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Some remarks on the generalized Gelfand-Levitan equation
JOURNAL
OF MATHEMATICAL
Some Remarks
ANALYSIS
AND
APPLICATIONS
92, 410-426
on the Generalized Equation
(1983)
Gelfand-Levitan
ROBERT CARROLL University
of Illinois at Champaign-Urbana,
Urbana, Illinois
61801
Submitted by V. Lakshtnikantham
1. INTR~OUCTION
The Gelfand-Levitan (GL) and MarEenko (M) equations arise in quantum scattering theory as an important part of the machinery used to recover the potential q(x) from a Sturm-Liouville operator D2 -q(x) via inverse methods (cf. [32, 39, 551). Modifications of this procedure for handling inverse problems also arise in other applied fields (see, e.g., [21-24,32,61]) and lead in particular to various types of GL equations. In all of this work the transmutation concept plays an important role (although often disguised). Now in earlier papers we developed an extensive theory and framework for transmuting an operator of the form J% = (dPu’)‘/dP + pi (pP= f lim A;/AP as x -+ co) into another operator of the same type (cf. [6-9, 12-16, 18-20, 231). In particular this theory applies to situations where P = P -pi is modeled on the radial Laplace-Beltrami operator in a rank one noncompact Riemannian symmetric space and the spherical functions arising from equations &: = -I’p<, (Q<(O)= 1, D,pl;(O) = 0 involve various special functions of interest in diverse contexts. In [8,9] we derived a general GL and M equation for transmutations D* + p (p as above). Given the fundamental role of such equations in the inverse scattering and inverse Sturm-Liouville problems and the nature of their derivation in [8,9] from eigenfunction kernel expressions,it is clear that the version of [8,9] contains valuable information about special functions (cf. also [ 28-3 1, 44-461, where some relations between discrete forms of the GL and M equations of quantum scattering theory and orthogonal polynomials have been studied-although some version of this approach can be envisioned for singular operators of the type P above, we shall not pursue this here). In this paper we shall first extend (in Section 2) the GL equation of [8, 91 to the context of transmutations $-+ $ and clarify its structure. In particular we shall represent it as a formula in spherical functions. Then (in Section 3) we take a model operator P, = D2 + ((2m + l)/x)D (A, =x2”‘+‘) and give 410 0022-247x/83
$3.00
Copyright Q 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.
GELFAND-LEVITAN
EQUATION
411
an explicit formula for the extended GL equation (for D* and P, = $) by evaluating various kernels in terms of pseudofunctions. This will serve as a guide to interpret such equations and seemsto be the first such result in this direction. The operator P, is particularly appropriate as a model here since the special functions which arise are Bessel functions about which information is readily available and also because of its importance in physics where angular momentum is represented by I= m - + in the transformed operator pm = Dz - (m’ - :)/x2 = D* - 1(1+ 1)/x* (cf. [ 14, 6, 7, 25, 32, 36, 37, 39, 41, 42, 47-49, 52, 53, 58-60, 62, 631). Our techniques for P, = Q can then be extended virtually automatically to other operators P as above by using the general transmutation machinery of [6-9, 12-16, 1g-201; this will be done in [ 171.
2. THE GL EQUATION
In [8, 91 we derived a GL equation (and an M equation) in a general context (for transmutations 0’ + Q) ” by a unified method based on transmutation operators. For the special case of quantum scattering theory this method goes back to Fadeev [39] but our context was considerably more general and both technical and conceptual changes were used in the proofs. Our framework is intimately related to various special functions as they arise in the form of spherical functions in harmonic analysis on rank one noncompact Riemannian symmetric spaces and thus the GL and M equations automatically have something to say about special functions. In the situation treated here (which extends [8, 91) special functions are displayed as the ingredients for a general extended GL equation (for transmutations P -+ &) and this equation becomes then in fact a formula in spherical functions. Let us refer to [6-9, 12-16, 18-241 for some of the recent development of transmutation ideas and to [32, 39,51,53,55] for background information. For notation we recall briefly that we are concerned with operators P and Q of the form P(D)u = (Ap u’)‘/Ap,
P(D)=P(D)+p;,
pP = + lim AL/d,. x-cc
(2.1)
For the interaction with special functions we take A, so that P(D) is modeled on the radial Laplace-Beltrami operator in a rank one noncompact Riemannian symmetric space (cf. [33-36,40, 50,62,64]-examples are indicated below). One considers eigenvalue equations p(D)u = -1’~ and isolates in particular two types of solutions. First, solutions o;(x) of this equation satisfying o:(O) = 1 and D,p~(O) = 0 are generalized spherical functions, and we call Jost solutions those @c,\(x) w A;“*(x) exp(f Ux) or -exp(fiA -p)x as x+ co (p =pP). One writes (p:(x) = c,(A) @f;(x) + 409’92,2-8
412
ROBERTCARROLL
c,(-1) @c,\(x) and the spectral measure arising for the cpf; eigenfunction theory is dvp(l) = dL/2n Ic,(~))~.
2.1. To
illustrate the situation, consider A, = Ap4 = from 1401, which with A, = AckJ= (,x-,-x)2fktl(,x +e-")2Dtl from [SO] will provide a nice background of specific examples and information. Then pP = f( p + 2q) = p and one has the following facts: For x E (0, co), q:(x) is entire in ,l of exponential type x while @{(x) is analytic for Im 1 > 0, extends to be analytic in C/(-N}, and satisfies Q;(x) - exp(U - p)x as x + 03 (in particular for c > 0, E > 0, q > --E It/, and x E [c. co), Q:(x) = e”.t-“‘“{ 1 + e-‘“O(A, x)} with lD:Ol,< K,). Further, for q > --E l
(&
_
e-.r)P(e2x
_
e~2.r)4
c,(l) = (22bT(~)r(ik))/(f(c
- b) r(a)),
(2.2)
where a = f(~ + U), b = j(p - U), and c = i(p + q + I). EXAMPLE 2.2.
Take A, = A,, as indicated in Example 2.1. Then
q;(x) = F(f(p + U), f(p - U), a + 1, -sh2x)
(2.3)
(p = a +/I + 1) is a Jacobi function of the first kind while for 13+ -i, -2i,..., the Jost solution is (note A,, F Ap4)
@f;(x) = (ex - e-x)i.k-P X
(2.4)
F(f(j? - a + 1 - ii), f(p + a + 1 -U),
1 - iA, -sK2x),
which is a Jacobi function of the second kind. A special case of particular A, zA~,-,,~ = (ex -e-x)2mt’ = 22m+‘.yh2m+‘~. importance is Then AL/A, = (2m + 1) coth x and we are working with canonical equations associated with the radial Laplacian in SL(2, R)/S0(2) (cf. [ 10, 111). In this case the spherical functions rpi = v[; can be written as
#J(x) = 2”r(m + 1) SK”x&Y
,12(chx),
(2.5)
where P,i?! ,,* denotes the associated Legendre function of the first kind (cf. [56]) and CD:= @T is @T(x) = (2-“r(l
- U)/fiJ’(p
- U)) e-i”“sh-“xQ”,i2-i,~(chx)
(2.6)
(p = m + f), where Q?!,,2-i,, denotes the associated Legendre function of the second kind. Further
c,(A) = c,(i) = (22mT(m+ 1) r(iA))/r(p
+ U).
(2.7)
GELFAND-LEVITAN
413
EQUATION
Let us also record a form of the function @y(x)/c,(-A) which turns out to be the natural analogue of the exponential function (up to f); this can be written
where e(z) = e-‘U~Q~(z)/Z(~ + u + 1) is entire in ,Uand v (cf. [56]). Let us recall now some basic constructions from transmutation theory. Thus we consider transmutation operators B: p-+ Q, which means that &B = BP’, acting on suitable objects. When P = p = Dz with ylj;(x) = Cos Ix, @2(x) = e’,‘“, d,,(x) = 1, c,(A) = i, and dv = 2dl/n, we write BQ: D* + 0; it is this case which was treated in [8, 91. Here we generalize further the technique of [8,9] to produce the most general type of extended GL equation for transmutations B: B--P& with P and Q of the type Pu = Thus recall from [8, 9, 18, 191 that B = 2?$: p + Q with (4 u’)‘/A,. 9: = Bcp: and B= ClrQ with @(,I) & = Z&Y;, where I@) = l/W(A) = Here ‘WV) = (@(x), f(x)) (Ql = &PI;), W’(x) = IcP(4/c&)l’. (W), cpl;W,(v = VP)*Qf(~) = wi(x>~ f(x)), wx> = (Jv)7 &xx)), (w = ws - vc), and A?F(x) = (F(A), p!(x)), . One sets also 9 = Z?= .$?-’ with 3 = gE2 and then 93Wx= Z, where W” = QW(A)Q. Similarly Z?= B-’ = .$ with l?BF? = Z, where @ =$@(A)!@, and we recall also that dv,, = W(l) dw, with do, = I?‘(A) dv,. The general GL equation then is Bk@ = B’ when written out in terms of kernels and we recall (cf. [8,9, 19,201) /WY x) = Qx(x>~ P!%Y)L = AP(X) A,‘(YXdW9 = AAx) A,‘(Y)
QKYD,.
