ON SOME EXACTLY SOLVABLE SCHRÖDINGER
TYPE EQUATIONS
N. COTFAS1, L. A. COTFAS2
1Faculty
2
of Physic, University of Bucharest, PO Box 76-54, Post Office 76, Bucharest,
Romania (ncotfas@yahoo.com)
Faculty of Economic Cybernetics, Statistics and Informatics, Academy of Economic Studies,
Bucharest, Romania (liviu.cotfas@ase.ro)
Abstract. Hypergeometric type operators are shape invariant, and a factorization into a product
of first order differential operators can be explicitly described in the general case. Some
additional shape invariant operators, directly related to certain Schrödinger type operators, are
obtained by using a deformation of the operators occurring in this general factorization. The
mathematical properties of the eigenvalues and eigenfunctions of the operators thus obtained
depend on the values of parameters involved. We investigate the square integrability of
eigenfunctions and the monotony of the eigenvalue sequence.
Key Words: Schrödinger equation, hypergeometric type operators, shape invariance.
1 INTRODUCTION
Many problems in quantum mechanics and mathematical physics lead
to equations of the type
 (s) y(s)   (s) y(s)  y(s) = 0
(1)
where  (s) and  (s ) are polynomials of at most second and first degree,
respectively, and  is a constant. These equations are usually called
equations of hypergeometric type [9], and each of them can be reduced to the
self-adjoint form
[ (s)  (s) y(s)]   ( s) y(s) = 0
(2)
by choosing a function  such that ( ) =  . The equation (1) is usually
2
considered on an interval (a, b) , chosen such that
lim  ( s )  ( s ) = 0 and
sa,b
 (s) > 0,  (s) > 0 for all s  (a, b) . Since the form of the equation (1) is
invariant under a change of variable s  cs  d , it is sufficient to analyze the
cases presented in Table 1. Some restrictions are imposed on  and  in
order that the interval (a, b) exist.
1
 (s)
s  
s
s  
 (s)
 (s)
( a, b )
(, )
s 2/2  s
e
s  
(1 s)
s2 1
s  
(s  1)(  )/21 (s  1)(  )/21
s2
s  
s 2e  /s
1 s
2
s  
s 1
2
(  )/21
(1 s)
(   )/21
(1, )
 <  < 0
(0, )
 < 0, > 0
 <0
(, )
2 /21  arctans
(1 s )
(1,1)
 < 0, > 0
 <  < 
(0, )
s  1es
e
, 
 <0
Table 1
The main cases
It
is
l = 
well-known
 (s)
2
[9]
that
for
 = l ,
where
l
and
l (l  1)   (s)l the equation (1) admits a polynomial solution
 l =  l( , ) of at most l degree
 (s)l   (s)l  l  l = 0.
(3)
The function  l ( s )  ( s ) is square integrable [2,3,9] on (a, b) and
0=0 <1 << l for any l <  , where    for  (s) {0, 2} and
  (1   ) / 2 for  (s) = 2 . The system of polynomials { l | l < } is
orthogonal with weight function  (s) in (a, b) .
2. FUNCTIONS OF HYPERGEOMETRIC TYPE
Let l  N , l <  , and let m {0,1,..., l} . If we differentiate (4) m times
then we get
3
 ( s)
d m 2
d m1
dm
 l  [ ( s)  m ( s)] m1  l  (l  m ) m  l = 0.
m 2
ds
ds
ds
(4)
The equation obtained by multiplying this relation by  m (s) can be written
as H m  l ,m = l  l ,m , where H m is the differential operator
Hm =
d2
d m(m  2) ( ( s )) 2
  (s) 
2
ds
4
 (s)
ds
m ( s )  ( s ) 1

 m(m  2) ( s )  m ( s ).
2  ( s) 2
  ( s)
and the functions  l ,m ( s) =  m ( s)
(5)
dm
 l ( s) defined by using  ( s) =  ( s) are
ds m
called the associated special functions. If 0  m  l <  then  l ,m ( s )  ( s )
is square integrable [2,3] on (a, b) . One can prove that the functions  l , m are
related through the first order differential operators
d
 m ( s)
ds
d  ( s)
Am =  ( s) 
 (m  1) ( s).
ds  ( s)
Am =  ( s)
(6)
namely, we have
for l = m
0
Am  l ,m = 
 l ,m1 for m < l < 

