ESI
The Erwin Schrodinger International
Institute for Mathematical Physics
Boltzmanngasse 9
A-1090 Wien, Austria
Inverse Problem and Darboux Transformations
for Two{Dimensional Finite{Dierence
Schrodinger Equation
A.A. suzko
Vienna, Preprint ESI 665 (1999)
Supported by Federal Ministry of Science and Transport, Austria
Available via http://www.esi.ac.at
January 29, 1999
Inverse problem and Darboux transformations for
two-dimensional nite-dierence Schrodinger equation
A.A. Suzko
Joint Institute for Nuclear Research, Dubna
1
Abstract
A discrete version of the two-dimensional inverse scattering problem is
considered. On this basis, algebraic transformations for the two-dimensional
nite-dierence Schrodinger equation are elaborated. Generalization of the
technique of one-dimensional Darboux transformations for the two-dimensional
nite-dierence Schrodinger equation is presented. Ideas of Bargmann-Darboux
transformations for the dierential one-dimensional multichannel Schrodinger
equation are used for the two-dimensional lattice Schrodinger equation. Analytic relationships are established between dierent discrete potentials and
the corresponding solutions.
1 Introduction
Study of multidimensional and multi-particle objects is qualitatively more complicated than that of the one-dimensional case. When there is no symmetry in the
potentials of interaction between particles, these systems are described by partial
dierential equations with unseparable variables. Just for this reason the inverse
scattering problem in 2- and 3-dimensional spaces was formulated by Faddeev [1]
and Newton [2, 3], Novikov and Henkin [4] at a much later than the 1-dimensional
problem. The foundation for developing the 2-dimensional inverse problem in nite dierences was laid by Berezanskii [5] who developed the theory of orthogonal
polynomials for the Jacobi innite matrix. The Darboux transformations in quantum mechanics for the Schrodinger dierential and dierence 1-dimensional equation
have much in common with the spectral transformations for orthogonal polynomials [6]. It is therefore expedient to analyze the spectral inverse problem for the
discrete 2-dimensional Schrodinger equation on the basis of the technique of orthogonalization of polynomials. The formulae of the inverse problem with degenerate
kernels are closely related to Bargmann-Darboux transformations [7],[8, 9]. The
one-dimensional Darboux transformations has already found wide applications in
quantum mechanics and in the theory of nonlinear integrable systems [10, 11, 6, 12].
In the present paper, a discrete version of the Gelfand-Levitan inverse spectral
problem is considered for the two-dimensional lattice Schrodinger equation. The
two-dimensional nite-dierence inverse problem, based on the procedure of orthogonalization of polynomial vectors, is a generalization of the one-dimensional procedure given in [13]. On the basis of the obtained formulae of the inverse problem,
1
Radiation Physics and Chemistry Problems Institute, Academy of Sciences of Belarus, Minsk
1
discrete Bargmann-Darboux transformations are given in two dimensions. The suggested algebraic approach allows one to construct families of discrete potentials in
an explicit form and the corresponding solutions.
2 Inverse problem
Consider the Schrodinger equation whose Hamiltonian is tridiagonal in a certain
basis with respect to both the coordinate variables n and m
(H )nm = anm (n ? 1; m) + an m (n + 1; m)
+bnm (n; m ? 1) + bnm (n; m + 1) + cnm (n; m) = (n; m):
+1
+1
(1)
The coecients anm; bnm; cnm are assumed to be real and represent discrete potentials; (n; m) are discrete wave functions, (n; m) is an integer point of the half-plane,
n = 0; 1; 2; ::; m = :::; ?1; 0; 1; :::, is a spectral parameter. The index n can vary
from 0 to 1 and the index m can vary from ?1 to 1 or from 0 to 1; when
0 n N and 0 m M , it is a special class of restricted problems. The
equation in nite dierences (1) can be represented by the expression with operator
coecients
(H )n = An (n ? 1) + Vn (n) + An (n + 1) = (n);
+1
(2)
when we treat one of the variables, n, as the only discrete coordinate; and the other,
m, as the channel index. If we set (?1) = 0, the action of the Schrodinger discrete
operator H on the vector = f(0); (1); (2); :::(n); :::g is represented by the
action of the Jacobi block matrix J on 0 V A 0 0 0 ::: 0 1 0 (0) 1 0 (0) 1
BB A V A 0 0 ::: 0 CC BB (1) CC BB (1) CC
BB 0 A V A 0 ::: 0 CC BB (2) CC BB (2) CC
(J )n = B
BB 0 0 0 : : : 0 CCC BBB : CCC = BBB : CCC ; (3)
B@ : : : 0 An Vn An CA B@ (n) CA B@ (n) CA
:
:
: : : : : : :
0
1
1
1
2
2
2
3
+1
whose elements Vn and An are matrices at each xed n, and each element (n)
of the vector corresponds to the vector f m(n)g; m(n) (n; m), in the other
space variable "m"
0
1
c
b
0
0
0
:::
0
:
n
n
BB b c b 0
CC
0
::: 0
: C
BB n n n
0
::: 0
: C
BB 0 bn cn bn
CC
:
:
:
:
:
:::
0
:
CC ;
Vn = B
BB : : : 0 b
CC
c
b
:
nm?
