A Generalization of Ince’s Equation

We investigate the Hill differential equation ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ′′ ′ 1 0, A t y t B t y t D t y t + + + + = λ where ( ) , A t ( ) , B t and ( ) D t are trigonometric polynomials. We are interested in solutions that are even or odd, and have period π or semi-period π . The above equation with one of the above conditions constitutes a regular Sturm-Liouville eigenvalue problem. We investigate the representation of the four Sturm-Liouville operators by infinite banded matrices.


, A t y t B t y t D t y t
λ where ( ),

Introduction
The first known appearance of the Ince equation, is in Whittaker's paper ( [1], Equation ( 5)) on integral equations.Whittaker emphasized the special case 0 a = , and this special case was later investigated in more detail by Ince [2] [3].Magnus and Winkler's book [4] contains a chapter dealing with the coexistence problem for the Ince equation.Also Arscott [5] has a chapter on the Ince equation with 0 a = .One of the important features of the Ince equation is that the corresponding Ince differential operator when applied to Fourier series can be represented by an infinite tridiagonal matrix.It is this part of the theory that makes the Ince equation particularly interesting.For instance, the coexistence problem which has no simple solution for the general Hill equation has a complete solution for the Ince equation (see [6]).
When studying the Ince equation, it became apparent that many of its properties carry over to a more general class of equations "the generalized Ince equation".These linear second order differential equations describe important physical phenomena which exhibit a pronounced oscillatory character; behavior of pendulum-like systems, vibrations, resonances and wave propagation are all phenomena of this type in classical mechanics, (see for example [7]), while the same is true for the typical behavior of quantum particles (Schrödinger's equation with periodic potential [8]).

The Differential Equation
We consider the Hill differential equation   The real number λ is regarded as a spectral parameter.We further assume that will play an important role in the analysis of (2.1).For ease of notation we also introduce the polynomials ) is a natural generalization to the original Ince equation Ince's equation by itself includes some important particular cases, if we choose for example 0, we obtain the famous Mathieu's equation with associated pzlynomial where , q ν are real numbers, Ince's equation becomes Whittaker-Hill equation with associated polynomial ( ) ( ) (2.9)

Eigenvalues
Equation (2.1) is an even Hill equation with period π .We are interested in solutions which are even or odd and have period π or semi period π i.e. ( ) ( ).

y t y t + π = ±
We know that ( ) y t is a solution to (2.1) then ( ), y t + π and ( ) y t − are also solutions.From the general theory of Hill equation (see [9], Theorem 1.3.4);we obtain the following lemmas: Lemma 3.1.Let ( ) y t be a solution of (2.1), then ( ) y t is even with period π if and only if ( ) ( ) Therefore, (3.8) can be written as η ≥ the function can be computed explicitly using Maple.For example, let us consider the case 2,  to correspond to the boundary conditions in lemma 3.1 respectively.This notation is consistent with the theory of Mathieu and Ince's equations (see [4] [10]).Lemma 3.1 implies the following theorem.

Theorem 3.2. The generalized Ince equation admits a nontrivial even solution with period π if and only if
m ∈  it admits a nontrivial even solution with semi-period π if and only if , , m ∈  it admits a nontrivial odd solution with semi-period π if and only if = a b d for some 0 ; m ∈  it admits a nontrivial odd solution with period π if and only if subject to the boundary conditions of lemma 3.1.We have the following for the eigenvalues λ in terms of 0,1, 2, m =  .
1) Even with period π we have .
2) Even with semi-period π we have ( ) 3) Odd with semi-period π we have ( ) 4) Odd with semi-period π we have ( ) The formal adjoint of the generalized Ince equation is By introducing the functions ∑ we note that the adjoint of (2.1) has the same form and can be written in the following form: For ease of notation, let  ω is a solution to its formal adjoint.Since the function ω is even with period π , the boundary condition for y and y ω are the same.Therefore we have the following theorem.
Theorem 3.5.We have for , , From Sturm-Liouville theory we obtain the following statement on the distribution of eigenvalues.Theorem 3.6.The eigenvalues of the generalized Ince equation satisfy the inequalities The theory of Hill equation [4] gives the following results.
; , , It is uniquely determined up to a constant factor.We denote this Ince function by   The adopted normalization of Ince functions is easily expressible in terms of the Fourier coefficients of Ince functions and so is well suited for numerical computations [6]; However, it has the disadvantage that Equations (4.13) and ( 4 ; , t ω a b is very simple, then it is possible to evaluate the integrals in (4.17), (4.18) in terms of the Fourier coefficients of the generalized Ince functions.This provides another way to to calculate m c and m s .Once we know m c and m s , we can evaluate the integrals on the left-hand sides of the following equations  The integrals on the right-hand sides of (4.19) and (4.20) are easy to calculate once we know the Fourier series of Ince functions.