I+, 4’1,
(2.9)
where /3= ker B, y’= ker .$, and 7(x, y) = 0 for x > y. Similarly, Y(X,Y) = (d(x), Q%9), = A,(Y) 4*(x)&~,
= A,(Y) A,‘W?&)~
P?(Y)), (2. IO)
x>,
where y = ker 9, p = ker Z?,and y(x, y) = 0 for y > x. We recall also that the generalized translation rc - P has the form W(x)
= od(49J(4
rp%-)),.~
m
= W)~ Jmw
= w-(n).
(2.11)
We write also &(A) = (Q(r), @(c)) = ‘J3G(A) (i.e., e(r) = 9\ l?(r)) and note here that W differs by a factor of two from the W used in [8,9] under Fourier transformation. Now
414
ROBERTCARROLL
Then from (2.11) and @ = q &‘, @(JY,x) can also be written @CY,xl = 44~) C @x> = 44xKd(~) = 4(xKM(Y)
d(xX (ri/(Ov Qp,K>>>, (2.13) PXXh JTx3>,. 7 W)>7
which displays the kernel /?(x, y, <) = (o:(y) C&(X),~~(~)),. of Tq (i.e., Ti: %d = (P(x, .Y,0, *KM N ow write out BWK = B in terms of kernels to get the extended GL equation in the form
!p/KY,63m9 x>& =B(.hx).
(2.14)
Again recall p(y, 0 = 0 for < > y and &y, x) = 0 for y > x, so J’F = !‘i in (2.14) and the right side is zero for y > x. We shall display this as a formula in spherical functions in THEOREM
2.3. The general extended GL equation is
&w-e T; *CC>= r”,*(x)
= (~~~03,d(x)),
.
Formula (2.15) suggests various consequences.For example, since dv = IV(n) du (W = mP ‘) we have formally
from which one concludes that
d(x) =WA) jam Q:(t) T;*tt> 45 Since dP(x) T; p(r) = @(a x), this is equivalent to
(2.17)
GELFAND-LEVITAN
EQUATION
415
and is easily confirmed by other means. Indeed (recall dv = C(z) dz),
and multiplying by W(A) = @-‘(A) we get (2.18). Conversely, from (2.18) or equivalently (2.17) one obtains (2.16) or (2.15). Similarly from (2.15) we write now (do = @dv)
from which follows (2.21)
Of course, this is just the expression &(JT) = 1393rpy(y) (where y is a parameter) or q!(y) = Ji #‘(0, CO:(Y)), &(
The above constructions were developed around P and Q of the form Pu = where the inversion theory is founded upon spectral measures (4u’)‘/& dv, = dA/2n ]cp(A)12. More generally, one wants to deal with operators p = P +pi - q(x), where q(x) is a suitable (complex-valued) potential (cf. [35, 36,41,42,55,62]). In this event the inversion theory must be expressed more generally through distributions in the spectral variables and one achieves this nicely with the generalized spectral function (distribution) of Marcenko (cf. [6, 7, 17, 18, 26, 27, 41, 42, 551). We do not give details here (see [ 17, 26, 271 for a complete exposition) but remark that general Parseval
416
ROBERTCARROLL
formulas and expansion formulas for such situations can be proved and take the form (cf. (6, 7, 17. 18, 26, 27, 41, 42, 551)
(RQ, QfDg),l = (A;zf, Al;“gj,
f(Y) = (RQ, WV) W:(Y))., (2.22)
(note that we express (F, G), as (RQ, FG),k). Here RQ is the generalized spectral function, RQ E Z’, where the product &lflg E Z, Z being a certain space of entire functions of exponential type for f, g suitable functions of compact support. Further, one can define B’ more generally by adjointness (cf. [ 14, 26,481 in the form g= .Y (3 = B-‘), where the adjoint # is determined by @p(X) U(X), 3V(X>) = (U(X), AP(X)W
Y) A,‘(Y)AQ(Y~
V(Y)))
= @Q(Y) V(Y), (l’(x~ Y)Ap(x) A, ‘(4’h U(x)))
(2.23)
= (A,(Y) V(Y), ANY)). This determines B with kernel as in (2. IO) (where p^= p)
IT(vvxl = 44~) A; ‘(~9~6, Y).
(2.24)
In [26] we also show how B’ is determined via a partial differential equation, but this is not needed here. Now extending the derivation of GL equations as in [39], we write following [ 261
‘Pm = 04x309Pm;
d(v) = MY9f), 4(t)).
(2.25)
Then compose these equations with RQ~f;(y) and RQ&(x) respectively to obtain (using the relation (RQ, Q:(x) &(t)) = 6(x - t))
(RQvCP:(Y)&N
(2.26)
= Wx, 0, (RQ, CP%Y)d(O)>,
(RQvd(Y) &x)> = (Yb’, t), (RQ9V?(X) d(t))) Interchanging x and y in (2.26)-(2.27) then (using (2.24)) Y(X, Y) A,‘W4W
=h
= YtYvx)/d~tx).
(2.27)
and equating these equations we have
x) = (Ptrv t), (Ray @tx) v:(O)).
(2.28)
Consequently, defining @tt, xl = CR’, vf;tt> f-X(x)>,
(2.29)
THEOREM 2.5. In the more general situation where the inversion theory for P and & is given via generalized spectral functions R ’ and RQ, one can write the general extended GL equation again as B’ = B W”, where g is
GELFAND-LEVITAN
EQUATION
417
defined as A?# with ker i? = B given by (2.24) and W” is the operator with kernel @t, x) given by (2.29). Thus as in (2.15) p’
Remark 2.6. As in Theorem 2.4 one can produce formally a formula (2.21), but (2.17) does not make any sensenow.
3. EXPLICIT FORMULAS The model situation which we shall investigate now involves the operator Q(D)u = Q,(D)u = u” + ((2m + 1)/x) U’
(3.1)
which corresponds to d,(x) = x*~+’ and p = 0. We shall give an explicit formula for the general extended GL equation of Theorem 2.3 for the case p = D* and & = Q, in providing pseudofunction evaluations for the various kernels (see Theorem 3.3 and Corollary 3.5). The operator Q, is related to the operator Q, = D* - (m’ - 4)/x’ which occurs in quantum physics, where I = m - 4 represents angular momentum (the hookup with physics occurs for m > 4); references are indicated in Section 1. In fact there is a transmutation from Q, to Q, effected by noting that xm+ “*em u = QmIXmt lDu) cXm+l/* =Al/* a (x)), and we refer again to [6, 7,58,60]. Let us write now q?(x) = Bm(x, A) = zmr(rn + 1)(1x)-” J,(lx), l2,(x, A) = 2-*“T(m
+ l)(~~)*~+’ &(x)
(3.2)
There is a slight departure here from what is our normal notation, namely,
Q:(x) = A&) &4,
P(J)= (f(x), J-=%(x)) = wv)~
f(x) = elf(x) = (.m, v:(x)),,
= JornA4 d?(x) d%W
(3.3)
Before we settled on this notation, however, considerable material was published with the notation (3.2) for the particular operator Q = Q,. It is shown in [7,9] how everything fits together and we recall this here briefly. Thus one has dwQ(l) = dil/2n 1c&)l’ = &,(A) dA and setting R,(A) = c$ 2mf’ with c, = 1/2mr(m + l), it turns out that Q,(1) = RO()L)(R,(l) is a generalized spectral function in the senseof MarEenko [55]). Thus in (3.2) we have Q,(x, A) = R,(k) A&) &(x) = R,(A) .nf(x>. so Q0.f = (f(x), n,(x, A)) = R,(A) Zlf and f(x) = I (aof) o:(x) dA from (3.3). In any
418
ROBERT
CARROLL
event we want to examine the kernel expressions which occur in transmuting Dz into Q, (note here a reversal of y and p from some previous work where we employed B: Q, + D’). Thus consider BQ: D2 + Q, = & so that
=z
-% ^m
7[ o R (~‘,~)Cos~xd~=2R”(y,x) !