Am  l ,m1=(l m ) l ,m for 0  m < l < .
(7)
and
 l ( s)

 l ,m ( s) =  Am
Am1
Al1
l
       ...     ( s)
m
l
m1
l
l 1
 l
for
m=l
for
0 < m < l < .
(8)
The operators H m satisfy the intertwining relations [2,7,8]
H m Am = Am H m1
and are shape invariant.
Am H m = H m1 Am
(9)
4
H m  m = Am Am
H m1  m = Am Am .
(10)
For each m <  , the functions  l ,m with m  l <  are orthogonal [2,3] with
weight function  (s) in (a, b) , and ||  l ,m1 ||= l m ||  l ,m || .
The normalized associated special functions l ,m =  l ,m / ||  l ,m || satisfy the
relations
for l = m
0
Am l ,m = 
 l  m l ,m1 for m < l < 

Am l ,m1 = l  m l ,m for 0  m < l < 
Am
Am 1
Al1
l ,m =
...
l ,l .
l  m l  m1
l  l 1
(11)
3. SHAPE INVARIANT OPERATORS RELATED TO H m
Some additional shape invariant operators directly related to H m can be
obtained in the cases when  and  are such that there exists k  R with
 (s) =  k (s) (see Table 2).
 (s)
 (s)
k
(a, b)
s  1
 1
(0, )
(1  s 2 ) /21


(1,1)
 ,
 >0
 <0
(1, )
 <0
(0, )
 <0
(, )
 <0
1  s2
 (s )

s
s2 1
s
(s  1)

s2
s
s  2

s
s2  1
s
2
/21
2
2
(s 2  1)/21

2
2
1
1
1
1
Table 2
The cases when  (s) is a power of  (s) .
From ( ) =  we get  (s) = (k  1) (s) = 2( k  1) (s) (s) , and
5
Am =  (s)
d
 m (s)
ds
Am =  (s)
d
 (2k  m  1) (s).
ds
(12)
~
~
For any constants  m the deformed operators Am = Am   m and Am = Am   m
which we can consider for any m  R satisfy the relations
( Am   m )( Am   m ) = H m  m   m (2 m  2k  1) ( s)   m2
( Am   m )( Am   m ) = H m1  m   m (2 m  2k  1) ( s)   m2 .
If we choose  m = /(2m  2k  1) with  an arbitrary constant, then the
~
d
operator H m = H m  
satisfy the intertwining relations [4,5]
ds
~ ~
~ ~
Am H m = H m1 Am ,
~ ~
~ ~
H m Am = Am H m1
(13)
~ ~
~
~
Am Am = H m1  m
(14)
and is shape invariant, namely, we have
~ ~
~
~
Am Am = H m  m ,
2
~
where m = m 
(2 m  2k  1) 2
. Following the analogy with (7) we consider
~
the function  m,m obtained, up to a multiplicative constant, by solving the
~ ~
equation Am  m,m =0 ,
2 s



( s ) m e 2 m 2  1

 arcsin s


( 1s 2 ) m e 2 m 1


~

 m,m ( s)=
( s 2 1) m ( s s 2 1) 2 m 1


m

2 m  1
s




2
m
2
2 m  1
(
s

1
)
(
s

s

1
)

~
if
 ( s) = s
if
 ( s)=1s 2
if
 ( s)=s 2 1
if
 (s) = s 2
if
 ( s)= s 2 1.
The mapping m  m is an increasing function on the set {m |
The set M = { m |
(15)
d ~
m > 0} .
dm
b~
d ~
m > 0 and   2m,m (s)  (s)ds <  } of all the values of m for
a
dm
6
which
~
d ~
m > 0 and  m,m  is square integrable on (a, b) is presented in
dm
Table 3.
 (s)
M
s
 (s )

