Acta Polytechnica Vol. 47 No. 2–3/2007 Root Asymptotics of Spectral Polynomials

We have been studying the asymptotic energy distribution of the algebraic part of the spectrum of the one-dimensional sextic anharmonic oscillator. We review some (both old and recent) results on the multiparameter spectral problem and show that our problem ranks among the degenerate cases of Heine-Stieltjes spectral problem, and we derive the density of the corresponding probability measure.


QES sextic AHOs
Our model is a 1D sextic anharmonic oscillator, which is frequently used to approximate various situations in quantum mechanics.Due to the quasi-exact solvability a part of the spectrum is easily tractable and two parameters of the potential leave enough space for the choice of application.To be more precise, we have been studying the solutions of the where J is a non-negative integer.During the last decade there has been considerable interest in non-selfadjoint problems, e.g.[1], [2].We recognize that b Î C is a challenge, but it goes beyond the scope of this paper, and we confine ourselves to b Î R.
Adopting the idea of Bender and Dunne [3], we seek the solution y of the Schrödinger equation We are interested in even solutions only, and the odd ones can be treated by analogy.For y to be a solution, the coefficients P n must fulfil the recurrence relation We see that the wave function y serves as a generating function for the polynomials P n (E).Choosing a particular value of J we get the particular solution of our problem, but for any non-negative value of J the infinite system of P n forms an orthogonal system of monic polynomials with respect to a probability measure (Favard's theorem [4], [5]).The peculiarity is that the polynomials of degree J or higher contain a common factor P J -1 , i.e. they can be factorized.The roots of the P J polynomial are the eigenvalues of the Schrödinger equation.These polynomials have other interesting properties [3], but we shall not use them.
The algebraic part of the spectrum, i.e. the quasi-exact energy eigenvalues, are solutions of P E J ( ) = 0 and the corresponding eigenfunctions are products We change the variable x t 2 ® and reduce the Schrödinger We are thus faced by a linear differential operator and its polynomial solutions.This is a problem that has been studied by many prominent mathematicians since the first half of the 19 th century, and many nice results have been achieved.

Heine-Stieltjes spectral problem
Generally, a linear differential operator with poynomial coefficients is called a (higher) Lamé operator (e.g.[6]).An important number is called the Fuchs index of L( ) z .The case r = 0 is called exactly solvable, and it has been well studied.These operators and their polynomial eigensolutions have many interesting properties.The operator L( ) The multiparameter spectral problem where a polynomial V(z) of degree at most r is sought so that the above equation will have a polynomial solution S(z) of degree n.This is called the (higher) Heine-Stieltjes spectral problem, V(z) a (higher) Van Vleck polynomial, and S(z) a (higher) Stieltjes polynomial.

Non-degenerate cases
Two important results are worth mentioning at this point.First, a generalization of the result by Heine (cf [6] ).
with a 1 <a 2 <… a l real and b 1 , …, b l positive.Then the following theorem holds.
Theorem (Stieltjes, Van Vleck, Bôcher, [6]) For any integer n ³1 1) there exist exactly n l n such that equation (2) has a polynomial solution S of degree exactly n.
2) each root of each V and S is real, simple, and belongs to the interval (a 1 , a l ).
3) none of the roots of S can coincide with some of a i 's.More- Recently, Borcea and Shapiro [7] found the density of the asymptotic root distribution for the Stieltjes polynomials.

The degenerate case
Inspecting the structure of (1) we see that it is a degenerate case of the r =1operator.The roots of Van Vleck polynomials are not confined to a finite interval as n ® ¥.In order to find their limitng distribution we have to scale them.To this end we must know their rate of growth.Thus, we are interested in the roots of the maximal modulus and their dependence on parameter b.To find this dependence we follow the main idea of the Gräffe-Lobachevskii method.Let and we find expressions for the sums of the powers of the roots Then for the root of the maximal modulus E max holds Because s k are related to p l by we need explicit expressions for p l up to l n k = -.These formulae can be derived from the recurrence relation.

We arrive at
where and l = 0 1 2 , , ,K.Thus, lim lim ( ) lim ( ) i.e. the limit is independent of b.Our conjecture is that lim ( ) Polynomials Sp n that have the same roots as P n but scaled by C Cn n = 3 2 do not fulfil any finite recurrence relation.However, we can rewrite them as a determinant of an n×n where x a g x a g x a and This enables us to extend the polynomial sequence by in- , where M n,i is the upper i×i principal submatrix of M n and this enables us to use the result of Kuijlaars and Van Assche concerning the three-term recurrence relations with variable coefficients [8].

This relation reads
Sp with initial conditionsSp n,0 1 = and Sp n,-= 1 0. Now, it remains to go to the polynomials Tp n,i that have the same roots as Sp n,i and fulfil the relation ETp a Tp Tp a Tp x , with The limits determine the density of the asymptotic root-counting measure: We immediately see that the integral does not converge for s = 0.The polynomial under the square root has three real roots t t t Schrödinger equation H E y y = , where y Î L 2 ( )