nm?
nm
BB
0
bnm cnm bnm C
@ : : : :
A
: : : :
:
:
:
:
2
0
1
1
1
2
2
2
3
1
1
+1
0 (n) 1
0 a 0 0 ::: 0 1
n
BB (n) CC
B
0 an 0 ::: 0 C
B
CC
BB (n) CC
B
B 0 0 an ::: 0 CC
CC :
BB
;
(
n
)
=
An = B
B
CC
:
:
:
:
:
:
B
B
B@ m(n) CCA
B
@ 0 0 0 ::: anm CA
:
: : : : :
It is clear that the operator H is Hermitian. Note that the coupling of equations (1)
with respect to m occurs only between immediate neighbours and for this reason
the symmetric coupling matrix Vn mm = Vmm (n) in (3) is tridiagonal and relates
the functions (n; m) at m; m 1, unlike the conventional multichannel case.
>From the Jacobi block matrix (3) tridiagonal in the variable n it is seen that
every vector of the solutions (n) is connected with the vectors of solutions (n 1)
at the neighboring points n 1. As a result, if we take nonhomogeneous boundary
conditions at one end of the interval 0 n N , we can obtain the solutions on the
whole interval by moving by subsequent steps from that end. It turns out that these
vectors of solutions as functions of the spectral parameter are polynomials of with matrix coecients [5] which can be orthogonalized in the spectral measure.
The spectral inverse problem is reduced to the construction of potential matrices Vn , An and an unknown system of orthonormal polynomials with the use of
the known system of orthonormal polynomials
corresponding
to the nite-dierence
equation (2) but with the known matrices V n and An .
0
0
1
1
2
2
;(
0)
0
2.1 Orthogonalization of polynomials
Introduce auxiliary solutions 'ms(n) 's(; n; m) and 'ms (n) to eq.(1) with the
help of the boundary conditions
'ms (?1) = 0; 'ms (0) = ms ;
'ms (?1) = 0; ' ms (0) = ms;
(4)
where s is a point on the m axis. The solutions 's satisfy the same equation (1)
with the known potentials a nm ; cnm=V mm (n); bnm =V mm (n); bnm=V mm? (n)
+1
a nm ' ms (; n ? 1)+ a n
1
+1
mX '
(
;
n
+
1)
+
V mm (n) 'm s (; n)
m ms
+1
+1
m =m?1
00
00
00
= 'ms (; n):
(5)
As a rst boundary condition, we can change the zeroth condition to a more
general one corresponding to the homogeneous boundary conditions ('ms(?1) ?
'ms(0))= = Ds 'ms (0). In what follows we assume the step of nite-dierence
dierentiation to equal 1. According to the conditions (4), the functions 'ms(n)
and 'ms (n) are equal to zero on the lines (?1; m), (0; m) except for the points
3
with m = s on the line n = 0 where 'ss(n) ='ss (n) = 1. From these val
ues of 's and 's one can nd the functions 'ms(n) and 'ms (n) at all other
points of the coordinate network by using the recurrence equation (1). Since the
matrix equation (2) is tridiagonal in the variable n, and owing to the boundary conditions (4), the vectors s (n) f:::' s(n); ' s(n); :::'ms(n):::g of solutions
(n) = ( (n); (n); :::s(n); ::) in the spectral variable are polynomials of the
nth degree. At n = 1 the vector s (1) is a polynomial of the rst degree; at n = 2,
the vector s(2) is a polynomial in of the second degree, and so on. It appears,
however, that dierent elements 'ms (n; ) of the vector of solutions s(n) have different degrees of polynomials. Because of the three-point coupling with respect to m
the functions 'ms (n; ) are polynomials in of the degree n ?js ? mj. The maximum
nth degree belongs to the elements with m = s and the elements ms (n; ) vanish at
all s lying out of the values (n ? m; n + m). The distribution of polynomial degrees
is shown in Fig.1.
