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/^^ -r-^ -r^. NEW YORK UNIVERSITY



mi

'^ >»-^ ,s Division of Electromagnetic Research



CO

^ I ^ I Y ^ Institute of Mathematical Sciences



NEW YORK UNIVERSITY

INSTITUTE OF M-XTMEMATIC/iL SCFENCFS

LIL/L-vji'i'

RESEARCH REPORT No. CX-20 25 W.verly Pbc., New York 3, N. Y.



The Determination of the Scattering Potential
from the Spectral Measure Function

Part III: Calculation of the Scattering Potential from the Scattering
Operator for the One-Dimensional Schrddinger Equation



IRVIN KAY anci HARRY E. MOSES



NEW YORK UNIVERSITY

INSTITUTE OF MATHEMATICAL SCIENCES

L'BriARY

25 W., r.'y Place, New York 3, N. Y.



O CONTRACT NO. AF19(122»-463



SEPTEMBER, 1955



NEW YORK UNIVERSITY

Institute of Mathematical Sciences
Division of Electromagnetic Research

Research Report No, CX-20



THE DETERMINATION OF THE SCATTERING POTENTIAL
FROM THE SPECTRAL MEASURE FUNCTION



Part III; Calculation of the Scattering
Potential from the Scattering
Operator for the One -Dimensional
Schr'odinger Equation



Irvin Kay and Harry E. Moses



J/n^j^f 0. Then if

J J u

the equation



X

>



(i.7) < x|K|x* > = - < x|Q|x' > - -^ (x+x'+2a)r < x|K |x" >dx"< x"|0 |x

-^(2a+x')



.1 .1 .

can be solved for < x|K |x > (x < x) where

-. /°° -.1 .> ^.(x+x'+aa)

(1.8) < x|n|x'> - ^^ J b(k)e-^(-*^' W - i^ r/^

/-co "^

and '?(x) is the usual Heaviside unit function

1 for X >

(1.8a) ^ (x)

'' ' for X < ,

then the scattering potential V(x) is given by



(1.9) V(x) - 2 ^ < x|R|x > .



Furthermore

(1.10) ®^^' '^'

/-co



which satisfy the boundary condition



(1.12) lira T (x) - e^^^ .
x-^-oo ^

2

The eigenfunctions corresponding to the point eigenveO-ues - '? . denoted by Y .«< (x)

J

are given by

X

(1.13) T_^^ - e ^ +1 < x|K|x' > e "^ dx».

J /.CO

The normalization of the eigenfunctions T. (x) of the continuous spectrum given
by (1.11) is

+ 00 +00

(l.IU) ^ I T*(x)Yj^,(x)dx + ^^ I Y*j,(x) Tj^,(x)dx - 6(k-k').

The bound states are orthogonal to eigenfunctions of the continuous spectrum,
they satisfy the following orthogonality relation with respect to each other

r -1 '^'^A'^

(1.15) Y .^ (x) Y ,^ (x)dx - ±6. r.^e ^ .



- 5 -



The eigenstates T,^(x) and T ,^ form a complete set and satisfy the compieteness

J

relationship

+ 00 +00

(1.16) ^[f Tj^(x)T*(x')dic + I Yj^(x)b(-k)T*j^(x')dKl

/-co /-co



2ra

J J «J



2, The one -dimensional eigenfunctionsj the scattering operator
The unperturbed Hamiltonian H is given by

,2

(2.1) I^ 2" » -00 < X < +00,

dx

where the superscript x on H^ signifies that the operator is expressed in the x-
representation. It is convenient to use the eigenf unction |H ,A ;E,a > given by

(2.2) -l£2li^ e^^ V^ ,

where a is restricted to having the values +1 or -1. The degeneracy variable
a corresponds to the direction of the momentum. It is easily verified from (2,2)
that

+ 00

(2.3) < H^,A^;E',a'|H^,A^;E,a > « T < H^,A^jE> ,a« |x > dx < x|H^,A^;E ,a >

/-oo

- 6(E-E') 5^^^, ,



where 5 . is the usual Kroneclcer 5. It can also be verified that
a,a*



- 6 -



(2.U) ^ f |Ho,A^jE,a > dE < H^,A^;E,a| - ^ (H^),
^ /o

where 7(H ) is the identity operator in Hilbert space and maps remainder of the
extended vector space into zero.