2zym + 1)4’-2m (4,2 -,2)/yI..2 = f(+)f(m++)
(3.4)
(cf. [25, 381). Thus &(r, x ) is in fact a function (but certainly not of the form 6(y - x) + a function), and setting k, = 2Qm + l)/fir(m + f) we have (x2 = r, y2 = q) (&(y,x),f(x))=k,y-““1;
(~~~-x~)“‘-“~f(x)dx
=~J~~-.(n-T)“-‘~2f(~)~.
(3.5)
Now recall the pseudofunctions (cf. [43, 571) Yn = (l/r(p)) Y_, = 6’“). Thus for a function g defined on [0, co) Yfn+K!*g=
’ r(m+f)-o
1’ g(O(rl - W-“‘2
PfxO-’
with
dt,
so that (3.5) can be expressed as
(PQ(YJ)lf(X))=
mm + 1)/h>
* cfmlm
viy*+l..2
On the other hand, y&x, y) will be quite different anticipated from formula (3.7)). Thus (cf. (2.10)) Y&
Y)
=
(A,(Y)
dib9,
Cm
Ax>,
=
A,(Y)
j,
(O:(Y)
(3.7)
(as can be already
Cos
lxRo@)
dl
“03 = 10
Q,( y, A) Cos Ax dA
=2-mT(m
+ l)-’
= 2-“I-@2 + I)-’
*cc(~~~)m+lJ,(~~)Cos~xd~ J0 ym+‘~21H,(P+“2
&SAX},
(3.8)
GELFAND-LEVITAN
419
EQUATION
where IH, denotes the Hankel transform. This formula is studied extensively in [6, 7, 161,for example, and we simply cite results here as needed.One can show that ya will be a distribution of order n, where -f < m < n - i (cf. [6, 7, 16, 531). A nicer result can be obtained by recalling a formula expressing Cos Ix in terms of km@, y) from EPD theory (cf. [25]-recall Cos 1x = c%?Q~4). Thus for -4 < m < n - 4
(Y&Y Y), g(v)) = 30 g(x) = YLX (+ Dxy jr g(y)y’““(x’
- yym-3/2
dy, (3.9)
where y”, = &2”-‘T(m + 1) r(n - m - 4). This can be rewritten (taking n > m + 5 if desired and setting x2 = r, y2 = q)
appears formally as the natural inverse to (3.7). Since we had not expressly written the inversion down before in this form. let us state 3.1. Given Q = Q, = D2 + ((2m + l)/x)D
THEOREM
one can write for -f
s’V,f)(fi)= i(Q)(&)=
fi
and Bo: D2 -+ Q,
< m < n- f
Ym+
fi
f(m+
1) D;iYn-,-l/2
I/Z
*.f -& I3 (3.11) *s”s(di)l-
As a guide now to interpret the general extended GL equation (2.15) we continue with these calculations. Thus referring to (2.12) first we compute @(c, x) = + jam @A) Cos Ax Cos At d/l $ -00 =A! A2m+ ’ (Cos ,l(x - 6) + Cos A(x + t)} dL. 2 !0
Recall
here from
do+(A) = &&I)
d/l
(3.12)
[9] for example that one has the relation with da(A) = R,(A) = c;/12*+ ‘, where c, =
420 1/2”f(m
ROBERT
CARROLL
+ I). We recall also a calculation
in [7 ] which gives for CI# -1.
-2,... .cc 1 1” cos Ayd
Jo
= $(F/q
f.Fll
}
= -+a
+ 1) Sin(ncr/2)
= -r(a
+ 1) Sin(7ca/2)
4.7 IA Ia = -r(a
+ 1) Sin(7ra/2)
(see [ $43, 571 for distribution calculations and notation). Setting /I,,, = 2 \/sr/ r(m + 1) r(--f - m), we obtain then formally (note ]y]-’ - y;’ + y14) ~(~,X)=f~~(~X-~~-*~-2+~X+~~-2m-2} Since (see (6, 71 for some calculations). (x4, p(x)) = (x”, ,9(-x)), we have formally ((x - ry-‘9
P(O) = ((x - r,“,-‘7 d-r))
(3.14) Y, = (l/r(p))
= WRY,
x”,- ’
and
* B/(x),
(3.15)
where g(x) = cp(-x). Thus formally (Ix - tl-Zm-2, p(t)) = Q-h
(3.16)
- 1) YpZm-, * {u, + dl.
On the other hand, setting v(x) = ((x + <)“,-‘, p(r)) we have J(x) = ((r - x>t- ‘3 aI)
(3.17)
= WN q3 * PI.
Similarly, if R = ((x + <)!-I, V(r)), we have R = ((x + <)t- ‘, cp(-r)) so si = Q/3)( YD * fj). Consequently, (Ix +
- 1)[ tj + ri]’
=r(-2m-l)Ypzm-,* since (S * 7)- = s * f Hence (recall r(2z) = 2”-‘T(z) f(z) l-(-z) = -7r/z Sin 7rz) LEMMA
3.2.
[cp+fj]
(3.18)
T(z + j)/\/;r
and
With the stipilations above,for 2m # integer we have * la,+fiqNx)
(~(~,X),~P(r))=P*{Y-2m-,
= r(-;;where a,,,= r(-m)/2*“+‘T(m
1) M-2m-2 * v/(xX
+ 1) = --n Csc m~/22mc’~2(m + 1).
(3.19)
GELFAND-LEVITAN
421
EQUATION
On the other hand, for 2m = integer one can find formulas for the term I: lzm+l Cos AJJd,I in [5], for example, to use in place of (3.13) and we omit this. Now looking at (3.19), we let I$‘(& x) act on P&J, l) in (2.14) in the same manner (i.e., &(JF, r) - p(r)). Since Pc(y, <) is even in c, we have p + 6 - 2~, but it is simpler to use the absolute value and take ,8- a, with
~~&(A 0 @Kx) & = (a,lr(-2m - 1))J’)’ Ix - x (we shall eventually write all such integrals as in (3.20) as j,“-cf. Theorem 3.3 and remarks later). Let us rewrite (2.14) here for convenience as
I ’ Pa(~,4 @Kx) & = &(Y,x) =&Y, x) 0
= ~YY)
Y~(x, Y) = 0~3~4 cos w,
(3.21)
(yc(x, y) = 0 for y > x). We put appropriate variables in (3.1 l), (3.20), and (3.21) and note first that since P&J, 0 = 0 for c > y in fact in (3.21) one has
= (f%,/~(-2m - 1)){151-‘“-’ *&(y,
<)1(x)
(3.22)
(the factor of f arises since &( y, -<) = &,(y, <)). Hence THEOREM 3.3. The extended GL equation for the above situation Q, = & can be expressed as (2m # integer)
($z,/T(-2m
- 1))(~~~-2”-2 * P&G 8 t(x) = q(Y)
1 D2+
r&7 Y). (3.23)
Remark 3.4. Note that in Theorem 3.1 if one writes out Y--m-1/2 * rl”@cJ)(fi~ = fhk)l& th e * action is on the variable q in ~p(fi,~), whereas in (3.23) we work on c in pc(y, r). Thus in particular (3.23) is different than the inversion of Theorem 3.1. Let us write out (3.23) using (3.4) and (3.11). Thus from (3.11) first we have formally f<-“‘~c(&, \/i7) ~-“-“* - (&/r(m + l))(c - v)-~~‘/~/ r(-m - 4) and hence (since ~c(x, y) should be even in x)
Y&5 Y)=
(3.24)
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ROBERTCARROLL
while Pc(y, <) is already in a similar form from (3.4). Hence (3.23) becomes formally (do(y) = yzm + ’ ) x sgn x(-y2 - y2);mm3’2 = ~m(l~1-2”-2 * (y’ - r2)~~‘,2}(-~)y-2”,(3.25) where
fir(m
Frn= -Csc mrcf(-m - +)/22”+ 9-(m+9l-(-2m+ 4).
l)=r(m+
l)/
COROLLARY 3.5. The extended GL equation (3.23) can be written in the form of (3.25) (2m f integer).
