Let us consider the ordinary differential operator D of order ,
where is a set of polynomials, for all , where p is a prescribed nonnegative integer called the height of D, with
This paper is concerned with the solution of the following problem: Given and , construct two polynomials, of degree n and of degree p such that the pair satisfies the following ordinary differential equation exactly
When , Equation (3) is called quasi exactly solvable (QES). This class of QES problems has applications in various fields of engineering, chemistry and quantum mechanics. Many different techniques to solve QES equations are reported in the literature: Among these are the Functional Ansatz Method, Constraint polynomial approach, asymptotic iteration method and Lie algebraic method (see  -  ). The case was also discussed very recently in , where the authors developed a new approach based on a special set of polynomials associated with the differential operator D called canonical polynomials. The main objective of this paper is to extend that canonical polynomials approach to solve equations of the form (3) with arbitrary order . More precisely, we present a procedure to construct a pair of polynomials based on the canonical polynomial associated with D. While the existing method for solving QES requires the solution of a nonlinear algebraic system with dimensions depending on the desired degree of y, the canonical polynomial approach presented in  requires a nonlinear algebraic system of dimensions depending on p only. This advantage is due to the fact that the sequence of canonical polynomials enjoys the permanence characteristic .
The canonical polynomials (to be explained shortly) appeared for the first time in  wherein Lanczos developed an efficient method, called the Tau method, to approximate the exact solution of differential equations in terms of a finite number of canonical polynomials. Later on, the concept of the canonical polynomial was generalized in  to develop a recursive approach of the Tau method that can apply to more complex differential equations. And it was due to the computational efficiency of the canonical polynomials that makes the Tau method more competitive compared to other existing approximation methods (more details can be found in  -  ).
Section 2 will concentrate on the construction of the canonical polynomials associated with the th differential operator (1) and on their computation. In Section 3 we present an algorithm that allows to obtain the pair of polynomials in an effective way. Two examples confirming our results are discussed in Section 4.
2. The Canonical Polynomials
Let D be the differential operator defined in (1). In this section we recall the main features of the canonical polynomials associated with D (see  ), and we give an algorithm for computing them. First rewrite (3):
where and . So, for the sake of simplicity, we shall hide the asterisk "*" and carry out the analysis for
keeping in mind that involves the unknown coefficients of .
Definition 1. For any integer , is called a kth canonical function of D if .
The following notation will enable us to formulate the next theorem:
Theorem 1. Under the above assumptions and notation, the canonical functions associated with the differential operator (4) are formally generated by the recursion:
In particular, if and , then
Since D is linear, the latter yields:
If , then is an exact solution.
If , then we obtain the desired formula for :
In particular, if then
If , then is an exact solution.
If , then
This completes the proof.
For illustration, when , Equation (7) gives:
Proceeding this way, we find that for
We are able now to formulate one of the main results of this paper:
Theorem 2. For all , each can be written in the form
where is a polynomial of degree k, called a canonical polynomial associated with D, and generated by the self starting recursive formula:
and where is a linear combination of the undefined canonical polynomials , called residual, and written as ,
where are sequences of constants given by the self starting recursion
Proof. This follows by an induction argument once (9) is inserted in (10) and the terms are rearranged:
yielding Equations (10) and (11) as required.
3. Construction of Solution
This section is concerned with the construction of the two polynomials that satisfy Equation (4).
Theorem 3. The above notation and assumptions hold. Suppose that the
coefficients of satisfy the following system
consisting of algebraic equations:
where are given by (11). Then
is an exact polynomial solution of Equation (4) where are parameters determined in terms of as defined in (5)-(6), and is a sequence of canonical polynomials associated with D and recursively generated by (10)
Proof. Let . Setting in Equation (8) we get
If condition (12) holds, then (5) implies that and consequently the right hand side of (15) vanishes:
In other words,
becomes an exact solution, but not necessarily an exact polynomial due to the appearance of the undefined canonical polynomials . However, in order to be an exact polynomial, must be independent of the p
undefined canonical functions . This can be achieved by an appropriate
adjustment of the p coefficients of as explained next. Using (9) in we can write:
Working out the coefficients of in (17) we find that due to Equation (13) we get:
Thus reduces to the polynomial (16)
The following corollary follows immediately from the previous theorem:
Corollary 4. If and
is an exact polynomial solution of Equation (4) where
(note that if and therefore all ’s are defined).