The perturbed Hamiltonian H is given by



(2.5) H - H^ + 6 V.



Initially, we shall assume that < x|V|x' > is generally not diagonal. Later
we shall give conditions under which < x|V|x» > is diagonal, i.e., has the form

(2.6) < x|V|x' > - V(x) 5(x-x«).

We shall assume, however, that < x|V|x' > is such that a scattering operator exists,
ITie outgoing, and incoming eigenf unctions of H corresponding to the continuous
spectrum satisfy the integral equations

+ 00 +00

(2.7) < x|H,A}E,a >j ■= < x|Hp,A^jE,a > ± | 6(E)""'^^ I ( exp(+ i /E |x-x'|)dx«

/-co /-oo

< X' |V|x" > dx»< X" |H,AjE,a > ,
In (2,7) we use the well-known result



(2.7a) < x|y^(E-H^)|x' > = + J (E)'^/^ exp(- i ^^ |x-x'|) .



The scattering operator and its inverse can be represented in the H -representation
by (see [6], Part I, Eqs. (U.12), (U.20), and [6], Part II, Eqs. (3.7), (3.8))



and



where



- 7 -



(2.8) < H^,A^}E,a|S|H^,A^;ESa' > - 6(E-E«) < a|S(E)|a' >



(2.9) < H^,A^;E,a|S"^|H^,A^jE',a« > = 6(E-E') < a|S"^(E) |a' >,



(2.10) < a|S(E)|a' > - 6 - 2ni€ < H^,A^jE,a|V|H,A-,E,a' >

+ 00 +00



lie I I < H^,A^;E,a|x > dx < x|V|x« >
/•oo /-oo



dx» < x' |H,a;E,a' >



and



.-1



(2.1i) < a|S'-^(E)|a' > - 6^^^, + 2nie < H^,A^}E,a|V |H,AjE,a' >^



+ 00 +00
^00 ''-oo



^a at + 2ni6 < H ,A^}E,a|x > dx < x|V|x' >



a, a'

-00 ''-oo



dx' < x'|H,AjE,a">^ ,
For future reference we note the i^ciprocity theorem
(2.12) < H^,A^;E,a|V|H,A;E,b >_ - ^< H,A;E,a|V|H^,A^;E,b > -



- < H^,A^jE,b|V|H,AiE,a >* ,



where the asterisk means complex con^gate.



- 8 -



If < x|V|x« > dies down sufficiently rapidly, the asymptotic forms of
< x|H,A;E,a > are related to < a|S(E)|b > and < a|S (E)|b > by



(2.13) lira < x|H,AjE,a >_ - < x|Hjj,AQjE,a > + < x|H^,A^jE, t 1 >



[-6^^^^]



(2.1U) lira < x|H,A;E,a >^. - < x|HQ,AQjE,a >

x-> ±00



+ < a|H^,A^jE, 7 1 > [^< ; l|s'-^(E)|a > -6, ^^^J,



as can be shown from (2,10) and (2,11).



3, Expression for the weight operator in terms of the scattering operator
We shall now express in terms of S and S" the operators W , "M associ

C X

with those operators U such that the eigenfunctions of H given by



(3,1) - < x|U|H ,A„;E,a >



o' o'



satisfy the boundary condition



(3.2) iim < x|H,A;E,a > - < x| Hq,A^j E,a > -
x->-oo



or, symbolically.



(3.2a) lim < xlH,A}E,a > - < x|H^,A^;E,a > .

x-^-oo



Since we shall work in the continuous spectrxon of H, we shall drop the factor



- 9 -



^(H ) for the time being, since it will always act like the identity operator.



o

From U =• U M we can write



(3.3) < x|H,AjE,a > -2I3< ^l^»A;E,a' >_ < a' |ti_(E; |a > ,

where < a|ix_(E)|a» > is given by (of. ^] , Part I, Eq. (5.5))
(3.U) < H^,A^}E,a|M^|H^,Ag;E',a' > - 6(E-E') < a|n_(E)|a' > .