Remark 3.6. It is possible to play an interesting game with Fourier transforms of distributions in order to “check” (3.25). Referring to [5] for formulas, we note first (1 # fl, *2,...) ,,7(y2 -x2): = 4 f-(,l + l)p’.‘+‘(y ]sl/2)-~‘-“‘J.‘+ l,2(y Is]) and .7(sgn x(x’ - y’)-k } = ifif(i + 1) sgn s(]s]/2y)).‘-‘!’ J-.I-I,2(yIsI). Thus e.g. under Fourier sgn e~(v~2-y2);m-3i2 + i fi r(-m - f) sgn s(]s]/2y)“+’ transform J,+,(vIsl). Also from [S] .,7]xl-t = -2 Sin(?rJ/2) r(J. + 1) ]s]-.‘-‘; .i7(]x]-‘sgnx} = 2iCos(nA/2) r(1+ 1) ]s]-~‘-’ sgns. Consider now formally .z. .iT(sgnx(x’ - y’)-:} = 2i! o (x2 - y’).k Sin xs dx = I, (3.26) .F( sgn x(x2 - y?)-: } = 2 lz x(x’ - y’)-: Cos xs d,u= J.
-0
Then dI/ds = iJ. Now in general D,.XT=
iY-( x:+ ’ } and DI,Fxf one has Therefore iT(x!+‘} =.X(jx]I+‘}. y,Lz-{lp-2
*(XT), of course, so D,.Fx.i = = D,(x:, emisx)= +(x.1+‘, emisx)= -W(X~+‘}. D,.F(sgn x Ix]‘} = D,.F(x-: -x-l} = W(x.l”} +
Now observe that (cf. (3.25) and set y, = 13,yP2”‘), * (y2 -<‘y-‘/7
= ~ISlm”4’-mJ,(4’Isl)T(-m
- i) 2-“-l
(3.27)
the with -2y, Sin mC(-2m - 1) fiT(m + f) (using above r(-m - f)y’“2” = 2Pm-’ \/?r y-“‘T(-m - f)). Now from [54] we know D, = yD,) (z = ys, Hence that (l/z) D,(z”J,(z)} = z”-‘J”-,(z). For s negative, DS(Islm+’ J,+,(yIsl)= DSlS“+‘J,,,+‘(PN = YS““J,(ys). -Y Islm+’JAY Isl) since D, = -yD, for z = -ys. Consequently, from the above
Z=ifir(-m-f)sgns(l~l/2y)~~’ J= -idI/ds = &2-m-‘y~mT(-m
J,+l(yIsl); - i) lsjm+’ J,,,(y IsI).
(3.28)
GELFAND-LEVITAN
423
EQUATION
This says that J= (3.27) = r,,,F{l~l-‘“-* * (y’ - <*)~-r”), which is (3.25) and confirms (3.23). Let us consider again extended GL equation (3.23). We saw that the integral action in (3.21) for Pa, j?c, etc. could be expressed through convolution in our typical problem where Q = Q, and in fact this can be done for general &. Thus recall m(<, x) = F~'(x - 0 + fi(x + 0, where l&‘(t) = (1/2n) I?, @(A) e’,” dl = (I/X) j-r F&l) Cos At dA (@(A) = ~&&)/2). Now, recalling that rp’(r)= cp(-0, we have jr tt’(x - 0 rp(<)dr = I”!,
fi(x + <) g(t) dt,
I’,” l@tx + <) (P(C)& = i’!!,
@3x - 6) (P(l) d<,
and
Ii? a, I&(-Y+ r) o(c) dr = {?, &(x - r) G(c) d& Thus for ~1even (rp= $), one has
THEOREM 3.7. can be written as
The extended GL equation (2.14) for D* and general 0 (3.30)
REFERENCES I. B. L. J. BRAAKSMA,A singular Cauchy problem and generalized translations, in “Bit. Conf. Diff. Eqs.,” pp. 40-52, Academic Press, New York, 1975. 2. B. L. J. BRAAKSMAAND H. S. V. DESNOO,Generalized translation operators associated with a singular differential operator, in “Ordinary and Partial Differential Equations, Dundee Conf.,” Lecture Notes in Mathematics No. 415, pp. 62-77, Springer-Verlag, Berlin, 1974. 3. D. BRESTERS,On the equation of Euler-Poisson-Darboux, SIAM J. Math. Anal. 4 (1973), 31-41. 4. D. BRESTERS,On a generalized Euler-Poisson-Darboux equation, SIAM J. Marh. Anal. 9 (1978). 924-934. 5. Y. BRY~KOV AND A. PRUDNIKOV,“Integral Transforms of Generalized Functions,” Moscow, 1977. 6. R. CARROLL,“Transmutation and operator differential equations”, Notas de Matematica 67. North-Holland, Amsterdam, 1979. 7. R. CARROLL,Transmutation. Generalized Translation, and Transform Theory, I and II, Osaka Jour. Math., to appear. 8. R. CARROLL.Remarks on the Gelfand-Levitan and Marcenko equations, Applicable Anal. 12 (1981). 153-157. 9. R. CARROLL, The Gelfand-Levitan and MarEenko equations via transmutation, RockJ Mountain J. Math. 12 (1982), 393427. IO. R. CARROLL,Singular Cauchy problems in symmetric spaces. J. Math. Anal. Appl. 56 (I 976), 4 l-54.
424 II. 12. 13. 14. 15. 16. 17. 18. 19. 20.
21. 22. 23. 24.
25. 26. 27. 28. 29. 3 I. 32. 33.
34. 35. 36.
ROBERT CARROLL R. CARROLL, Group theoretic nature of certain recursion relations for singular Cauch) problems. J. Math. Anal. Appl. 63 (1978), 156-167. R. CARROLL. Some inversion theorems of Fourier type. Rw. Roumaine Math. Pures Appl., to appear. R. CARROLL. Elliptic transmutation. I. Proc. Roy. Sot. Edinburgh 91A (1982). 3 15-334. R. CARROLL, Some remarks on singular pseudodifferential operators, Comm. Partial D~$?kentiul Equations 6 (1981). 1407-1427. R. CARROLL, Some remarks on transmutation, Applicable Anal. 9 (1979), 291-294. R. CARROLL. Transmutation and separation of variables, Applicable Anal. 8 (1979), 253-263. R. CARROLL. “Transmutation. Scattering Theory, and Special Functions.” North-Holland. Amsterdam. 1982. R. CARROLL, A survey of some recent results in transmutation. in “Spec. Theory Diff. 0~s.“. North-Holland. 198 I. R. CARROLL AND J. GILBERT. Some remarks on transmutation. scattering theory, and special functions, M&h. Ann. 258 (1981). 39-54. R. CARROLL AND J. GILBERT. Scattering techniques in transmutation and some connection formulas for special functions, Proc. Japan Acad. Ser. A Math. Sci. 57 (1981). 34-37. R. CARROLL AND F. SANTOSA. Scattering techiques for a one-dimensional inverse problem in geophysics, Math. Meth. Appl. Sci. 3 (1981), 145-171. R. CARROLL AND F. SANTOSA, Inverse scattering techniques in geophysics, Applicable Anal. II (1980). 79-81. R. CARROLL AND F. SANTOSA. On complete recovery of geophysical data. Math. Meth. Appl. Sci. 4 (1982). 33-73. R. CARROLL AND F. SANTOSA, R&.olution d’un probleme inverse qui determine complttement les donntes gkophysiques, C. R. Acud. Sci. Paris Sk. A-B 292 (1981), 23-26. R. CARROLL AND R. SHOWALTER, “Singular and Degenerate Cauchy Problems,” Academic Press, New York, 1976. R. CARROLL, On the characterization of transmutations, An. Acad. Basil. Ci&c. 54 (1982). 27 I-280. R. CARROLL, On the Canonical Development of Parseval Formulas for Singular Differential Operators, Rend. Accud. Lincei, to appear. K. CASE, Inverse scattering, orthogonal polynomials, and linear estimation, Advan. in Math., Supp. Stud. 3, 1978, 25-43. K. CASE, Orthogonal polynomials from the viewpoint of scattering theory, J. Math. Phys. 15 (1974), 21662174.K. CASE, K. CASE AND M. KAC. A discrete version of the inverse scattering problem, J. Math. Phys. 14 (1973). 594-603. K. CHADAN AND P. SABATIER, “Inverse Problems in Quantum Scattering Theory,” Springer, New York, 1977. M. CHAO. “Harmonic Analysis of a Second Order Singular Differential Operator Associated with Noncompact Semisimple Rank One Lie Groups,” Thesis, Washington Univ.. St. Louis. MO., 1976. H. CHEBLI. Operateurs de translation gtntralis& et semigroupes de convolution, Lecture Notes in Mathematics No. 404, pp. 35-39, Springer-Verlag, Berlin, 1974. H. CHEBLI. Sur un thitorbme de Paley-Wiener associe g la di?composition spectrale d’un opbrateur de Sturm-Liouville sur (0, co), J. Funct. Anal. 17 (1974), 447-461. H. CHEBLI. Thborkme de Paley-Wiener associb i un optrateur diffkrentiel singulier sur (0. 00). J. Math. Pures Appl. 58 (1979). I-19.
