Quasi-Exactly Solvable Differential Models: A Canonical Polynomials Approach ()
1. Introduction
Let D be a linear differential operator of degree
with polynomial coefficients, and let
be an exact solution of the differential equation:
(1)
where
are polynomials in x for
.
Among the very efficient techniques to solve equations of the form (1) is the Tau Method. This method was invented by Lanczos in [1] to solve simple equations and it was extended later by Ortiz [2] to treat more sophisticated problems with applications in a wide range of scientific fields. The basic idea of the Tau Method consists of finding polynomial solutions to Equation (1) in terms of a special set of polynomials associated with D called canonical polynomials, denoted by
. These are defined as the pseudo inverse image of the set of the power functions
by D.
In this paper, we use the canonical polynomials to develop a new method to solve quasi-exactly solvable (QES) second order ordinary differential equations. The QES problem, considered in this paper, can be stated as follows: Let p and n be two nonnegative integers, and suppose that
are polynomials with expressions
The coefficients
are found in terms of
and
so that the function
defined implicitly by the differential equation
(2)
is an exact polynomial solution of degree n and written as
, where
are determined in terms of
.
Quasi-exactly solvable problems have applications in engineering, chemistry and physics among many other fields. This includes a wide range of mathematical settings that involve Schrödinger equations describing problems in quantum mechanics, for example, anharmonic singular potentials, coulombically repelling electrons on a multidimensional sphere, Planer Dirac electron in magnetic fields and Kink stability analysis among many other problems (see [3] [4] [5] [6] and the references given therein).
QES problems are usually solved by means of the Functional Bethe Ansatz method or by a constraint polynomial approach (see [7] ). With the latter, we derive a
system of algebraic equations, which gives the n roots of
, and afterward the p coefficients of
are calculated. If one wishes to increase the order n of polynomial
, then a new nonlinear algebraic system of higher dimensions has to be resolved. With our proposed method that employs the canonical polynomials, we solve first a
system of algebraic equations giving the coefficients of
and then the coefficients of
are computed. With this method, a polynomial solution
of higher order n can be obtained without increasing dimensions
of the algebraic system.
Section 2 introduces the canonical polynomials in the context that fits the problem under consideration. In Section 3 the main results of this paper are given and discussed. Sections 4 and 5 illustrate the implementation of the proposed method.
2. The Canonical Polynomials
In this section we recall the main features of the canonical polynomials associated with D (see Ortiz [2] ). We begin with this definition.
Definition 1. For any integer
,
1)
is called the kth generating polynomial of D if
.
2)
is called a kth canonical function of D if
satisfies the differential equation
.
In [2] , Ortiz presented an algorithm in the form of a self-starting recursive formula for computing the
's associated with linear differential operators of arbitrary degree
. The following theorem gives a variation of this algorithm for second order differential operators of the form (2) in which this paper is concerned:
Theorem 1. The canonical functions associated with the differential operator (2) can be generated by the following recursion:
(3)
where
for any
with the convention that
for
.
Proof. For any
, let us expand the kth generating polynomial
in terms of
:
(4)
Since, by Definition 1,
, and since D is a linear operator, (4) becomes:
(5)
which implies that
from which follows the desired recursion (3):
For instance, when
, Equation (3) gives
where
(6)
That is,
has a polynomial component
of degree 0 and a residual component
, which is the subspace generated by
.
For
, we have
where
(7)
is a polynomial of degree 1 and the residual
.
Equations (6) and (7) suggest that any canonical function is the sum of a polynomial
, and a residual
. This is formulated in the following theorem:
Theorem 2. For all
,
can be written as
where
are called polynomials given by the recursion:
(8)
with
for all
. And
where
is a sequence of constants given by the recursion
(9)
with
for all i and
.
Proof. Equations (6)-(7) show that this theorem holds for
and
. The use of an induction argument allows to prove it for all k. To this end let us assume that
(10)
with
and let us take
. Combining (10) with (3) gives:
as required.
Corollary 1. Suppose that the above assumptions and notation hold true.