From Eqs. (3.2a) and (2,13) we obtain an equation for < a||j, (E) |a' >, namely



(3.5) < x|H^,A^}E,a > '22^ x|H^,A^jE,a' > < a'|n_(E)|a >

a'



+ < X



|H^,A^jE,-i >2 ^ -i|S(E)|a' > - 6._l^3, 1< a' ||i_(E) |a >.



On multiplying (3.5) through by < H ,A }E",a"|x> and integrating with respect
to x, and then using the orthogonality relation (2.3), one finds that



(3.6) 6(E"-E)5^„^^ - 6(E"-E)^6^„^^, < a'|njE)|a >

+ 6(E"-E)5^,,^_^^ ^< -l|S(E)|a' >-6.^^a, ]< a'|njE)|a >,
which on renaming some of the variables, may be rewritten as

^^'"^^ ^a,a' " < a|ME)|a' > * 6^^.j,Z! p -■L|S(E) |a"> -6.j^,a«]< a"|iijE)|a' >.

Eq. (3.7) may be regarded as an equation for the matrix < a||i._(E)|a' >. The solu-
tion of (3.7) can be shown to be



- 10 -



(3.8) < a|ti (E)|a'



> a



< -1|S(E)|-H > i

< -±|S(E)i-l > < -1|S(E)-1 >




where we adopt the convention that the first row and column are labeled by +1 and
the second by -i, i.e., < +i|n_(E) (-1 > » 0. Eq. (3»8) gives us M_ in the H -
repre sentation .

From [6], Part I, Eq. (5.l5) we have



(J. 9) W^ - M^""- M*"-*- .

-1 »-l

To find W , therefore, we shall have to obtain M_ and M .We shall express

these operators in terms of the H -representation. Let us define < a|n2 (E)|a' >
by



(J.IO) < H^,A^jE,a|M2"*-|HQ,AQjE'>a» > - 6(E-E') < alM-^-^CE) |a« >.

From vrh/i_ « M_M""'' - ^CH^)* (where we recall ^(H^) is the identity in Hilbert
space) it can be shown that for fixed E the quantities < a|ji~ (E)|a' > are just
the elements of the matrix which is the inverse of the matrix whose elements are
< a|p. (E)|a' >, It is then shown that



(3.il) < a|ix"-^(E)|a' >



1





Let us now define < a|M (E)|a« > by



- n -



(3.12) < H^,A^;E,a|M^^|HQ,A^jE«,a' > - 5(E-E') < a|n*'-^(E) |a« > .



From the fact that M " - (M )" « (M~ ) is the Hermitian adjoint of M we have



(J. 13) < a|/"^(E)|a' > - < a- jn^-^CE) |a >* .



If, as in [6] WB define < a|a) (E)a' > by



(3.1ii) < H ,A^jE,a|W^|H , A^;E«,a' > - 5(E-E') < ajoo (E) |a' > ,



ve have from (3«9)



(3.15) < alo)^(E)|a' > -JjJ < a|n;-^(E) |an> < a" |n*'^(E)|a« >,



or, on using (3.13) and (3.11)



< -±|S(E)|+± >







(3.16) < a|co^(E)|a» >



c



< -1|S(E)|+1 >
In deriving (3.16) from (3.15) we have used the relation



(3.17) I < -1|S(E)|-1 > 1^ + I < -1|S(S)|+1 > 1^ - 1,



which follows from the fact that S* - s" , i.e., SS* - ^(H^).

For the sake of completeness, we shall also give the expression for
< a|p. (E)|«' > defined by

(3.18) < x|H,A;E,a > -^ < x|H,AjE,a' >^ < a'|lA^(E)|a > ,



- 12 -



where < x|H,AjE,a > satisfies the boimdary condition (3.2a). The expression can
be shown to be



- < +1|S (E)l-i >



< +i|S'-'-(EJ|+l > < +1|S""'-(E) l+i >



(3.19) < a(ii^(E)|a' > -



U. The complex energy plane

In this sec^^ion we consider the analytic continuation of various quantities

introduced above, and for this purpose we introduce the complex E-plane which shall

have a cut along the positive real axis. The results obtained here will be useful

in later sections.