GELFAND-LEVITAN
EQUATION
425
37. E. COPSONAND A. ERD~LYI, On a partial differential equation with two singular lines. Arch. Rational
Mech. Anal. 2 (1958), 76-86.
38. V. DITKIN AND A. PRUDNIKOV,“Formulaire pour le Calcul Optrationnel,” Masson, Paris. 1967. 39. L. FADEEV, The inverse problem of quantum scattering theory, Uspekhi Mat. Nauk 14 (1959) 57-l 19. (Russian Math. Surueys) 40. M. FLENSTED-JENSEN. Paley-Wiener type theorems for a differential operator connected with symmetric spaces, Ark. Mat. IO (1972). 143-162. 41. M. GASYMOV. The expansion in eigenfunctions of a nonselfadjoint second order differential operator with a singularity at zero, in “Trudy Letnej Sk. Spektral. Teorii Operator....” pp. 20-45, Izd. Elm. Baku, U.S.S.R., 1975. 42. M. GASYMOV.On the eigenfunction expansion of a nonselfadjoint boundary problem for a differential equation with a singularity at zero, Dokl. Akad. Nauk. SSSR 165 (1965). 26 l-264. 43. I. GELFANDAND G. SILOV.“Generalized Functions,” Vols. 1-3, Moscow, 1958. 44. J. GERONIMO.A relation between the coefftcients in the recurrence formula and the spectral function for orthogonal polynomials, Trans. Amer. Math. Sm. 260 (1980). 65-82. 45. J. GERONIMOAND K. CASE, Scattering theory and polynomials orthogonal on the unit circle, J. Math. Phys. 201 (1979), 299-310. 46. J. GERONIMOAND K. CASE, Scattering theory and polynomials orthogonal on the real line, Trans. Amer. Math. Sot. 258 (1980), 467494. 47. V. KATRAKHOV,Transmutation operators and pseudodifferential operators, Sibirsk. Mat. i. 21 (1980), 86-97. 48. V. KATRAKHOV,Isometric transmutation operators and the spectral function for a class of one dimensional pseudodifferential operators, Dokl. Akad. Nauk SSSR 251 (1980), 567-570. 49. I. KIPRIYANOVAND V. KATRAKHOV,On a class of one dimensional singular pseudodifferential operators, Mat. Sb. 104 (1977), 49-68. 50. T. KOORNWINDER,A new proof of a Paley-Wiener type theorem for the Jacobi transform, Ark. Mat. 13 (1975), 145-159. 51. B. LEVITAN. “The Theory of Generalized Translation Operators,” Izd. Nauka, Moscow, 1973. 52. B. LEVITAN. The expansion in Bessel functions for Fourier series and integrals, Uspekhi Mat. Nauk 6 (1951), 102-143. 53. J. LIONS,Operateurs de Delsarte et probltmes mixtes, Bull. Sot. Math. France 84 (1956), 9-95. 54. W. MAGNUS,F. OBERHEIXNGER,AND R. SONI,“Formulas and Theorems for the Special Functions of Mathematical Physics,” Springer, New York, 1966. 55. V. MAREENKO,“Sturm-Liouville Operators and Their Applications,” Izd. Nauk. Dumka, Kiev, U.S.S.R., 1977. 56. L. ROBIN. “Fonctions sphtriques de Legendre et fonctions spheroidales,” Vols. l-3, Gauthier-Villars, Paris, 1957-1959. 57. L. SCHWARTZ.“Thtorie des distributions,” Hermann (Papillon). Paris, 1966. 58. J. SIERSMA,“On a Class of Singular Cauchy Problems,” Thesis, Univ. Groningen, The Netherlands, 1979. 59. I. SPRINKHUIZEN-KUYPER,A fractional integral operator corresponding to negative powers of a certain second order differential operator, J. Math. Anal. Appl. 72 (1979). 674-702. 60. V. STASEVSKAYA, The inverse problem of spectral analysis for a differential operator with a singularity at zero, Wen. Zap. Kharkov. Mat. ObE. 25 (1957). 49-86.
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CARROLL
6 I. W. SYMES. Inverse boundary value problems and a theorem of Gelfand-Levitan, J. Marh. Anal. Appl. 71 (1979), 379402. 62. K. TRIM~CHE. Transformation integrale de Weyl et theoreme de Paley-Wiener associes a un operateur differentiel singulier sur (0. co), J. Marh. Put-es Appl. 60 (1981). 5 l-98. 63. V. VOLK. On inversion formulas for differential operators with a singularity at x = 0, Uspekhi blat. Nuuk. 8 (1953). 141-151. 64. J. WALKER. “Conjugate Hankel Transforms and H’ Theory,” Thesis. Univ. Texas, Arlington, 1980.
OF MATHEMATICAL
Some Remarks
ANALYSIS
AND
APPLICATIONS
92, 410-426
on the Generalized Equation
(1983)
Gelfand-Levitan
ROBERT CARROLL University
of Illinois at Champaign-Urbana,
Urbana, Illinois
61801
Submitted by V. Lakshtnikantham
1. INTR~OUCTION
The Gelfand-Levitan (GL) and MarEenko (M) equations arise in quantum scattering theory as an important part of the machinery used to recover the potential q(x) from a Sturm-Liouville operator D2 -q(x) via inverse methods (cf. [32, 39, 551). Modifications of this procedure for handling inverse problems also arise in other applied fields (see, e.g., [21-24,32,61]) and lead in particular to various types of GL equations. In all of this work the transmutation concept plays an important role (although often disguised). Now in earlier papers we developed an extensive theory and framework for transmuting an operator of the form J% = (dPu’)‘/dP + pi (pP= f lim A;/AP as x -+ co) into another operator of the same type (cf. [6-9, 12-16, 18-20, 231). In particular this theory applies to situations where P = P -pi is modeled on the radial Laplace-Beltrami operator in a rank one noncompact Riemannian symmetric space and the spherical functions arising from equations &: = -I’p<, (Q<(O)= 1, D,pl;(O) = 0 involve various special functions of interest in diverse contexts. In [8,9] we derived a general GL and M equation for transmutations D* + p (p as above). Given the fundamental role of such equations in the inverse scattering and inverse Sturm-Liouville problems and the nature of their derivation in [8,9] from eigenfunction kernel expressions,it is clear that the version of [8,9] contains valuable information about special functions (cf. also [ 28-3 1, 44-461, where some relations between discrete forms of the GL and M equations of quantum scattering theory and orthogonal polynomials have been studied-although some version of this approach can be envisioned for singular operators of the type P above, we shall not pursue this here). In this paper we shall first extend (in Section 2) the GL equation of [8, 91 to the context of transmutations $-+ $ and clarify its structure. In particular we shall represent it as a formula in spherical functions. Then (in Section 3) we take a model operator P, = D2 + ((2m + l)/x)D (A, =x2”‘+‘) and give 410 0022-247x/83
$3.00
Copyright Q 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.
GELFAND-LEVITAN
EQUATION
411
an explicit formula for the extended GL equation (for D* and P, = $) by evaluating various kernels in terms of pseudofunctions. This will serve as a guide to interpret such equations and seemsto be the first such result in this direction. The operator P, is particularly appropriate as a model here since the special functions which arise are Bessel functions about which information is readily available and also because of its importance in physics where angular momentum is represented by I= m - + in the transformed operator pm = Dz - (m’ - :)/x2 = D* - 1(1+ 1)/x* (cf. [ 14, 6, 7, 25, 32, 36, 37, 39, 41, 42, 47-49, 52, 53, 58-60, 62, 631). Our techniques for P, = Q can then be extended virtually automatically to other operators P as above by using the general transmutation machinery of [6-9, 12-16, 1g-201; this will be done in [ 171.