0
1
1
2
It is clear that the vectors of solutions, whose elements are polynomials
in can be orthogonalized. Berezanskii has shown [5] that if the elements
anm ; bnm and cnm in (1) are real and
anm > 0 (n = 1; 2; :::; m = ::: ? 1;
0; 1; 2; :::), one can introduce polynomials of the rst kind ('ms (; n) =
Ps n;m ()) which are orthogonal with
the weight of the spectral matrix
() = jjss ()jj
;(
Fig.1. Nonzeroth ms (n; ) are inside and on boundaries of the cone with
its vertex at (0; s) [5]; the numbers at the
nodes of the net indicate the degrees of
'ms (n) polynomials.
1 Z
X
ss =?1
0
)
0
'ms(; n)dss ()'s m (; n0) = nn mm
0
0
0
0
0
(6)
and similarly for 'ms (; n)
X Z '
0
ms (; n)d ss () 's m (; n ) = nn mm :
ss
0
0
0
0
0
(7)
0
The spectral matrix elements ss (or ss ) are determined by the boundary values of
the special solutions (n; m) (or (n; m)) obeying the zeroth condition (?1; m) =
m (?1) = 0 (m = :: ? 1; 0; 1:::). For example, for p states of the discrete spectrum,
4
0
0
the matrix () is the sum of p terms formed from productions of the column vectors
?( ) fs( )g and the row vectors ?y( ) (s ( ))
p
X
ss () = ( ? )s ( )s ( );
0
0
=1
where the elements s( ) = s(0; ) are dened by (n = 0; m) (m = :: ? 1; 0; 1:::)
and ( ? ) is the Heaviside step function equal to 1 at = and to zero when
6= . A solution (n; m) to eq.(1) with the zeroth condition (or a more general
homogeneous condition ) can be obtained by multiplying the matrix (; n) by the
vector ?() = (0; ):
1
X
'ms (n; ) s(0; ):
(8)
(n; m; ) =
2
s=?1
At every xed n and m the summation over s is nite owing to 'ms(n; ) = 0 outside
the interval (n ? m; n + m).
2.2 A discrete version of the Gelfand{Levitan inverse
problem
Using the procedure of orthogonalization of polynomials we construct unknown polynomial solutions 'ms (; n) normalized with the spectral weight ss () as a linear
combination of the known polynomial solutions 'ms (; n), orthogonal with respect
to the measure ss ()
0
0
'ms(; n) =
n
X
m+(X
n?n )
0
n =0 m =m?(n?n )
0
0
K (n; m; n0; m0) 'm s (; n0):
0
(9)
0
As one can see above, the validity of this relation is a consequence of the fact that
both the functions, 'ms (; n) and 'ms(; n), obey the same discrete Schrodinger
equation (1) and the same boundary conditions (4), given on the lines n = ?1 and
n = 0. Therefore, the solutions 'ms (; n) and 'ms (; n) are polynomials in of the
same degree n ?js ? mj, but with dierent coecients. The matrix of the polynomial
solutions (; n), which is of degree n, is orthogonal to every polynomial matrix of
degree lower than n and hence to every (; n0) for n0 < n
XZ
'ms (; n)dss () 's m (; n0) = 0:
(10)
ss
0
0
0
0
Equation (9) is a discrete analog of the Volterra integral equation, in which the
coecients K (n; m; n0; m0) are determined by the condition of orthogonality of the
Two solutions of the same equation of second order dier from each other by a normalization
factor at those points where both of them exist when one of the boundary conditions is the same.
