AP07_2-3.vp


1 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 poten-
tial leave enough space for the choice of application. To be
more precise, we have been studying the solutions of the
Schrödinger equation H E� �� , where � � L2( )� and

H
x

x bx
b

J x� � � � � � �
�

�
�
�

	



�
�

d
d

2

2
6 4

2
2

4
4 3( )

where J is a non-negative integer. During the last decade
there has been considerable interest in non-selfadjoint prob-
lems, e.g. [1], [2]. We recognize that b �� is a challenge, but it
goes beyond the scope of this paper, and we confine ourselves
to b ��.

Adopting the idea of Bender and Dunne [3], we seek the
solution � of the Schrödinger equation H E� �� in the form

�( )
( )

!
x e

P E

n n
x

x b x n
n n

n

� �
�
�
�

	


�

�
�

�
�

	



�

� �

�

�
4 2

4 4 2

0

1
4 1

2
�


 .

We are interested in even solutions only, and the odd ones
can be treated by analogy. For � to be a solution, the coeffi-
cients Pn must fulfil the recurrence relation

P E E
n b

P E

n n J n

n n( )
( )

( )

( )( )

� �
��

�
�

	



�

� � � � �
�

�

�
4 3

2

16 1 2
3
2

1

�
	



� �P En 2( )

with the initial conditions P E� �1 0( ) and P E0 1( ) � .

We see that the wave function � serves as a generating
function for the polynomials Pn(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 Pn
forms an orthogonal system of monic polynomials with re-
spect to a probability measure (Favard’s theorem [4], [5]). The
peculiarity is that the polynomials of degree J or higher
contain a common factor PJ�1, i.e. they can be factorized.
The roots of the PJ polynomial are the eigenvalues of the
Schrödinger equation. These polynomials have other inter-
esting 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 EJ ( ) � 0 and the corre-
sponding eigenfunctions are products

�n x Q x T x( ) ( ) exp( ( ))�
2 2 .

We change the variable x t2 � and reduce the Schrödinger
equation to

4 4 2 2 4
2

0
2

2
2t

f

t
t bt

f
t

Jt E
b

f
d

d

d
d

� � � � � � �( ) ( ) (1)

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
19th century, and many nice results have been achieved.

2 Heine-Stieltjes spectral problem
Generally, a linear differential operator

�( ) ( )z Q z
z

i

i

i
i

k

�
�

 dd1

with poynomial coefficients is called a (higher) Lamé opera-
tor (e.g. [6]). An important number

r Q z ii k i� ��max (deg ( ) ), ,1 �

is called the Fuchs index of �( )z . The case r � 0 is called ex-
actly solvable, and it has been well studied. These operators
and their polynomial eigensolutions have many interesting
properties. The operator �( )z is called non-degenerate if
deg ( )Q z k rk � � .

The multiparameter spectral problem

�( ) ( ) ( ) ( )z S z V z S z� � 0,

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 de-
gree n. This is called the (higher) Heine-Stieltjes spectral
problem, V(z) a (higher) Van Vleck polynomial, and S(z) a
(higher) Stieltjes polynomial.

2.1 Non-degenerate cases
Two important results are worth mentioning at this point.

First, a generalization of the result by Heine (cf [6] ).

32 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 47 No. 2–3/2007

Root Asymptotics of Spectral
Polynomials
B. Shapiro, M. Tater

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.

Keywords: Lamé operator, Van Vleck polynomials, asymptotic root-counting measure.



©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 33

Acta Polytechnica Vol. 47 No. 2–3/2007

Theorem
For any non-degenerate higher Lamé operator �( )z with alge-

braically independent coefficients of Qi(z), i k�1, ,� and for any

n �1 there exist exactly
n r

n
��

�
��

	



�� Van Vleck polynomials V(z)’s and

corresponding degree n Stieltjes polynomials S(z)’s.
Now, we mention also a physically important case

( ) ( )z
S

z
z

S
z

VSi
i

l

j i
jj

l
� � � � �

� ��
� �
� � �dd

d
d

2

2
1 11

0 (2)

with �1<�2<… �l real and �1, …, �l positive. Then the follow-
ing theorem holds.

Theorem (Stieltjes, Van Vleck, Bôcher, [6])
For any integer n �1

1) there exist exactly
n l

n
� ��

�
��

	



��

2
polynomials V of degree l � 2

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 (�1, �l).

3) none of the roots of S can coincide with some of �i’s. More-

over,
n l

n
� ��

�
��

	



��

2
polynomials are in one-to-one correspon-

dence with
n l

n
� ��

�
��

	



��

2
possible ways to distribute n points

into l�1 open intervals ( , )� �1 2 , ( , )� �2 3 , …, ( , )� �l l�1 .

Recently, Borcea and Shapiro [7] found the density of the
asymptotic root distribution for the Stieltjes polynomials.

2.2 The degenerate case

Inspecting the structure of (1) we see that it is a degenerate
case of the r �1 operator. 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 mod-
ulus and their dependence on parameter b. To find this de-
pendence we follow the main idea of the Gräffe-Lobachevskii
method. Let

P E E p En
n

l
l

l

n
( ) � �

�

�


0

1

and we find expressions for the sums of the powers of the
roots

s E Ek
k

n
k� � �1 � .

Then for the root of the maximal modulus Emax holds

lim
maxk

k
k
s

E��
�1 .

Because sk are related to pl by

s p s k pk n l k l n k
l

k
� � �� � �

�

�


 0
1

1

we need explicit expressions for pl up to l n k� � . These
formulae can be derived from the recurrence relation.