2. THE GL EQUATION
In [8, 91 we derived a GL equation (and an M equation) in a general context (for transmutations 0’ + Q) ” by a unified method based on transmutation operators. For the special case of quantum scattering theory this method goes back to Fadeev [39] but our context was considerably more general and both technical and conceptual changes were used in the proofs. Our framework is intimately related to various special functions as they arise in the form of spherical functions in harmonic analysis on rank one noncompact Riemannian symmetric spaces and thus the GL and M equations automatically have something to say about special functions. In the situation treated here (which extends [8, 91) special functions are displayed as the ingredients for a general extended GL equation (for transmutations P -+ &) and this equation becomes then in fact a formula in spherical functions. Let us refer to [6-9, 12-16, 18-241 for some of the recent development of transmutation ideas and to [32, 39,51,53,55] for background information. For notation we recall briefly that we are concerned with operators P and Q of the form P(D)u = (Ap u’)‘/Ap,
P(D)=P(D)+p;,
pP = + lim AL/d,. x-cc
(2.1)
For the interaction with special functions we take A, so that P(D) is modeled on the radial Laplace-Beltrami operator in a rank one noncompact Riemannian symmetric space (cf. [33-36,40, 50,62,64]-examples are indicated below). One considers eigenvalue equations p(D)u = -1’~ and isolates in particular two types of solutions. First, solutions o;(x) of this equation satisfying o:(O) = 1 and D,p~(O) = 0 are generalized spherical functions, and we call Jost solutions those @c,\(x) w A;“*(x) exp(f Ux) or -exp(fiA -p)x as x+ co (p =pP). One writes (p:(x) = c,(A) @f;(x) + 409’92,2-8
412
ROBERTCARROLL
c,(-1) @c,\(x) and the spectral measure arising for the cpf; eigenfunction theory is dvp(l) = dL/2n Ic,(~))~.
2.1. To
illustrate the situation, consider A, = Ap4 = from 1401, which with A, = AckJ= (,x-,-x)2fktl(,x +e-")2Dtl from [SO] will provide a nice background of specific examples and information. Then pP = f( p + 2q) = p and one has the following facts: For x E (0, co), q:(x) is entire in ,l of exponential type x while @{(x) is analytic for Im 1 > 0, extends to be analytic in C/(-N}, and satisfies Q;(x) - exp(U - p)x as x + 03 (in particular for c > 0, E > 0, q > --E It/, and x E [c. co), Q:(x) = e”.t-“‘“{ 1 + e-‘“O(A, x)} with lD:Ol,< K,). Further, for q > --E l
(&
_
e-.r)P(e2x
_
e~2.r)4
c,(l) = (22bT(~)r(ik))/(f(c
- b) r(a)),
(2.2)
where a = f(~ + U), b = j(p - U), and c = i(p + q + I). EXAMPLE 2.2.
Take A, = A,, as indicated in Example 2.1. Then
q;(x) = F(f(p + U), f(p - U), a + 1, -sh2x)
(2.3)
(p = a +/I + 1) is a Jacobi function of the first kind while for 13+ -i, -2i,..., the Jost solution is (note A,, F Ap4)
@f;(x) = (ex - e-x)i.k-P X
(2.4)
F(f(j? - a + 1 - ii), f(p + a + 1 -U),
1 - iA, -sK2x),
which is a Jacobi function of the second kind. A special case of particular A, zA~,-,,~ = (ex -e-x)2mt’ = 22m+‘.yh2m+‘~. importance is Then AL/A, = (2m + 1) coth x and we are working with canonical equations associated with the radial Laplacian in SL(2, R)/S0(2) (cf. [ 10, 111). In this case the spherical functions rpi = v[; can be written as
#J(x) = 2”r(m + 1) SK”x&Y
,12(chx),
(2.5)
where P,i?! ,,* denotes the associated Legendre function of the first kind (cf. [56]) and CD:= @T is @T(x) = (2-“r(l
- U)/fiJ’(p
- U)) e-i”“sh-“xQ”,i2-i,~(chx)
(2.6)
(p = m + f), where Q?!,,2-i,, denotes the associated Legendre function of the second kind. Further
c,(A) = c,(i) = (22mT(m+ 1) r(iA))/r(p
+ U).
(2.7)
GELFAND-LEVITAN
413
EQUATION
Let us also record a form of the function @y(x)/c,(-A) which turns out to be the natural analogue of the exponential function (up to f); this can be written
where e(z) = e-‘U~Q~(z)/Z(~ + u + 1) is entire in ,Uand v (cf. [56]). Let us recall now some basic constructions from transmutation theory. Thus we consider transmutation operators B: p-+ Q, which means that &B = BP’, acting on suitable objects. When P = p = Dz with ylj;(x) = Cos Ix, @2(x) = e’,‘“, d,,(x) = 1, c,(A) = i, and dv = 2dl/n, we write BQ: D* + 0; it is this case which was treated in [8, 91. Here we generalize further the technique of [8,9] to produce the most general type of extended GL equation for transmutations B: B--P& with P and Q of the type Pu = Thus recall from [8, 9, 18, 191 that B = 2?$: p + Q with (4 u’)‘/A,. 9: = Bcp: and B= ClrQ with @(,I) & = Z&Y;, where I@) = l/W(A) = Here ‘WV) = (@(x), f(x)) (Ql = &PI;), W’(x) = IcP(4/c&)l’. (W), cpl;W,(v = VP)*Qf(~) = wi(x>~ f(x)), wx> = (Jv)7 &xx)), (w = ws - vc), and A?F(x) = (F(A), p!(x)), . One sets also 9 = Z?= .$?-’ with 3 = gE2 and then 93Wx= Z, where W” = QW(A)Q. Similarly Z?= B-’ = .$ with l?BF? = Z, where @ =$@(A)!@, and we recall also that dv,, = W(l) dw, with do, = I?‘(A) dv,. The general GL equation then is Bk@ = B’ when written out in terms of kernels and we recall (cf. [8,9, 19,201) /WY x) = Qx(x>~ P!%Y)L = AP(X) A,‘(YXdW9 = AAx) A,‘(Y)
QKYD,.
I+, 4’1,
(2.9)
where /3= ker B, y’= ker .$, and 7(x, y) = 0 for x > y. Similarly, Y(X,Y) = (d(x), Q%9), = A,(Y) 4*(x)&~,
= A,(Y) A,‘W?&)~
P?(Y)), (2. IO)
x>,
where y = ker 9, p = ker Z?,and y(x, y) = 0 for y > x. We recall also that the generalized translation rc - P has the form W(x)
= od(49J(4
rp%-)),.~
m
= W)~ Jmw
= w-(n).
(2.11)
We write also &(A) = (Q(r), @(c)) = ‘J3G(A) (i.e., e(r) = 9\ l?(r)) and note here that W differs by a factor of two from the W used in [8,9] under Fourier transformation. Now
414
ROBERTCARROLL
Then from (2.11) and @ = q &‘, @(JY,x) can also be written @CY,xl = 44~) C @x> = 44xKd(~) = 4(xKM(Y)
d(xX (ri/(Ov Qp,K>>>, (2.13) PXXh JTx3>,. 7 W)>7
which displays the kernel /?(x, y, <) = (o:(y) C&(X),~~(~)),. of Tq (i.e., Ti: %d = (P(x, .Y,0, *KM N ow write out BWK = B in terms of kernels to get the extended GL equation in the form
!p/KY,63m9 x>& =B(.hx).
(2.14)
Again recall p(y, 0 = 0 for < > y and &y, x) = 0 for y > x, so J’F = !‘i in (2.14) and the right side is zero for y > x. We shall display this as a formula in spherical functions in THEOREM
2.3. The general extended GL equation is
&w-e T; *CC>= r”,*(x)
= (~~~03,d(x)),
.
