodic problem with boundary conditions y(a, λ) = y(0, λ), y [1] (0, λ) = y [1] (a, λ) The semi-periodic problem with boundary conditions y(a, λ) = −y(0, λ), y [1] (a, λ) = −y [1] (0, λ) It is known [3 Theorem 4.4.1] and [5] that the numbers λ n and µ n occur in the order The intervals (λ 2m , µ 2m ) and (µ 2m+1 , λ 2m+1 ) are known to correspond to the stability intervals of (2.3) on R. Our primary objective is to investigate the location of these and to explore their relationship with the regularizing function α.We do this by deriving asymptotic estimates for the eigenvalues λ n and µ n of the periodic and semi-periodic problems.Our main tool is the following result due to Hochstadt [7].For 0 < τ < a we consider the problem of (2.Our approach to estimating the λ n and µ n is to derive approximations for the Λ τ,n .These depend explicitly on τ and we are able to approximate their maximum and minimum values. In considering (2.3) over [τ, τ + a], the value of τ , in the sense of which of a) b) or c) it satisfies is relevant to our argument in so far as it affects the position of the singularity.
estimate is too crude for our needs, in particular the dependence on τ is contained in the The detailed analysis of θ(τ + a, λ) depends on which of the three cases of §2 we are in.
We consider in some detail case a) and summarize the equivalent results for the other two.
The proof of these results is similar to the proof of the corresponding results in [6].The details may be found in [8].
Then, as λ → ∞, (5.4) Proof.We integrate (4.2) from τ to τ + a and see that EJQTDE, 1999 No. 7, p.6 ) cos(2θ(s, λ))ds. (5.5) We note that the first and third terms of (5.5) occur in (5.4).The fourth term is o(λ −1/2 ) by Lemma 2. The second term may be rewritten as (5.6) The first term on the right hand side of (5.6) occurs in (5.4).We rewrite the second as Integration by parts with the terms involving φ N and θ yields the integrated term of (5.4).
The proof is similar to the proof Theorem 1 and uses the fact that θ( a A problem with the use of Theorem 1 to compute eigenvalue approximations lies in the fact that the computation of φ N (t, λ) requires knowledge of θ( a 2 , λ).We require a secondary iteration, involving Corollary 1, to approximate θ( a 2 , λ) in terms of θ(τ, λ).
These functions were introduced in [6].The changes in the analysis of [6] are due to the appearance of δ(t) which takes into account the fact that q and α are extended beyond [0, a] by periodicity.The proof of the following result is similar to the proof of ([6] Theorem 4.5), for the details we refer to [8].
Theorem 2 If M and N are chosen as in (5.3) and (6.2) then as λ → ∞.

.
The situation now is somewhat different from that considered above and also from that of [6] in as much as q has singularities at both endpoints of the interval [τ, for j = 1, 2, ..., N where N is to be chosen later.We also define two sequences {ξ The corresponding result for the right hand singularity is as follows.
The proofs of these results are similar to the proof of Theorem 1.The details may again be found in [8].
We remark that unlike cases a) and b), there is no need to use a secondary iteration to approximate θ at the singularities since θ( a 2 , λ) and θ( 3a 2 , λ) for λ = Λ n, a 2 are known from the boundary conditions.