2
5
vector-functions (; n) = (::: (; n); (; n); :::; s(; n); :::) orthogonal with the
spectral measure () to the functions (; n0) orthogonal with the weight matrix
(), when n0 n
XZ
(11)
'ms (; n)(d ss () ? dss ()) 's m (; n0) = K (n; m; n0; m0):
1
ss
2
0
0
0
0
0
The Volterra equations (9) have a triangular form, K (n; m; n0; m0) = 0 for n0 > n.
It is easy to see from (9) and orthogonality of 'sm (; n) (7) that for n0 < n
XZ
K (n; m; n0; m0) =
'ms (; n)d ss () 's m (; n0):
ss
0
0
0
0
Inserting (9) into (11) for n0 < n we obtain the following system of equations for
the orthogonalization coecients K (n; m; n0; m0)
K (n; m; n0; m0) + K (n; m; n; m)Q(n; m; n0; m0) +
(12)
m
n
?
n
nX
?
X
K (n; m; n00; m00)Q(n00; m00; n0; m0) = 0;
+
where
n
00
00
+(
1
)
m =m?(n?n
=0
00
Q(n; m; n0; m0) =
00
)
X Z '
0
ms (; n)(dss () ? d ss ()) 's m (; n ):
ss
0
0
0
0
(13)
0
Equation (12) is a two-dimensional analog of the Gelfand-Levitan integral equations
for nite-dierence equation (1). However, this system is not sucient to determine
the coecients K . A supplementary system of equations is obtained upon substituting (9) into the complete relation (6) at n0 = n and m0 = m
K ? (n; m; n; m) = Q(n; m; n; m) +
n?n
nX
? m X
K ? (n; m; n; m)K (n; m; n00; m00)Q(n00; m00; n; m): (14)
+
2
+(
1
n
00 =0
00
)
1
m =m?(n?n
00
00
)
Now let us nd connections between the potential coecients and orthogonalization
coecients K (n; m; n0; m0). To this end, we insert expression (9) for the polynomial
functions 'ms(; n) in terms of 'ms (; n) into the Schrodinger dierence equation
(1)
? ?n
nX
? m nX
anm
K (n ? 1; m; n0; m0) 'm s (; n0)
+(
1
0
1
)
0
n =0 m =m?(n?1?n )
m+(nX
+1?n )
nX
+1
+an+1m
K (n + 1; m; n0; m0) 'm s (; n0)
n =0 m =m?(n+1?n )
n m +(
mX
+1
Xn?n )
X
K (n; m00; n0; m0) 'm s (; n0)
Vmm (n)
+
n =0 m =m ?(n?n )
m =m?1
n?n )
n m+(X
X
=
K (n; m; n0; m0) 'm s (; n0):
n =0 m =m?(n?n )
0
0
0
0
0
0
0
0
00
0
0
00
0
00
0
00
0
0
0
0
0
0
6
(15)
For brevity the following notation is used: Vm;m (n) = cnm ; Vm;m (n) = bnm and
Vm;m? (n) = bnm. We transform the r.h.s. of the above relation by means of the
substitution of 'm s (; n) from the nite dierence equation (5) for 'ms (; n)
with the known potentials a nm ; V m;m (n), (m0 = m ? 1; m; m + 1)
+1
+1
1
0
0
m+(X
n?n )
n
X
m =m?(n?n )
n?n )
n m+(X
X
n
+
0
0 =0
0
K (n; m; n0; m0) 'm s (; n0) =
0
n =0 m =m?(n?n )
0
0
+ an
0
0
0
0
K (n; m; n0; m0)( a n m 'm s (; n0 ? 1)
0
0
0
mX
0
'
V m m (n0) 'm s (; n0)):
m m s (; n + 1) +
0
+1
0
0
+1
m =m ?1
00
0
00
00
(16)
0
Further, we take advantage of the orthogonality relation (7) for the matrix functions
matrix (). Multiplying expression (15) with
(; n) orthogonal with the weight
its transformed r.h.s. (16) by 's m (; n+1), integrating over d ss (), and summing
0 , we arrive at the relationship between the potentials
up over
the
indices
s
and
s
anm; a nm and the coecients K (n; m; n0m0)
m; n; m) :
an m =a n m K (nK+(n;
(17)
1; m; n + 1; m)
The relations for the coecients cnm and bnm are established
in a similar manner. To
'
determine bnm = Vmm (n), eq.(15) is multiplied by s m (; n) with (16) taken
into account and integrated with the weight ss () by using the orthogonality (7).
