1. Introduction
The first known appearance of the Ince equation,
![](//html.scirp.org/file/5-1720218x11.png)
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
![](//html.scirp.org/file/5-1720218x15.png)
![](//html.scirp.org/file/5-1720218x16.png)
![](//html.scirp.org/file/5-1720218x17.png)
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
).