Operators and Banded Matrices
In this section we introduce four linear operators associated with Equation (2.1), and represent them by banded matrices of width 2 1.
η + It is this simple representation that is fundamental in the theory of the generalized Ince equation.We assume known some basic notions from spectral theory of operators in Hilbert space.
Let 1 H be the Hilbert space consisting of even, locally square-summable functions : f →   with period π .The inner product is given by By restricting functions to [ ] We also consider a second inner product We consider the differential operator S is given by the operator   be the space of square-summable sequences with its standard inner product , .Then ( ) ( ) x nt defines a bijective linear map ( ) Let n e denotes the sequence with a 1 in the th n position and 0's in all other positions, we also , ( )  We find that the operator 1 M can be represented in the following way, n n j n n j n j j j q e n M e r e q e q e n m Ic Now consider the operator 2 S that is defined as 1 S in (5.3) but in the Hilbert space 2 H consisting of even functions with semi-period π .This operator has eigenvalues , , , , : cos 2 1 , Consider the operator where Finally, consider the operator 4 S that is defined as : . (5.13)

Fourier Series
The generalized Ince functions admit the following Fourier series expansions Here η is a positive integer, the coefficients , We will at times represent the coefficients ,

= a b d for some 0 . m ∈  Example 3 . 3 .
To gain some understanding about the notation we consider the almost trivial completely solvable example, the so called Cauchy boundary value problem

≤
or λ belongs to one of the closed intervals with distinct endpoints , m =  then the generalized Ince equation is unstable.For all other real values of λ the equation is stable.In the case for some positive integer m and the parameters , a , b d the degenerate interval [ ] By theorem 3.2, the generalized Ince's equation with ( ) admits a non trivial even solution with period π .It is uniquely determined up to a constant factor.We denote this Ince function by when it is normalized by the conditions ( ) admits a non trivial even solution with semi-period π .It is uniquely determined up to a constant factor.We denote this Ince function by admits a non trivial odd solution with semi-period π .

4 . 1 .
admits a non trivial odd solution with period π .It is uniquely determined up to a constant factor.We denote this Ince function by Liouville theory ([11] Chapter 8, Theorem 2.1) we obtain the following oscillation properties.Theorem Each of the function systems with respect to the weight ( ) d are positive and independent of , t and

1 D 1 S
S of definition of consists of all functions 1 y H ∈ for which y and y′ are absolutely continuous and 1 y H ′′ ∈ , by restricting functions to [ ] 0, 2 π , this corresponds to the usual domain of a Sturm- Liouville operator associated with the boundary conditions (3.1).It is known ([12] Chapter V, Section 3.6) that is self-adjoint with compact resolvent when considered as an operator in ( )

1 M
Note that the factor 2 should appear only with 0 .e is self-adjoint with compact resolvent in y ω This inner product generates a norm that is equivalent to the usual d and the corresponding eigenvectors form sequences of Fourier coefficients for the functions 2 .

4 ) 8 )
We did not indicate the dependence of the Fourier coefficients on , mUsing relations (4.13) and (4.14), we can represent the generalized functions in a different way constant function y is an eigenfunction corresponding to the eigenvalue 0 0. α = The adopted normalization of Section 4 implies that It is a generalized Ince polynomial (even with period π ).