As a result, we have
K (n; m; n ? 1; m + 1)
n; m) + a
bnm =bnm K (n;Km(n;+m1;; n;
nm
m + 1)
K (n; m + 1; n; m + 1)
K (n + 1; m; n; m + 1) :
(18)
?an m K
(n; m + 1; n; m + 1)
0
+1
+1
0
+1
+1
0
+1
0
+1
+1
+1
+1
The relation for cnm = Vmm (n) is derived analogously, only (15) is multiplied by
' s m (; n)
(n; m; n ? 1; m) ? a K (n + 1; m; n; m) :
(19)
cnm =cnm + a nm KK
n m
(n; m; n; m)
K (n; m; n; m)
Substituting (17) into (19) we arrive at
(n; m; n ? 1; m) ? a
K (n + 1; m; n; m) :
cnm =cnm + a nm KK
(20)
n m
(n; m; n; m)
K (n + 1; m; n + 1; m)
0
+1
+1
At anm =a nm = 1, bnm =bnm= 1, the derived generalized expressions turn into more
simple ones presented in [7]. The two-dimensional nite-dierence inverse problem
7
under consideration is also a generalization of that [15] with the potential coecients
anm 6= 1, bnm 6= 1, connected nevertheless in a special way.
In principle, it is easy to formulate the problem of restoring the matrix Vmm (n) in
(2) with all nonzeroth elements, like in a multichannel problem [7]. The latter would
correspond to the potential being nonlocal with respect to one of the coordinate
variables (in our case "m"). If the consideration were made in the polar coordinate
system, nonlocality with respect to angles would occur. In the case of continuous
coordinates, the inverse problem for the potential, nonlocal relative to angles, was
considered by Kay and Moses [14].
0
3 Bargmann{Darboux transformations for the
two-dimensional discrete Schrodinger equation
In this section, we describe the algebraic procedure by taking into considaration
simple kernels Q in the form of a sum of several terms with a factorized coordinate
dependence
p X
0
0
( ; n; m) (; n0; m0)
Q(n; m; n ; m ) =
=
p X X
'
=1 s
=1
ms
(; n)s ( )
X
s
0
s ( ) 's m ( ; n0):
0
0
(21)
0
Here the functions ( ; n; m) m ( ; n) are combined as elements of the vector
( ; n) = (::: (n); (n); :::; m ( ; n); :::)y obtained as a product of the
matrix solutions ( ; n) taken at eigenvalues = of the reconstructed H and
the vector ?( )
X
(; n; m) = 'ms (; n)s ( ):
1
2
s
The elements s form the normalization matrix C () = ?( )?y( ) with elements
Css () = s ( )s () corresponding to the bound state (n; m) (; n; m),
= 1; 2; ::p.
Like Q, the ortogonalization kernel K (n; m; n0m0) is also presented as a sum of
several factorized terms. Really, substituting (21) into the Gelfand Levitan equation
(12), we obtain
0
0
K (n; m; n0; m0) =
m X
n?n
p X
n
X
? f
=1 n
+(
00
=0
00
m =m?(n?n
00
)
00 )
(22)
K (n; m; n00; m00) ( ; n00; m00)g ( ; n0; m0):
Noting that the expression in braces is the solution
8
(n; m) (9) at = of eq.(1)
with the desired potentials anm; bnm and cnm , we immediately receive
p
X
0 0
K (n; m; n0; m0) = ?
