Received 10 October 2014; revised 10 November 2014; accepted 17 November 2014

ABSTRACT

We investigate the Hill differential equation where and are trigonometric polynomials. We are interested in solutions that are even or odd, and have period or semi-period. 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.

Keywords:

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

(2.1)

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

(2.2)

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

(2.3)

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

(2.4)

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

(2.5)

with associated pzlynomial

(2.6)

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

(2.7)

with associated polynomial

(2.8)

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

(2.9)

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

(3.1)

is even with semi period if and only if

(3.2)

is odd with semi period if and only if

(3.3)

is odd with period if and only if

(3.4)

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

(3.5)

where

(3.6)

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

Proof. Let (3.5) can be written as,

(3.7)

which is equivalent to

(3.8)

Noting that

and

we see that

Therefore, (3.8) can be written as

(3.9)

Since is strictly positive, the lemma follows. □

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

(3.10)

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

(3.11)

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

(3.12)

By introducing the functions

we note that the adjoint of (2.1) has the same form and can be written in the following form:

(3.13)

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

(3.14)

(3.15)

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

(3.16)

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

(3.17)

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

(4.1)

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

(4.2)

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

(4.3)

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

(4.4)

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

Theorem 4.1. Each of the function systems

(4.5)

(4.6)

(4.7)

(4.8)

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

(4.9)

(4.10)

(4.11)

(4.12)

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

, (4.13)

, (4.14)

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

(4.15)

(4.16)

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

(4.17)

(4.18)

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

(4.19)

(4.20)

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

(5.1)

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

(5.2)

We consider the differential operator

(5.3)

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

(5.4)

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,

(5.5)

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

(5.6)

where

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

(5.7)

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

(5.8)

where

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

(5.9)

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

(5.10)

(5.11)

(5.12)

(5.13)

6. Fourier Series

The generalized Ince functions admit the following Fourier series expansions

(6.1)

(6.2)

(6.3)

(6.4)

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

(6.5)

(6.6)

(6.7)

(6.8)

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

(6.9)

(6.10)

where

Therefore, we can write

(6.11)

(6.12)

(6.13)

(6.14)

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

(6.15)

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