Let us first introduce the function < xlH ,A jE.a > • This function is de-

o o

fined as the analytic continuation of < x|H ,A }E,a > vben E is on the upper part
of the cut. From (2.2), we see that when E is in the lower part of the cut



(U.l) * - -i< x|H^,A^j;E,-a> .



One can also introduce < x|H ,A ;E,a >" which defined as being equal to < x|H ,A jE,a >
when E is on the lower side of the cut. Hence, from (U.l)



(U.2)



< x|H^,A^}E,a > - i



and



(U.3)



< x|H^,A^jE,a >" - i < x|H^,A^}E,-a >



- 13 -



when E is on the upper side of the cut.

We can also introduce < H ,A ;E,a|x > which equals < H ,A }E,a|x > when E is
on the upper side ol" the cut. Then since



(U.U) < Hg,A^;E,a|x > - < x|H^,Ag;E,-a >



we have



(U.5) < H^,A^;E,a|x >* - < x|H^,A^jE,-a >■*■ .



Similar iy one can define < H ,A jE,a|x > as being the analytic continuation of
< H ,A }E,a|x > when E is at the bottcm of the cut, where from (U.U) it can be
shown that



(U.6) < H^,A^;E,alx >" - < x|H^,A^}E,-a >' .



Now we define < x|H,A}E,a > as the analytic continuation of < x|H, A}E,a >
when E is on the top of the cut. From the integral eqiation (2.7) it is clear
that when E is at the bottom of the cut we have

+ 00 +00

(U.7) < x|H,AjE,a >* - -i < x|HQ,A^j}E,-a > + ^ t{zy^^^ f | dx'dx" e* ^ ""^^

/-co /-CD

< X' |V|x«> < x»« |H,A;E,a >*



from which it can be shown that



(U.8) < x|H,A}E,a >* - -i < x|H,A}E,-a >



- lU -



when E is on the lower part of the cut.

Likewise one can introduce < x|H,A}E,a >" which is the analytic continuation
of < x(H,AjE,a > defined on the lower part of the cut. It is clear from (U.8)
that



(U.9) < x|H,A;E,a >" - i < x|H,AjE,-a >^

and that when E is at the top of the cut

(U.IO) < x|H,A}E,a >' - i < x|H,AjE,-a >_ •

* +

The analytic continuation of < a|S(E) |a' > when E is on the top of the cut
will be denoted by < a|SfE)|a' >*. From (2.10), (U.8), (U.5), (U.l), (2.11),
(2.12) we see that when E is at the bottom of the cut



+ 00 ■♦

(U.U) < a|S(E)|a« >* - 5^ ^, + 2ni j dx j

/-oo y-c



+ 00 +00

dx«

00



< HQ,A^}E,-a|x > < x|V|x' > < x' |H,AjE,-a« >^



< -a|S"^(E)|-a' > - < -a'|S(E)|-a >* ,



where, as usual, the asterisk means complex conjugate. We denote by < a|S(E)|a' >'
the analytic continuation of < a|S(E)|a' > when E is at the bottom of the cut. It
can be shown that when E is at the top of the cut



(U.12) < a|S(E)|a« >" « < -a|S"-'-(E) |-a« > - < -a«|S(E)|-a >* ,



- 15 -

5. Extension of the \inperturbed Haiailtonian. Diagonal form for the scattering

potential .

In [6] we saw that it is necessary to extend the definition of the operator H
to a space larger than Hilbert space in order that the extended operator H have the
same spectmun as H. In [^^ J , Part II the extension was carried out by introducing
'eigenf unctions' of the extended operator H , Here wb specialize this procedure
to the present case.