For computational purposes, one can reduce the height of D from p to zero by differentiating (4) p times. This is due to the following trivial identity:
Applying this identity to (4) we get
Inserting the later in (18) we get
which is a differential operator of order with height 0. Therefore we can apply our results to (22) and reconstruct the solution of the original problem by an antiderivative process. This will reduce the computation cost because the residual subspace of the new operator will be 0.
In the section we solve two applied problems by means of Algorithm ((12)-(13)-(14)-(18)-(19)) formulated in Theorem 3:
Example 1. Modified Manning potential with parameters. Let us consider the Schrodinger’s equation
where the potential is given by
where , are given constant parameters and is an unknown parameter and E is the unknown eigenvalue. We wish to compute E and . This potential describes a double-well potential
whenever , , and which was discussed in ( ,  ).
Equation (23) can be written as a 2nd order QES in the form (4) with height . Setting and
allows to write (23) as
We can reduce the height of Equation (24) from to by taking its first derivative:
which implies that
We solved Equation (25) by Algorithm ((12)-(13)-(14)) with , and .
First we use Equations (10)-(11) to compute the canonical polynomials associated with Equation (25). Here are some of them:
From (12), we have
which gives .
From (26), and therefore where takes the six values:
which are the zeros of the following polynomial
whose the plot is given in Figure 1. Further, for , we obtain obtain for the six values of are
Figure 1. Plot of whose the zeros are . Here .
The graphs of the six functions are shown in Figure 2.
Example 2. The Schrodinger’s equation of invariant decatic anharmonic oscillator in N-dimensional spherical coordinates is
where R stands for the radial wave function. Setting transforms Equation (27) to
Figure 2. Plot of the wave function for the six values of that are the roots of that appear at the top of each plot. .
where , being a positive integer.
Further, consider the transformation
where and are parameters that depend on two unknowns and :
and should computed with the eigenvalue E. This yields a second order ODE of the form (1) with height :
In order to obtain an equation with height , we differentiate it twice to get:
The canonical polynomials are obtained by recursion (10)-(11):
We have applied Algorithm ((12)-(13)-(14)) for different sets of parameters:
1) For , , , , , , , , ; , the unknown are determined by solving the following system:
Note that Equation (31) and Equation (33) are linear in and E respectively. So we compute and E in terms of and substitute their expressions in Equation (32) which gives the values of . As a result we get for two sets of solutions:
For Set 1 we have
and the exact solution of Equation (29) when is
Then the wave function for is
which is plotted in Figure 3.
2) For ; ; ; ; ; ; ; ; we obtain a system of equations with unknown . This system has three sets of solutions:
Figure 3. Plot of wave function for the Set 1 ( ).
The exact polynomial solution of Equation (29) that corresponds to each set of the computed parameters above are:
Figure 4 shows .
3) For ; ; ; ; . Here are the results:
The exact solution that corresponds to Set 1:
Figure 4. Plot of wave function for the Set 1 ( ).
Figure 5. Plot of wave function for the Set 1 ( ).
Figure 5 shows .
In this paper we have extended the canonical polynomials approach that was developed in  to solve QES differential equations of arbitrary high order . While the existing methods for solving QESs require the solution of a nonlinear algebraic system whose dimensions depend on the desired degree of , our new approach requires solving a nonlinear algebraic system of dimensions depending on p, the height of the differential operator. This advantage is due to the fact that the sequence of canonical polynomials enjoys the permanence characteristic.
The financial support of the Public Authority for Applied Education and Training (PAAET), Kuwait, during this research is greatly appreciated.
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