A Generalization of Ince’s Equation

We investigate the Hill differential equation 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.

• Hill Equation
• Ince Equation
• Sturm-Liouville Problem
• Infinite Banded Matrix
• Eigenvalues
• Eigenfunctions

1. 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, 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.

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 equa- tion with periodic potential [8] ).

2. The Differential Equation

We consider the Hill differential equation

where

Here is a positive integer, the coefficients for are specified real numbers.

The real number is regarded as a spectral parameter. We further assume that Unless stated

otherwise solutions are defined for We will at times represent the coefficients for in the vector form:

The polynomials

will play an important role in the analysis of (2.1). For ease of notation we also introduce the polynomials

Equation (2.1) is a natural generalization to the original Ince equation

Ince’s equation by itself includes some important particular cases, if we choose for example we obtain the famous Mathieu’s equation

with associated pzlynomial

If we choose and where are real numbers, Ince’s equation becomes Whittaker-Hill equation

with associated polynomial

Equation (2.1) can be brought to algebraic form by applying the transformation For example when and we obtain

3. Eigenvalues

Equation (2.1) is an even Hill equation with period. We are interested in solutions which are even or odd and have period or semi period i.e. We know that is a solution to (2.1) then and 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 be a solution of (2.1), then is even with period if and only if

is even with semi period if and only if

is odd with semi period if and only if

is odd with period if and only if

Equation (2.1) can be written in the self adjoint form

where

Note that is even and -periodic since the function is continuous, odd, and - periodic.

Proof. Let (3.5) can be written as,

which is equivalent to

Noting that

and

we see that

Therefore, (3.8) can be written as

Since is strictly positive, the lemma follows. □

In the case of Ince’s Equation (2.4), we have the following formula for the function

When the function can be computed explicitly using Maple. For example, let us consider the case with Applying (3.6), we obtain

Equation (2.1) with one of the boundary conditions in lemma 3.1 is a regular Sturm-Liouville problem. From the theory of Sturm-Liouville ordinary differential equations it is known that such an eigenvalue problem has a sequence of eigenvalues that converge to infinity. These eigen values are denoted by and 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 if and only if for some it admits a nontrivial even solution with semi-period if and only if for some it admits a nontrivial odd solution with semi-period if and only if for some it admits a nontrivial odd solution with period if and only if for some

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

subject to the boundary conditions of lemma 3.1. We have the following for the eigenvalues in terms of.

1) Even with period we have

2) Even with semi-period we have

3) Odd with semi-period we have

4) Odd with semi-period 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:

Lemma 3.4. If is twice differentiable defined on then, is a solution to the generalized Ince equation if and only if is a solution to its adjoint.

Proof. We Know that

and

For ease of notation, let

then

Substituting for and and simplifying we obtain

□

From lemma 3.4 we know that if is twice differentiable, is a solution to the generalized Ince’s equation with parameters and if and only if is a solution to its formal adjoint. Since the function is even with period, the boundary condition for and 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.

Theorem 3.7. If or belongs to one of the closed intervals with distinct endpoints then the generalized Ince equation is unstable. For all other real values of the equation is stable. In the case

for some positive integer and the parameters the degenerate interval is not an instability interval: The generalized Ince equation is stable if

4. Eigenfunctions

By theorem 3.2, the generalized Ince’s equation with admits a non trivial even solution with period. It is uniquely determined up to a constant factor. We denote this Ince function by when it is normalized by the conditions and

The generalized Ince’s equation with admits a non trivial even solution with semi-period. It is uniquely determined up to a constant factor. We denote this Ince function by when it is normalized by the conditions and

The generalized Ince equation with admits a non trivial odd solution with semi-period. It is uniquely determined up to a constant factor. We denote this Ince function by when it is normalized by the conditions and

The generalized Ince equation with admits a non trivial odd solution with period. It is uniquely determined up to a constant factor. We denote this Ince function by when it is normalized by the conditions and

From Sturm-Liouville theory ( [11] Chapter 8, Theorem 2.1) we obtain the following oscillation properties.

Theorem 4.1. Each of the function systems

is orthogonal over with respect to the weight, that is, for

Moreover, each of the previous system is complete over.

Using the transformations that led to Theorem 3.5, we obtain the following result.

Theorem 4.2. We have

where and are positive and independent of and

with

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 (4.14) require coefficients and which are not explicitly known.

Of course, once the generalized Ince functions and are known we can express and in the form

If we square both sides of (4.13) and (4.14) and integrate, we find that

If 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 and.

Once we know and, 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.

5. 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 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 be the Hilbert space consisting of even, locally square-summable functions with period. The inner product is given by

By restricting functions to is isometrically isomorphic to the standard. We also consider a second inner product

We consider the differential operator

The domain of definition of consists of all functions for which and are absolutely continuous and, by restricting functions to, 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, and its eigenvalues are All eigenvalues of are simple. If we consider as an operator in the Hilbert space then its adjoint is given by the operator

on the same domain see ( [12] , Chapter III, Example 5.32). The adjoint is of the same form as but with replaced by respectively. By Theorem 3.5, we see that has the same eigen- values as Let be the space of square-summable sequences with its standard inner product Then

defines a bijective linear map Consider the operator defined on

Let denotes the sequence with a 1 in the position and 0’s in all other positions, we also define

i.e. and for We find that the operator

can be represented in the following way,

where and if and Note that the factor should appear only with

is self-adjoint with compact resolvent in equipped with the inner product This inner product generates a norm that is equivalent to the usual The operator has the eigenvalues and the corresponding eigenvectors form sequences of Fourier coefficients for the functions

Now consider the operator that is defined as in (5.3) but in the Hilbert space consisting of even functions with semi-period. This operator has eigenvalues with eigenfunctions Using the basis then,

defines a bijective linear map Consider the operator defined on

Let for we get the following formula for

where

Now consider the operator that is defined as but in the Hilbert space consisting of odd func-

tions with semi-period. This operator has the eigenvalues with eigenfunctions Using the basis functions

defines a bijective linear map Consider the operator defined on

Let for we have the following formula for

where

and

Finally, consider the operator that is defined as but in the Hilbert space consisting of odd functions with period. This operator has the eigenvalues with eigenfunctions Using the basis

defines a bijective linear map. Consider the operator defined on

Let for Then, the formula for is

where

Example 5.1. For the Whittaker-Hill Equation (2.7) in the following form [8]

the function from (3.6) is equal to 1, therefore the operators are self-adjoint on the

Hilbert spaces respectively. Hence the infinite matrices are sy- mmetric. They are represented by

6. Fourier Series

The generalized Ince functions admit the following Fourier series expansions

We did not indicate the dependence of the Fourier coefficients on The normalization of Ince functions implies

Using relations (4.13) and (4.14), we can represent the generalized functions in a different way

where

Therefore, we can write

where

and the Fourier coefficients and belong to the parameters Properties of the coefficients and follow from those of and

A generalized Ince function is called a generalized Ince polynomial of the first kind if its Fourier series (6.1), (6.2), (6.3), or (6.4) terminate. It is called a generalized Ince polynomial of the second kind if its expansion (6.11), (6.12), (6.13), or (6.14) terminate. If they exist, these generalized Ince polynomials and their corresponding eigenvalues can be computed from the finite subsections of the matrices of Section 5.

Example 6.1. Consider the equation

one can check that if we set any constant function is an eigenfunction corresponding to the

eigenvalue The adopted normalization of Section 4 implies that It is a generalized Ince polynomial (even with period).

References

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