Formula (2.15) suggests various consequences.For example, since dv = IV(n) du (W = mP ‘) we have formally
from which one concludes that
d(x) =WA) jam Q:(t) T;*tt> 45 Since dP(x) T; p(r) = @(a x), this is equivalent to
(2.17)
GELFAND-LEVITAN
EQUATION
415
and is easily confirmed by other means. Indeed (recall dv = C(z) dz),
and multiplying by W(A) = @-‘(A) we get (2.18). Conversely, from (2.18) or equivalently (2.17) one obtains (2.16) or (2.15). Similarly from (2.15) we write now (do = @dv)
from which follows (2.21)
Of course, this is just the expression &(JT) = 1393rpy(y) (where y is a parameter) or q!(y) = Ji #‘(0, CO:(Y)), &(
The above constructions were developed around P and Q of the form Pu = where the inversion theory is founded upon spectral measures (4u’)‘/& dv, = dA/2n ]cp(A)12. More generally, one wants to deal with operators p = P +pi - q(x), where q(x) is a suitable (complex-valued) potential (cf. [35, 36,41,42,55,62]). In this event the inversion theory must be expressed more generally through distributions in the spectral variables and one achieves this nicely with the generalized spectral function (distribution) of Marcenko (cf. [6, 7, 17, 18, 26, 27, 41, 42, 551). We do not give details here (see [ 17, 26, 271 for a complete exposition) but remark that general Parseval
416
ROBERTCARROLL
formulas and expansion formulas for such situations can be proved and take the form (cf. (6, 7, 17. 18, 26, 27, 41, 42, 551)
(RQ, QfDg),l = (A;zf, Al;“gj,
f(Y) = (RQ, WV) W:(Y))., (2.22)
(note that we express (F, G), as (RQ, FG),k). Here RQ is the generalized spectral function, RQ E Z’, where the product &lflg E Z, Z being a certain space of entire functions of exponential type for f, g suitable functions of compact support. Further, one can define B’ more generally by adjointness (cf. [ 14, 26,481 in the form g= .Y (3 = B-‘), where the adjoint # is determined by @p(X) U(X), 3V(X>) = (U(X), AP(X)W
Y) A,‘(Y)AQ(Y~
V(Y)))
= @Q(Y) V(Y), (l’(x~ Y)Ap(x) A, ‘(4’h U(x)))
(2.23)
= (A,(Y) V(Y), ANY)). This determines B with kernel as in (2. IO) (where p^= p)
IT(vvxl = 44~) A; ‘(~9~6, Y).
(2.24)
In [26] we also show how B’ is determined via a partial differential equation, but this is not needed here. Now extending the derivation of GL equations as in [39], we write following [ 261
‘Pm = 04x309Pm;
d(v) = MY9f), 4(t)).
(2.25)
Then compose these equations with RQ~f;(y) and RQ&(x) respectively to obtain (using the relation (RQ, Q:(x) &(t)) = 6(x - t))
(RQvCP:(Y)&N
(2.26)
= Wx, 0, (RQ, CP%Y)d(O)>,
(RQvd(Y) &x)> = (Yb’, t), (RQ9V?(X) d(t))) Interchanging x and y in (2.26)-(2.27) then (using (2.24)) Y(X, Y) A,‘W4W
=h
= YtYvx)/d~tx).
(2.27)
and equating these equations we have
x) = (Ptrv t), (Ray @tx) v:(O)).
(2.28)
Consequently, defining @tt, xl = CR’, vf;tt> f-X(x)>,
(2.29)
THEOREM 2.5. In the more general situation where the inversion theory for P and & is given via generalized spectral functions R ’ and RQ, one can write the general extended GL equation again as B’ = B W”, where g is
GELFAND-LEVITAN
EQUATION
417
defined as A?# with ker i? = B given by (2.24) and W” is the operator with kernel @t, x) given by (2.29). Thus as in (2.15) p’
Remark 2.6. As in Theorem 2.4 one can produce formally a formula (2.21), but (2.17) does not make any sensenow.
3. EXPLICIT FORMULAS The model situation which we shall investigate now involves the operator Q(D)u = Q,(D)u = u” + ((2m + 1)/x) U’
(3.1)
which corresponds to d,(x) = x*~+’ and p = 0. We shall give an explicit formula for the general extended GL equation of Theorem 2.3 for the case p = D* and & = Q, in providing pseudofunction evaluations for the various kernels (see Theorem 3.3 and Corollary 3.5). The operator Q, is related to the operator Q, = D* - (m’ - 4)/x’ which occurs in quantum physics, where I = m - 4 represents angular momentum (the hookup with physics occurs for m > 4); references are indicated in Section 1. In fact there is a transmutation from Q, to Q, effected by noting that xm+ “*em u = QmIXmt lDu) cXm+l/* =Al/* a (x)), and we refer again to [6, 7,58,60]. Let us write now q?(x) = Bm(x, A) = zmr(rn + 1)(1x)-” J,(lx), l2,(x, A) = 2-*“T(m
+ l)(~~)*~+’ &(x)
(3.2)
There is a slight departure here from what is our normal notation, namely,
Q:(x) = A&) &4,
P(J)= (f(x), J-=%(x)) = wv)~
f(x) = elf(x) = (.m, v:(x)),,
= JornA4 d?(x) d%W
(3.3)
Before we settled on this notation, however, considerable material was published with the notation (3.2) for the particular operator Q = Q,. It is shown in [7,9] how everything fits together and we recall this here briefly. Thus one has dwQ(l) = dil/2n 1c&)l’ = &,(A) dA and setting R,(A) = c$ 2mf’ with c, = 1/2mr(m + l), it turns out that Q,(1) = RO()L)(R,(l) is a generalized spectral function in the senseof MarEenko [55]). Thus in (3.2) we have Q,(x, A) = R,(k) A&) &(x) = R,(A) .nf(x>. so Q0.f = (f(x), n,(x, A)) = R,(A) Zlf and f(x) = I (aof) o:(x) dA from (3.3). In any
418
ROBERT
CARROLL
event we want to examine the kernel expressions which occur in transmuting Dz into Q, (note here a reversal of y and p from some previous work where we employed B: Q, + D’). Thus consider BQ: D2 + Q, = & so that
=z
-% ^m
7[ o R (~‘,~)Cos~xd~=2R”(y,x) !
2zym + 1)4’-2m (4,2 -,2)/yI..2 = f(+)f(m++)
(3.4)
(cf. [25, 381). Thus &(r, x ) is in fact a function (but certainly not of the form 6(y - x) + a function), and setting k, = 2Qm + l)/fir(m + f) we have (x2 = r, y2 = q) (&(y,x),f(x))=k,y-““1;
(~~~-x~)“‘-“~f(x)dx
=~J~~-.(n-T)“-‘~2f(~)~.
(3.5)
Now recall the pseudofunctions (cf. [43, 571) Yn = (l/r(p)) Y_, = 6’“). Thus for a function g defined on [0, co) Yfn+K!*g=
’ r(m+f)-o
1’ g(O(rl - W-“‘2
PfxO-’
with
dt,
so that (3.5) can be expressed as
(PQ(YJ)lf(X))=
mm + 1)/h>
* cfmlm
viy*+l..2
On the other hand, y&x, y) will be quite different anticipated from formula (3.7)). Thus (cf. (2.10)) Y&
Y)
=
(A,(Y)
dib9,
Cm
Ax>,
=
A,(Y)
j,
(O:(Y)
(3.7)
(as can be already
Cos
lxRo@)
dl
“03 = 10
Q,( y, A) Cos Ax dA
=2-mT(m
+ l)-’
= 2-“I-@2 + I)-’
*cc(~~~)m+lJ,(~~)Cos~xd~ J0 ym+‘~21H,(P+“2
&SAX},
(3.8)
GELFAND-LEVITAN
419
EQUATION
where IH, denotes the Hankel transform. This formula is studied extensively in [6, 7, 161,for example, and we simply cite results here as needed.One can show that ya will be a distribution of order n, where -f < m < n - i (cf. [6, 7, 16, 531). A nicer result can be obtained by recalling a formula expressing Cos Ix in terms of km@, y) from EPD theory (cf. [25]-recall Cos 1x = c%?Q~4). Thus for -4 < m < n - 4
(Y&Y Y), g(v)) = 30 g(x) = YLX (+ Dxy jr g(y)y’““(x’
- yym-3/2
dy, (3.9)
where y”, = &2”-‘T(m + 1) r(n - m - 4). This can be rewritten (taking n > m + 5 if desired and setting x2 = r, y2 = q)
appears formally as the natural inverse to (3.7). Since we had not expressly written the inversion down before in this form. let us state 3.1. Given Q = Q, = D2 + ((2m + l)/x)D
THEOREM
one can write for -f
s’V,f)(fi)= i(Q)(&)=
fi
and Bo: D2 -+ Q,
< m < n- f
Ym+
fi
f(m+
1) D;iYn-,-l/2
I/Z
*.f
As a guide now to interpret the general extended GL equation (2.15) we continue with these calculations. Thus referring to (2.12) first we compute @(c, x) = + jam @A) Cos Ax Cos At d/l $ -00 =A! A2m+ ’ (Cos ,l(x - 6) + Cos A(x + t)} dL. 2 !0
Recall
here from
do+(A) = &&I)
d/l
(3.12)
[9] for example that one has the relation with da(A) = R,(A) = c;/12*+ ‘, where c, =
420 1/2”f(m
ROBERT
CARROLL
+ I). We recall also a calculation
in [7 ] which gives for CI# -1.