In Hilbert space H is defined by the way it acts in the x-representation,
namely,

(5.1) hJ » - 1^ .

dx

We shall require that the extended space, H , be also given by (5»1)» In the
notation of [6], Part II, H is then an x-extended operator. With this extension
the potential V has the desirable property that it is diagonal in the x-representa-
tion, as we shall show shortly. First, however, we shall obtain the additional eigen-
functions necessary to complete the extended space. In accordance with (6], Part II
we shall look for solutions of



(5.2) H^lH^,A^jE,a> = E|H^,AQjE,a>



for values of E in the vicinity of each E. which is a point eigenvalue of H such
that



(5.3) lim < x|H^,A^;E,a > - 0,

X->-00



In terns of the x-representation Eq. (5*2) becomes



- 16 -



.2
(5.U) - -^ < x|H^,A^;E,a > - E < x|H^,A^}E,a > .

dx

For a fixed negative value of E, there is only one solution which satisfies (5.3) «
We conclude therefore that the spectnun of the extended HaJiiltonian H and the
point spectrum of H cannot be degenerate .

Let us denote by < x|H ;E > the suitable solutions of (5.U). Then ws can
write

(5.5) < x|H^}E > - A(E) ^"^ " ^ (-E)-^/^e>^ "" - A(E) < x|H^,A^jE,-l >*

2 ^n

where A(E) is an arbitrary function of E which has no zeros or singiilarities for
E < 0. With this choice of IH lE > we take for the bi-orthogonal vector |H^jE >_

■ O D

another solution of (5.2) or (5.U) which satisfies



(5.6) „ - [me)]-^ ^ (-Er"^/" e" ^^'^ - < H^,A^jE,-l|x >*



Also

(5.5b) < H }E|x > - A*(E) " . '^ ■ (-E)'-'/'* eV-"" « i A*(E) < H ,A ;E,+l|x >"



,-(E) !!!^ (.E) - ^A e>^ . i A*(E) < H„,A„;E,.1|
2 v^



In the general vector space all states < x|0 > will have the property that they
are quadraticaily integrable from x = -co to any finite limit,

we shall now prove V is diagonal in the x-representation. We have



- 17 -



f5.7) eV = H - H - DH U^ - H UU - (UH -H U)U - e(kH - H k) + 6^(KH -H K)K

O OO 00 000 vOo" 00



We shall first calculate (KH -H K) in terms of the x-representation. Since K is
triangular in the x-representation we can always write



(5.8) < x|k1x' > = -^(x-x') < x|P|x' > ,



where



< x|P|x» > = < x|K|x« > for x« < X,
(5.9)

< x|P|x» > ■ an arbitrary function, for x« > x

V/e choose < x|P|x' > so that it is continuous at x = x'. Since H is Hermitian,
we have

2 2

< xjKH |x' > « - < x|K |x« > -^ ^ < x|K |x' >

° ax' ax«

(5.10) ^2

■ 2_^'77(x-x') < xjPlx' >.

ax'



Now

^^(x-x') < x|P|x' > = -6(x-x') < x|P|x' > + '?(x-x') ^ < x|P|x' >
(5.11)

« -5(x-x') < xjPjx > + '^(x-x') ~ < x|P|x' >,



and



- 18 -



^2 a

-2— 7(x-x') < x|P|x' > - 6'(x-x') < x|P|x > - 6(x-x') ~ < x|P|x >

9x1^ °^

(5.12) ^2

+ -T^Cx-x') -2_ < x|P|x« >.
Sx'



Likewise



2 2

(5.13) < x|H K|x' > . - 1-p. < x|K|x' > - - i-, 7(x-x') < x|P|x' >
° 8x dx



But



(5Ji;) l.'7^(x-X«) < x|P|x' > - 5(x-x') < x|P|x > +^(x-x') ^ < x|P|x' >



and



s2 H

£ - ^(x-x«) < x|F|x« > « 5«(x-x») < x|P|x > + 5(x-x') ^ < x|P|x >

8x



(5.15) ;, ^2

+ 5(x-x') ~ < x|P|x > + ??(x-x') •^< x|P|x'



ax



In (5.15) the expression ^< x|P|x > means that after x» has been set equal to
X in < x|P|x' >, one takes the derivative of < x|P|x > with respect to x» In
contrast, the expression -^ < x|P|x > means that one has taKen the derivative of
< x|P|x' > with respect to x and then set x' equal to x. In (5.12) ^~j < x|P|x >
means that one is to take the derivative of < xjpjx' > with respect to x' and then
set x' equal to x. From (5.11) - (5.15), (5.8), and the fact that



|j< x|P|x> + Jt.< x|P|x> = 4 - 26(x-x') ^< x|K|x > + '^(x-x')