(n; m) ( ; n ; m ):
=1
(23)
It is evident now that the new wave functions 'ms(; n), determined by (9) with the
kernel K taken in the form (23), are related to the old ones 'ms (; n) by
'ms(; n) = ?
p
X
(n; m)
m+(X
n?n )
n
X
0
m =m?(n?n )
n
0 =0
0
(n0; m0; ) 'm s (; n0):
0
(24)
0
In view of (23) for K (n; m; n0; m0) in eqs. (17), (18) and (19), one can immediately
write expressions for discrete potentials in the closed form
Pp
(n; m) ( ; n; m)
an m =an m Pp
;
(25)
(n + 1; m) ( ; n + 1; m)
bnm
=1
+1
+1
+1
=1 = bnm Pp
Pp
+ a n;m Pp
Pp
? an m Pp
Pp
+1
=1
+1
=1
=1
=1
+1
=1
and
=1 (n; m)
(; n; m)
(n; m + 1) ( ; n; m + 1)
(n; m) ( ; n ? 1; m + 1)
(n; m + 1) ( ; n; m + 1)
(n + 1; m) ( ; n; m + 1)
(n; m + 1) ( ; n; m + 1)
Pp
(n; m) ( ; n ? 1; m)
cnm =cnm + anm Pp
(
n;
m
)
( ; n; m)
Pp a
? P p n m (n + 1; m) (; n; m) :
(n + 1; m) ( ; n + 1; m)
=1
(26)
(27)
=1
=1
+1
=1 The solutions (n; m) have to be found from the Gelfand-Levitan equations (12)
and (14) taking account of (21) and (23). Thus, the operator K (n; m; n0; m0) dened
by (23) transforms the solutions 'sm (n) of eq. (5) into the solutions 'sm(n) of eq.(1),
determined by (9) or (24), with the potentials anm ; bnm and cnm dened by (25), (26)
and (27).
It is not dicult to see from the denitions (13) and (11) that the kernels Q and
K like (21) and (23) can be obtained provided that the spectral weight functions ()
and () for both the sets of potentials coincide except, for instance, p eigenvalues
9
at = . This permits one to construct potentials with p new bound states by
using (25), (27) and (26) or generate the family of spectral-equivalent potentials
whose spectra coincide = and it is only the normalization factors C 6=C that
are dierent. In the latter case Q is taken in the form
p X X
'ms (; n)(Css ( )? C ss ( )) ' s m (; n0) (28)
Q(n; m; n0; m0) =
=1 ss
0
0
0
0
0
and the above procedure can be used to construct spectral-equivalent operators H
and H . In spite of a complicated form of the expressions for the potentials (25),
(26) and (27), they are simplied for a large set of particular cases. For example, if
we deal with the freediscrete Schrodinger equation as a reference one, xed by the
choice cnm 0, a nm =bnm 1.
Let us now consider another simple case when the Hamiltonians dier only by
spectral data at one bound state. In this case, the summation over in all formulae
(24) { (27) vanishes. General solutions 'sm (; n) from (24) at arbitrary can be
written as
n?n
n m X
0 0 X
'ms(; n) = ? (n; m)
(n ; m ) 'm s (; n0);
(29)
+(
0
)
0
n =0 m =m?(n?n )
0
0
0
where (n; m) is a special solution of the Schrodinger equation (5) for the value of
the spectral parameter = and (n; m) is the solution of eq.(1) with the same
eigenvalue = . New potentials
are expressed in terms of the known old potentials
anm , cnm , and bnm , functions (n; m) and functions (n; m) that can be determined
from the second Gelfand-Levitan equation (14)
an
m
+1
bnm
+1
=an
(n; m) (n; m) ;
(n + 1; m) (n + 1; m)
m
+1
(n; m) (n; m + 1) bnm
=
+ a nm
+1
and
(n; m)
(n; m + 1)
!
(n; m) :
(n ? 1; m + 1) ? a
n m (n; m + 1)
(n + 1; m)
+1
(30)
+1
(31)
cnm =cnm + anm (n ? 1; m) ? a n m (n; m) :
(32)
(n; m)
(n + 1; m)
It should be noted that relationships between potentials and functions can be obtained within the Darboux transformation method or factorised method without
using formulae of the inverse problem.