1  s2
s
s2 1
s
s2
s
1 
,   for any   R

 2

1


for   

2



1

  ,1    
for ( ,0]
 2 2


2
2



  1      1   1    

 for   (0, 1 )
  ,

,




2  2
2
2 
2
 2 2

1 1  
1



,

for   [ , )
 2


2
2
2


 for any   R
s2  1
s

for   0

(   1 , ) for  > 0

2


  , 1    |  |  for any   R

2
2 

Table 3
The set M .
If l  M and n  N are such that {l  n, l  n  1,..., l}  M then for each
m {l  n, l  n  1,..., l  1} the function
~
~
~
~
A
A 1
A
A 1 ~
~
 l , m = ~ m ~ ~ m~
 ~ l ~2 ~ l ~
 l ,l
l  m l  m1 l  l 2 l  l 1
has the form
(16)
7
2 s
l  m

c j ( s ) l  j e 2l  2  1
 j =0
 arcsin s
l  m

 c s j ( 1s 2 ) l  j e 2l 1
 j
~
 l ,m = j = 0


l  m j
2
l j
2
2 l 1
c
s
(
s

1
)
(
s

s

1
)

j

 j =0


l  m j
2
l j
2
2 l 1
c
s
(
s

1
)
(
s

s

1
)
 j
 j =0
if
 (s) = s
if
 ( s )=1s 2
if
 ( s )=s 1
if
 ( s )= s 2 1
(17)
2
l  M and n  N are such that
~
{l , l  1,..., l  n}  M , then the function  l ,m  is square integrable for any
~
m {l , l  1,..., l  n} . The definition (16) of  l ,m can be re-written as
~
~ ~
~ ~ ~
Am
~
~
 l ,m = ~
~  l ,m 1 , whence Am  l ,m1 = (l  m ) l ,m .
l   m
where
cj
are real constants. If
~
~
~
~
~
~
~
Al3
Al1
Al 2
 l ,l 3 

  l ,l 2 
  l ,l

  l ,l 1 
~
~
~
~
~
Al3
Al 2
 l 1,l 3 

  l 1,l 2 


l 1,l 1
~
~
~
Al3
 l 2,l 3 

  l 2,l 2
~
 l 3,l 3
Fig. 1
The functions
~~
~ ~
~ ~
~
 l ,m
~~
Since H l  l ,l =( Al Al  l ) l ,l =l  l ,l and
~ ~
~ ~
~~
~~
~ ~
~ ~
H A ~
A H ~
H m1 l ,m1=l  l ,m1  H m  l ,m = ~ m ~m  l ,m1= ~m m~1  l ,m1=l  l ,m
l  m
l  m
~~
~ ~
we get by recurrence H ml ,m =l l ,m . We have
~ ~
~
~
~ ~
Am Am ~
H m1  m ~
~
Am  l ,m = ~ ~  l ,m1 = ~ ~  l ,m1 =  l ,m1
l  m
l  m
8
~ ~
~
that is, Am  l ,m =  l ,m1.
Fig. 2
The boundary of M in the case  (s)= s 2 1 .
4. APPLICATION TO SCHRÖDINGER TYPE EQUATIONS
~~
~ ~
If we use in equation H ml ,m = l l ,m a change of variable
(a, b)  (a, b) : x  s( x) such that ds/dx =  ( s( x)) and define the new
~
~
functions l ,m ( x) =  ( s( x))  ( s( x))  l ,m ( s( x)) then we get an equation of
Schrödinger type

~
~~
~
d2 ~
l ,m ( x)  Vm ( x)l ,m ( x) = l l ,m ( x).
dx 2
~
(18)
~
The operators corresponding to Am and Am are
~
~
~
d
A m = [ ( s)  ( s)]1/2 Am [ ( s)  ( s)] 1/2 |s= s ( x ) = 
 Wm ( x)
dx
~
~
~
d
A m = [ ( s)  ( s)]1/2 Am [ ( s)  ( s)] 1/2 |s= s ( x ) = 
 Wm ( x)
dx
~
where the superpotential Wm ( x) is given by the formula
(19)
9
~
 (s( x)) 
1  d