(5.16)



2 2

-^< x|K|x' > - -^
9x dx«



x|K|x'>



Hence from (5.7) and the fact that K is triangular in the x-representation we
have



2 2

< x|V|x' > « 26(x-x') ^< x|K|x > + ^ (x-x') ^ -^ < x|K|x« > - -^ < x|K|x« >

9x 8x'



(5.17)



+ 2l~ < x|K |x >.< x|Kq|x' >



X —



2 2

•^ < x|Klx"> ^ < x|K |x">

ax 3x'



dx" < X" |K |x' >



Now from the general theory, V is Hermitian if the generalized Gelfand-Levitan
equation of [bj can be solved. Furthermore, it will later be shown that
< x|K(x' > is real. Therefore the right-hand side of (5.17) is the sum of a
Hermitian operator which is diagonal in the x-representation, and an operator
which, because it is triangular in the x-representation, cannot be Hermitian.
Since the right-hand side must be Hermitian, the expression in brackets must
vanish and we are left >a.th



(5.18)



< x|V|x' > - 6(x-x') V(x),



where



- 20 -



(5.19) V(x) - 2 ^ < x|K|x >



Hence V is diagonal in the x-representation and can be obtained from K very simply,
by means of (5.19).



6. The weight operator and the Gelfand-I^vitan equation

We shall now develop expressions for the weight operator in terras of the H -
representation. In accordance with the general theory, the equation for K is



X

< x|K |x' > - - < x|r2|x' > - 6 I < x|K|x" > dx" < X" [Hlx* >j

•-00



where



(6.2) cO. - W - '^(H )



^c 'iKmv * Wd^^(v«o^^(-»o)-



we have already expressed W in terms of the H -representation and have seen that
it can be obtained in terms of the reflection coefficient of the scattering
operator (Eqs. (3.1U) and (3.16)). We shaH.l now obtain the expression for W.
in terms of the H -representation.

From the general procedure discussed in [6], Part II, W, as given in the H -
representation has the form



(6.3) B< H^,A^jE,a|W^|H^,A^;E',b >b - 5(E-E') < a|a)j(E) |b > ,



where |H ,A jE,a >„ are the bi -orthogonal eigenf unctions (5.5a) of H defined for
negative eigenvalues E in the neighborhoods of the point eigenvalues E. of H.
Since, in the present case, these eigenfunctions have no degeneracy, we shall
write



- 2J. -



(6.U) g< H^,E|W^|H^,E' >g - 5(E-E')co^(E) - 6(E-E') [c(E)l"-^ ,

where, in accordance with (6 j , Part II, the C(E. ) are the normalization constants
of the proper eigenstates of H, that is,

(6.5) - C(E^) 5^^ .

We shall now obtain e < x|Xi-|x' > in the form



(6.6) 6 < xlAjx' > - < x|(w^-^(H^))^(H^)|x« > + < x|W^£]5(E^-K^)7(-H^)|x' >,



We note that

< ^l(V^^"oO^(»o)l^' ^ -Z^r< ^iV^oJ^*^ ^f a|c.^(E)|b > - 5 J

(6.7)

fl{E)dE < H^,A^jE,blx' >.

On using (3. -1-6) and (2.2) we have


/o

00

+ I dF < x|H^,Aq',E,+1 > < -1|S(E) |+± >*< H^,A^;E,-±Ix' >
/o



(6.8) ° ^





- 2 Re I dE < xlH^,A^;E,-l > < -1|S(E)|+1 > < H^,A^jE,+l|x' >,



0* o» » ^,w,_,,.-. "o'-o'



where Re means 'real part', Eq. (6.8) leads to



- 22 -

00 1



(6.9) < x|Cw^-'!^(H^))^(H^)|x« > = Re ^ f dE E expUi yf(x+x')J< -1|S(E)|+1 >.


1

Online LibraryIrvin W KayThe determination of the scattering potential from the spectral measure function. Part III → online text (page 1 of 2)