-2,... .cc 1 1” cos Ayd
Jo
= $(F/q
f.Fll
}
= -+a
+ 1) Sin(ncr/2)
= -r(a
+ 1) Sin(7ca/2)
4.7 IA Ia = -r(a
+ 1) Sin(7ra/2)
(see [ $43, 571 for distribution calculations and notation). Setting /I,,, = 2 \/sr/ r(m + 1) r(--f - m), we obtain then formally (note ]y]-’ - y;’ + y14) ~(~,X)=f~~(~X-~~-*~-2+~X+~~-2m-2} Since (see (6, 71 for some calculations). (x4, p(x)) = (x”, ,9(-x)), we have formally ((x - ry-‘9
P(O) = ((x - r,“,-‘7 d-r))
(3.14) Y, = (l/r(p))
= WRY,
x”,- ’
and
* B/(x),
(3.15)
where g(x) = cp(-x). Thus formally (Ix - tl-Zm-2, p(t)) = Q-h
(3.16)
- 1) YpZm-, * {u, + dl.
On the other hand, setting v(x) = ((x + <)“,-‘, p(r)) we have J(x) = ((r - x>t- ‘3 aI)
(3.17)
= WN q3 * PI.
Similarly, if R = ((x + <)!-I, V(r)), we have R = ((x + <)t- ‘, cp(-r)) so si = Q/3)( YD * fj). Consequently, (Ix +
- 1)[ tj + ri]’
=r(-2m-l)Ypzm-,* since (S * 7)- = s * f Hence (recall r(2z) = 2”-‘T(z) f(z) l-(-z) = -7r/z Sin 7rz) LEMMA
3.2.
[cp+fj]
(3.18)
T(z + j)/\/;r
and
With the stipilations above,for 2m # integer we have * la,+fiqNx)
(~(~,X),~P(r))=P*{Y-2m-,
= r(-;;where a,,,= r(-m)/2*“+‘T(m
1) M-2m-2 * v/(xX
+ 1) = --n Csc m~/22mc’~2(m + 1).
(3.19)
GELFAND-LEVITAN
421
EQUATION
On the other hand, for 2m = integer one can find formulas for the term I: lzm+l Cos AJJd,I in [5], for example, to use in place of (3.13) and we omit this. Now looking at (3.19), we let I$‘(& x) act on P&J, l) in (2.14) in the same manner (i.e., &(JF, r) - p(r)). Since Pc(y, <) is even in c, we have p + 6 - 2~, but it is simpler to use the absolute value and take ,8- a, with
~~&(A 0 @Kx) & = (a,lr(-2m - 1))J’)’ Ix -
I ’ Pa(~,4 @Kx) & = &(Y,x) =&Y, x) 0
= ~YY)
Y~(x, Y) = 0~3~4 cos w,
(3.21)
(yc(x, y) = 0 for y > x). We put appropriate variables in (3.1 l), (3.20), and (3.21) and note first that since P&J, 0 = 0 for c > y in fact in (3.21) one has
= (f%,/~(-2m - 1)){151-‘“-’ *&(y,
<)1(x)
(3.22)
(the factor of f arises since &( y, -<) = &,(y, <)). Hence THEOREM 3.3. The extended GL equation for the above situation Q, = & can be expressed as (2m # integer)
($z,/T(-2m
- 1))(~~~-2”-2 * P&G 8 t(x) = q(Y)
1 D2+
r&7 Y). (3.23)
Remark 3.4. Note that in Theorem 3.1 if one writes out Y--m-1/2 * rl”@cJ)(fi~ = fhk)l& th e * action is on the variable q in ~p(fi,~), whereas in (3.23) we work on c in pc(y, r). Thus in particular (3.23) is different than the inversion of Theorem 3.1. Let us write out (3.23) using (3.4) and (3.11). Thus from (3.11) first we have formally f<-“‘~c(&, \/i7) ~-“-“* - (&/r(m + l))(c - v)-~~‘/~/ r(-m - 4) and hence (since ~c(x, y) should be even in x)
Y&5 Y)=
(3.24)
422
ROBERTCARROLL
while Pc(y, <) is already in a similar form from (3.4). Hence (3.23) becomes formally (do(y) = yzm + ’ ) x sgn x(-y2 - y2);mm3’2 = ~m(l~1-2”-2 * (y’ - r2)~~‘,2}(-~)y-2”,(3.25) where
fir(m
Frn= -Csc mrcf(-m - +)/22”+ 9-(m+9l-(-2m+ 4).
l)=r(m+
l)/
COROLLARY 3.5. The extended GL equation (3.23) can be written in the form of (3.25) (2m f integer).
Remark 3.6. It is possible to play an interesting game with Fourier transforms of distributions in order to “check” (3.25). Referring to [5] for formulas, we note first (1 # fl, *2,...) ,,7(y2 -x2): = 4 f-(,l + l)p’.‘+‘(y ]sl/2)-~‘-“‘J.‘+ l,2(y Is]) and .7(sgn x(x’ - y’)-k } = ifif(i + 1) sgn s(]s]/2y)).‘-‘!’ J-.I-I,2(yIsI). Thus e.g. under Fourier sgn e~(v~2-y2);m-3i2 + i fi r(-m - f) sgn s(]s]/2y)“+’ transform J,+,(vIsl). Also from [S] .,7]xl-t = -2 Sin(?rJ/2) r(J. + 1) ]s]-.‘-‘; .i7(]x]-‘sgnx} = 2iCos(nA/2) r(1+ 1) ]s]-~‘-’ sgns. Consider now formally .z. .iT(sgnx(x’ - y’)-:} = 2i! o (x2 - y’).k Sin xs dx = I, (3.26) .F( sgn x(x2 - y?)-: } = 2 lz x(x’ - y’)-: Cos xs d,u= J.
-0
Then dI/ds = iJ. Now in general D,.XT=
iY-( x:+ ’ } and DI,Fxf one has Therefore iT(x!+‘} =.X(jx]I+‘}. y,Lz-{lp-2
*(XT), of course, so D,.Fx.i = = D,(x:, emisx)= +(x.1+‘, emisx)= -W(X~+‘}. D,.F(sgn x Ix]‘} = D,.F(x-: -x-l} = W(x.l”} +
Now observe that (cf. (3.25) and set y, = 13,yP2”‘), * (y2 -<‘y-‘/7
= ~ISlm”4’-mJ,(4’Isl)T(-m
- i) 2-“-l
(3.27)
the with -2y, Sin mC(-2m - 1) fiT(m + f) (using above r(-m - f)y’“2” = 2Pm-’ \/?r y-“‘T(-m - f)). Now from [54] we know D, = yD,) (z = ys, Hence that (l/z) D,(z”J,(z)} = z”-‘J”-,(z). For s negative, DS(Islm+’ J,+,(yIsl)= DSlS“+‘J,,,+‘(PN = YS““J,(ys). -Y Islm+’JAY Isl) since D, = -yD, for z = -ys. Consequently, from the above
Z=ifir(-m-f)sgns(l~l/2y)~~’ J= -idI/ds = &2-m-‘y~mT(-m
J,+l(yIsl); - i) lsjm+’ J,,,(y IsI).
(3.28)
GELFAND-LEVITAN
423
EQUATION
This says that J= (3.27) = r,,,F{l~l-‘“-* * (y’ - <*)~-r”), which is (3.25) and confirms (3.23). Let us consider again extended GL equation (3.23). We saw that the integral action in (3.21) for Pa, j?c, etc. could be expressed through convolution in our typical problem where Q = Q, and in fact this can be done for general &. Thus recall m(<, x) = F~'(x - 0 + fi(x + 0, where l&‘(t) = (1/2n) I?, @(A) e’,” dl = (I/X) j-r F&l) Cos At dA (@(A) = ~&&)/2). Now, recalling that rp’(r)= cp(-0, we have jr tt’(x - 0 rp(<)dr = I”!,
fi(x + <) g(t) dt,
I’,” l@tx + <) (P(C)& = i’!!,
@3x - 6) (P(l) d<,
and
Ii? a, I&(-Y+ r) o(c) dr = {?, &(x - r) G(c) d& Thus for ~1even (rp= $), one has
THEOREM 3.7. can be written as
The extended GL equation (2.14) for D* and general 0 (3.30)
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