1) If
, then
are generated by the recursion:
And
where
2) If
, then
are generated by the recursion:
And
where
are constants given by the recursion
3. Solutions of Quasi-Exactly Solvable Models in Terms of {Qn}
Having obtained Algorithm (8)-(9) that generates the canonical polynomials that fit to our context, we can now formulate the main result of this paper:
Theorem 3. Let
and
be two given polynomials of degree
and
respectively, and suppose that
is an unknown polynomial of degree
. Let
. If
satisfy the following system of algebraic equations
(11)
(12)
where
are given by (9), then the polynomial
(13)
is an exact solution for the differential Equation (2), where
are given by (8).
Proof. Let
. If (11) holds true, then
and therefore setting
in (5) gives
This, in turn, implies that
That is,
(14)
is an exact solution for the differential Equation (2). For Expression (14) to be a polynomial, it must be independent of the residual terms. To this end, we proceed as follows:
Now, for y(x) to be independent of the p undefined elements
, the coefficients of the latter must be set equal to zero leading to the following algebraic system that consists of p equations:
Explicitly,
with the p unknowns
. This completes the proof of the theorem.
For illustration let us consider particular cases:
Case
. Then we have
where
satisfy the algebraic equation
where
are given by,
with
. And
where
are given by the recursion:
for
, with
.
Case
. In this case,
where
and
satisfy the
algebraic system
where
are given by the recursion
And the exact solution
is given as:
where
are given by the recursion:
4. Two Coulombically Repelling Electrons on a Sphere
This is a system governed by the ODE
(15)
where s denotes the inter-electronic distance between two electrons interacting via a Coulomb potential constrained to remain on the surface of a D-dimensional sphere (
) of radius R,
,
, E is the unknown energy, and
stands for the unknown inter-electronic wave function (see [8] [9] ).
Setting
and
, then Equation (15) can be written in the form (2) with
:
(16)
Let us apply the Algorithm (11)-(12) developed in the previous section to this example. Here
(17)
Let us determine
and
, and thereafter obtain R and E. The implementation of Algorithm (11)-(12) yields the following algebraic system that consists of two nonlinear equations with unknowns
:
(18)
(19)
where
are obtained by,
which is equivalent to the explicit recursion
Then the desired solution
follows from Equation (13):
where
are given by the recursion:
or, equivalently,
To illustrate let us consider different values for n.
Case
: In this case
and the values of
are the roots of the equation:
which are
.
When
we obtain:
The value
implies that
and if
lead to
. For these reasons, they are discarded.
Case
: In this case
and the values of
are the roots of the equations
which are
where
When
we obtain:
The values
lead to
and therefore they are discarded.
Case
: In this case
and the values of
are the roots
of the equation
which are
where
When
we obtain:
5. Planer Dirac Electron in Coulomb and Magnetic Fields
Such systems are modeled by the covariant Dirac equation (see [10] [11] [12] ),
which can be transformed to the following second order ODE
(20)
which is of the form (1) with
where
with
being integer.
This problem was solved in [13] using Bethe Ansatz approach. We will solve it by implementing Algorithm (11)-(12) with
. The main task is to find expressions for E, Z, B and the exact solution
as a polynomial of degree n.
Applying Algorithm (11)-(12) we find:
Case
: Let
. Then
, and
and
are the roots of the system:
Coefficient of Q0:
(21)
Coefficient of Q1:
(22)
First we solve Equation (21) and obtain
Substitute the latter in (22) we arrive to the equivalent cubic equation:
Using a computer algebra program such as Mathematica [14] , we find that this cubic equation has one real root only which is:
where
Further
This implies that
Case
: For this value of n we find that
, and
and
are the roots of the system:
which can be solved using Mathematica.
6. Conclusion
In this paper we proposed an efficient method that allows finding closed expressions for the solution of QES models in terms of the canonical polynomials associated with the given differential operator. Unlike the existing method, our method involves solving a system of nonlinear algebraic equations of which the dimensions depend on p and do not increase with n. Several examples were implemented to testify the efficiency of the proposed method.