We arrive at

s C k b n nk � �
�( ) ( )� � �� 1

where

� �

� �

� � � � �

� � � � �
�
�
�

3 2 1 2 1
3 4 0 2 2

l k l
l k l

&
&

if
if

and l � 0 1 2, , ,�. Thus,

lim lim ( ) lim ( )
k

k
k

k
k

k
ks C k b n n C kk k

�� �� ��
� �

� � 3
2

i.e. the limit is independent of b. Our conjecture is that

lim ( )
k

k C k
��

� �.

Polynomials Spn that have the same roots as Pn but scaled
by C Cnn �

3
2 do not fulfil any finite recurrence relation.

However, we can rewrite them as a determinant of an n×n
tridiagonal matrix Sp Mn n� det( ), where

M

E
E

E

n

n n

n n n

n n n

:

, ,

, , ,

, ,

�

�

�

�

	 �


 	 �


 	 �

1 2

2 2 3

3 3

0 0 0 0
0 0 0

0

�

�

,

,

, , ,

4

1

1 1 1

0 0

0 0 0 0
0 0 0
0 0

�

� � � � � � �

� �

�

�


 	 �

n n

n n n n n nE
�

� � ��

0 0� 
 	n n n nE, ,�

�

�

�
�
�
�
�
�
�
�
�

	




�
�
�
�
�
�
�
�
�

and

	 n i
n

n i
C,

�
� �4 4 1
2

, �n i
n

i
C,
( )

�
�4 1

, 
 n i
n

n i n i
C,

( )( )
�

� � � �2 2 1 2 2 2



This enables us to extend the polynomial sequence by in-
troducing a subsequence Sp Mn i n i, ,det( )� , where Mn,i is the
upper i × i principal submatrix of Mn 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 E E Sp Spn i n i n i n i n i n i, , , , , ,( ) ( )� � �� �	 � 
1 2
with initial conditions Spn,0 1� and Spn,� �1 0. Now, it remains
to go to the polynomials Tpn,i that have the same roots as Spn,i
and fulfil the relation

ETp a Tp Tp a Tpn i n i n i n i n i n i n i, , , , , , ,� � � �� � �1 1 1 2	 ,
with

a
i n i

Cn i n
,

( )( )
�

� � �4 1 2 2 1

The limits

	 � 	
�

( ) lim ,� �
�i n

n i 0 , a a Ci n n i n
( ) lim ,�

��� 
 �

�
� �

�
4

1 2

determine the density of the asymptotic root-counting mea-
sure:

� �� �� ��� 2 2
0

1

a a( ), ( ) d ,

where

� �
� �

� �x y y s s x
s x y

, ( )( )
,

� � �
�

�

�
�

��

1

0

if

otherwise

The sought density � is then expressed as

C

C s
�

�

� �

d

64 1 2 2 20

1

( )� �
�� .

We immediately see that the integral does not converge
for s � 0. The polynomial under the square root has three real
roots � � �1 2 3( ) ( ) ( )s s s� � if s

C C
� �
�

��
�

��
16

3 3
16

3 3
, and �3 1� for

all s ��. Thus � can be written as

�
�

�

� � � � � �
�

�

�
� � ��

C d
64 1 2 3

1

2

( )( )( )
,

which can be evaluated as [9]

�
� �

� �

� �
( ) , , ,s

C
F�

�

�

�

�

�
��

	



��8

1 1
2

1
2

1
3 1

2 1
2 1

3 1
s

C C
� �
�

��
�

��
16

3 3
16

3 3
, .

Acknowledgments
This research was partially supported by project LC06002.

References
[1] Shin, K. C.: Schrödinger Type Eigenvalue Problems with

Polynomial Potentials: Asymptotics of Eigenvalues, math.
SP/0411143.

[2] Trinh, D. T.: Asymptotique et analyse spectral de l’oscil-
lateur cubique, Th se, Université de Nice, 2002.

[3] Bender, C. M., Dunne, G. V.: Quasi-Exactly Solvable Sys-
tems and Orthogonal Polynomials, J. Math. Phys., Vol. 37
(1996), p. 6–11.

[4] Chihara, T. S.: An Introduction to Orthogonal Polynomials.
Gordon and Breach, New York, 1978.

[5] Marcellán, F., Álvarez-Nodarse, R.: On the “Favard’s
Theorem“ and its Extensions. J. Comp. Appl. Math.,
Vol. 127 (2001), p. 231–254.

[6] Borcea, J., Brändén, P., Shapiro, B.: Higher Lamé
Equations, Heine-Stieltjes and Van Vleck Polynomials, in
preparation.

34 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 47 No. 2–3/2007

Fig. 1: a) Dependence of the three real roots of 64 1 02 2 2� �( )� � �C s on s (for C � �). If s
C C

 �
�
��

�
��

16
3 3

16
3 3

, , there is only one real root

� 3 1( )s ! ; b) Comparison of density � with a numerical example for n � 100 and b � 2 66. .



©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 35

Acta Polytechnica Vol. 47 No. 2–3/2007

[7] Borcea, J., Shapiro, B.: Root Asymptotics of Spectral
Polynomials for the Lamé Polynomials, math.
CA/0701883.

[8] Kuijlaars, A. B. J., Van Assche, W.: The Asymptotic Zero
Distribution of Orthogonal Polynomials with Varying
Recurrence Coefficients, J. Approx. Theory, Vol. 99
(1999), p. 167–197.

[9] Prudnikov, A. P., Brychkov, Yu. A., Marichev, O. I.:
Integrals and Series. Elementary Functions. Nauka, Moscow,
1981.

Prof. Boris Shapiro
e-mail: shapiro@mat.su.se

Department of Mathematics

University of Stockholm
S-10691 Stockholm, Sweden

RNDr. Miloš Tater, CSc.
e-mail: tater@ujf.cas.cz

Nuclear Physics Institute, Academy of Sciences
CZ-25068 Řež near Prague