On the High-Order Quasi Exactly Solvable Differential Equations

In this paper, we present a new method for solving a class of high-order quasi exactly solvable ordinary differential equations. With this method, the computed solution is expressed as a linear combination of the canonical polynomials associated with the given differential operator. An iterative algorithm summarizing the procedure is presented and its efficiency is demonstrated through considering two applied problems.


Introduction
Let us consider the ordinary differential operator D of order  method (see [1]- [7]). The case 2 ν = was also discussed very recently in [8], 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 2 ν ≥ . 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 [8] 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 [9].
The canonical polynomials (to be explained shortly) appeared for the first time in [10] 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 [11] 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 [12]- [17]). Section 2 will concentrate on the construction of the canonical polynomials associated with the ν th differential operator (1) and on their computation. In Section 3 we present an algorithm that allows to obtain the pair of polynomials

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 [11]), and we give an algorithm for computing them. First rewrite (3): 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 ( ) The following notation will enable us to formulate the next theorem: where and : Theorem 1. Under the above assumptions and notation, the canonical functions associated with the differential operator (4) are formally generated by the recursion: Since D is linear, the latter yields: is an exact solution.
This completes the proof.
For illustration, when 0 k = , Equation (7) gives: Proceeding this way, we find that for 0,1, 2,3, We are able now to formulate one of the main results of this paper: is a polynomial of degree k, called a canonical polynomial associated with D, and generated by the self starting recursive formula: and where k p R + is a linear combination of the undefined canonical polynomials 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.

Construction of Solution
This section is concerned with the construction of the two polynomials that satisfy Equation (4).
are given by (11). Then ( ) is an exact polynomial solution of Equation (4) where is a sequence of canonical polynomials associated with D and recursively generated by (10) Proof. Let If condition (12) holds, then (5) implies that ( ) 0 n p α = and consequently the right hand side of (15) vanishes: and therefore all k Q 's are defined).

Computational aspects
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 From (20), which is a differential operator of order p ν + 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.

Applications
In the section we solve two applied problems by means of Algorithm ( (12) where the potential ( ) V x is given by ( ) v is an unknown parameter and E is the unknown eigenvalue. We wish to compute E and 3 v . This potential describes a double-well potential allows to write (23) as We can reduce the height of Equation (24) from 1 p = to 0 p = by taking its first derivative: where R stands for the radial wave function. Setting ( ) ( )  ( )  , , E λ λ are determined by solving the following system:   which is plotted in Figure 3.