Wm ( x) = 
 m  
(s( x)) 
2 (s( x)) 
2  ds
2m  2k  1
~
~
~
(20)
~
and Vm ( x) = Wm2 ( x)  Wm ( x)  m . Since
b
~
~
~
b
~
~
a l ,m ( x)k ,m ( x)dx  a l ,m (s)k ,m (s) (s)ds
the functions l ,m ( x) are square integrable (resp. orthogonal) if and only if
~
the corresponding functions  l ,m ( s) are square integrable (resp. orthogonal).
Particular cases [1, 6, 8]. Let  m = (2m    1)/2 and  m = (2m    1)/2 .
1. Coulomb type potential. In the case  (s) = s , the change of variable
(0,)  (0,) :x  s( x) = x 2 /4 leads to
11

~

Wm ( x) =    m   
2
x
2
m

2  1


~
1 
3 1
1

Vm ( x) =    m     m   2  
2 
2 x
x

~
2
m = 
.
(2 m  2  1) 2
(21)
2. Trigonometric Rosen-Morse type potential. In the case  (s) = 1  s 2 , the
change of variable (0,  )  (1,1) :x  s( x) = cos x leads to
~

Wm ( x) =  m cotan x 
2m    1
~
2
2
Vm ( x) =  m   m cosec2 x   cotan x   m  m(m    1)
2
~

m = m(m    1) 
(2 m    1) 2


(22)
3. Eckart type potential. In the case  (s)=s 2 1 , the change of variable
(0, )  (1, ) :x  s( x) = cosh x leads to
~

Wm ( x) =  m cotanh x 
2m    1
~
2   cosech2 x   cotanh x   2  m(m    1)
Vm ( x) =  m
m
m
~
2
m = m(m    1) 
.
(2 m    1) 2


(23)
10
4. Hyperbolic Rosen-Morse type potential. In the case  (s)=s 2 1 , the change
of variable RR :x  s( x) = sinh x leads to
~

Wm ( x) =  m tanh x 
2m    1
~
2   sech2 x   tanh x   2  m(m    1)
Vm ( x) =   m
m
m
2
~

m = m(m    1) 
.
(2 m    1) 2


(24)
The exactly solvable Schrödinger type equations play an important role in
quantum mechanics. In this paper, we have explored the properties of some
additional shape invariant operators directly related to hypergeometric type
operators. Particularly, we have identified new exactly solvable Schrödinger
type equations satisfying the usual requirements concerning their eigenvalues
and eigenfunctions. They may play a role in some future applications.
Acknowledgment
NC acknowledges the support provided by CNCSIS under the grant IDEI 992
- 31/2007.
References
[1] F. COOPER, A. KHARE, U. SUKHATME, Supersymmetry and quantum mechanics, Phys.
Rep., Vol. 251, pp. 267—385, 1995.
[2] N. COTFAS, Shape invariance, raising and lowering operators in hypergeometric type
equations, J. Phys.A: Math. Gen., Vol. 35, pp. 9355-9365, 2002.
[3] N. COTFAS, Systems of orthogonal polynomials defined by hypergeometric type equations
with application to quantum mechanics, CEJP, Vol. 2, pp. 456-466, 2004.
[4] N. COTFAS, Shape invariant hypergeometric type operators with application to quantum
mechanics, CEJP, Vol. 4, pp. 318-330, 2006.
[5] N. COTFAS, Gazeau-Klauder type coherent states for hypergeometric type operators, CEJP,
Vol. 7, pp. 147-159, 2009.
[6] J.W. DABROWSKA, A. KHARE, U. SUKHATME, Explicit wavefunctions for shape-invariant
potentials by operator techniques, J. Phys. A: Math. Gen. , Vol. 21, pp. L195-L200, 1988.
[7] L. INFELD AND T.E. HULL, The factorization method, Rev. Mod. Phys., Vol. 23, pp. 21—
68, 1951.
[8] M.A. JAFARIZADEH AND H. FAKHRI, Parasupersymmetry and shape invariance in
differential equations of mathematical physics and quantum mechanics, Ann. Phys., NY, Vol.
262, pp. 260—276, 1998.
[9] A.F. NIKIFOROV, S.K. SUSLOV, V.B. UVAROV: Classical Orthogonal Polynomials of a
Discrete Variable, Springer, Berlin, 1991.