10
+1
Connection between Darboux transformations and inverse problem ones. It is
interesting to note that transformation (29) with one bound state corresponds to
Darboux transformation for the nite-dierence equation (1). Indeed, let us search
for a solution 'ms(; n) of eq.(1) with some initially unknown potentials in the form
(29). Next it is necessary to nd conditions for the potentials anm; bnm and cnm and
special functions (n; m) at which general solutions 'ms (; n) specied by (29) will
satisfy the discrete Schrodinger equation (1). Substitute (29) into (1)
an
m
+1
(n + 1; m)
n+1
X?n
nX
+1
0
n =0 m =m?(n+1?n )
?1?n ) nX
?1 m+(nX
0
0
0
0
0
+anm (n ? 1; m)
(n0; m0) 'm s (; n0)
(n0; m0) 'm s (; n0)
0
n =0 m =m?(n?1?n )
n m +(
mX
+1
Xn?n ) 0 0 '
X
00
(n ; m ) m s (; n0) =
Vmm (n) (n; m )
+
n =0 m =m ?(n?n )
m =m?1
m
+(
n
?
n
)
n
0 0 X
X
(n ; m ) 'm s (; n0):
(33)
= (n; m)
n =0 m =m?(n?n )
Transform the r.h.s. of (33) substituting 'm s (; n) from (5)
n?n ) n m+(X
X
(n; m)
(n0; m0) 'm s (; n0) =
n =0 m =m?(n?n )
n?n ) n m+(X
X
(n; m)
(n0; m0)[a n m 'm s (; n0 ? 1)+ a n +1m 'm s (; n0 + 1)
n =0 m =m?(n?n )
(34)
+ cn m 'm s (; n0)+ bn m 'm ?1s (; n0 )+ bnm +1'm +1s (; n0)]:
0
0
0
00
0
0
00
0
00
0
00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Further, to obtain the relations for potentials an m , bnm and cnm, multiply eq.(33)
with its transformed r.h.s. (34) by 'ms (; n + 1), 'm s (; n) and 'ms (; n) and
take into considaration the completeness relation (7) for the functions ' (). The
expressions for anm , bnm and cnm thus derived coincide with formulae (30), (31) and
(32), correspondingly, obtained from the formulae of the inverse problem.
+1
+1
+1
4 Conclusion
The Gelfand-Levitan spectral inverse problem for the discrete two-dimensional
Schrodinger equation is considered on the basis of the Berezanskii technique of orthogonalization of polynomial matrices. By using the derived formulae of the inverse
problem, discrete Bargmann{Darboux transformations in two dimensions are given.
Analytic relationships are established between the solutions for two dierent sets of
discrete potentials and the potentials themselves.
11
I thank Dr.V.M.Muzafarov and E.P.Velicheva for helpful discussions.
A part of this work was done in June of 1998 at the Erwin Schrodinger International
Institute for Mathematical Physics, Wien, Austria.
References
[1] Faddeev L.D. 1974 Sov.Probl.Math. 3, 93-180.
[2] Newton R.G. 1980 J.Math.Phys. 21 1698; 1981 22, 631; ibid, 2191.
[3] Newton R. 1989 Inverse Schrodinger Scattering in Three Dimensions (SpringerVerlag. Berlin Heidelberg/New York).
[4] Novikov R.G. and Henkin G.M. 1987 Usp.Mat.Nauk 42 93-151 (in russian).
[5] Berezanskii Yu.M. 1965 The Expansion into the Set of Eigenfunctions of Selfadjoint Operators (Naukova Dumka, Kiev) 798 p.
[6] Spiridonov V. and Zhedanov A. 1995 Methods Appl. Anal. 2, 369-98
[7] Zakhariev B.N. and Suzko A.A. 1990 Direct and inverse problems. (Potentials
in quantum scattering) (Springer-Verlag. Berlin Heidelberg/New York, 223p.)
[8] Suzko A.A. 1985 Physica Scripta 31 447-449; 1986 34 5-7.
[9] Suzko A.A., 1997 Int. J.Mod.Phys. A, 12 277.
[10] Matveev V.B. and Salle M.A. 1979 Lett.Matt.Phys. 3 425.
[11] Matveev V.B. and Salle M.A. 1991 Darboux transformations and Solitons
Springer Series in Nonlinear Dynamics (Springer-Verlag, Berlin Heidelberg/New
York);
[12] Spiridonov V. 1997 J.Phys.A, Math.Gen. 30 L15-L20.
[13] Case K.M. and Kac M., 1973 J.Math.Phys., 14 594-603.
[14] Kay I. and Moses H.E. 1961 Nuovo Chim. 22 689; 1961 Comm.Pure Appl.Math.
14 435.
[15] Suzko A.A., 1978 Dissertation (1 degree), JINR, Dubna, 106.
12