IL NUOVO CIMENTO VOL. X X X I V , N. 6 16 Dicembre 1964

A Convolution Integral for the Resolvent of the Sum of Two Commuting Operators.

L. BIANCHI and L. FAVELLA

Istituto di Fisica dell' Universith - Torino Istituto Nazionale eli Fisica Nucleate - Sezio~,e di Torino

(ricevuto il 6 0 t t o b r e 1964)

In dealing with many problems in scattering theory (me is often faced with the determination of the resolvent of the sum of two independent operators, their own resolvents being known.

The general case is very difficult to manage. If, however, we limit ourselves to the case in which the operators are bounded and commute, it is rather easy to show that a compact integral representation can be given for the complete resolvent in terms of the partial resolvents (1). The usefulness of this representation stems from the fact that it is often possible to evaluate it approximately and, above all, that the analytic properties of the total operator are most easily recognizable under this form, owing to the flexibility of the representation itself.

Let A 1, A 2 be two bounded commuting operators, and R(2; At), R(2; A 2) their resolvents, defined by R(~; A i ) = ( 2 - - A ~ ) - L We will prove the following

The vren~:

(1) R ( ~ ; A I + A 2 ) = ( ~ _ _ A I _ _ A ~ ) _ I = 1 ; R '2i~ __( ~' ; A~) R( ) ' - - ~' ; A2) d).'

o

where the path of integration C, in the complex ,U-plane, encircles in a counterclockwise direction the whole spectrum (discrete+continuum) of R(,~'; A1), or that of R(~--2~; A 2) in a clockwise direction.

spectrum of R (~-~f;A 2) . . . .

complex ~r--plane C f I �9

spectrum of R (~r;A~) , , , I t

) , Fig. 1. - The path of integration C.

(1) This theorem has been proved for a particular ease by M. VERDE: Nuovo Cimento, 28,547 (1963).

1826 L. BIANCHI and L. FAVELLA

In order to pro~ge the theorem let us consider the operators T~(z)=exp [iAkz], defined by their power series expansions. For real z t h e y give rise to two families of bounded operators, which form also two strongly cont inuous semigroups (2). Then , for Re (2) sufficiently large, we can write

R(2; A,i) = / e x p [-- 2z] Tk(z) dz = (2 - - iAk) -1 (2) (k = ] , 2)

0

and similarly c o

/ t

R(2; i(Ax + A2) ) = ] e x p [-- 2z]T(z) dz = ( 2 - - i(A~ + A2))-1 , (2')

0

where T(z) = e x p [i(A 1 + A2) ]. On the other hand all the resolvents R(2; iA~), R ( 2 ; i ( A I + A 2 ) ) , as funct ions

of 2 are analy t ic everywhere, except on their spectra: hence the integrals (2) can be ana ly t ica l ly cont inued into the whole complex 2-plane, spectra excluded (s). Now, the relat ions between the R 's and the T's are reversible (a), so t ha t by means of the same techniques used for ord inary Laplace transforms, we obta in

(3) R ( 2 ; A I + A 2 ) = ~ ( 2 ' ; A x ) R ( 2 - - 2 ' ; A 2 ) d 2 ' ,

where the in tegra t ion pa th C' is i l lustrated in the figure. In any case C' can be deformed to encircle ei ther spect rum (C" or C"), since R(2 ' ; Ak), as 2'--* c~ (except for possible s ingular directions) is 0(2 '-1) (a), and this proves the representa t ion (1).

--Ctt ,~

C'

I m (.,l. f ) . . . . .

spect rum of R (~ -Z ' ;A 2) (

( Re(). r)

spectrum of R (;tt;A 1)

Fig. 2. - The integration paths C', C', C".

We wan t to i l lustrate this theorem by considering the Green 's funct ion for the =(A~+A~) . We know tha t Laplace operator in six dimensions, A ~ 2

G ( 2 ; x , x ' ) = exp[ i~/X l x - x'p ]

(2) DUNFORD-SCHWARTZ: Linear Operators, part I (New York, 1958), p. 614. (a) DU~ORD-SCHWARTZ: Linear Operators, part I (New York, 1958), p. 566. (~) M. A. NAIMARK: Normed Rings (Amsterdam, 1959), p. 117.

A CONVOLUTION I N T E G R A L F O R THE R E S O L V E N T ETC. 1827

Then, according to the previous s ta tement , we have for Im ( 2 ) # 0

G(2; x, x , y , y ) = 2 'G(2 ' ;

o

Since, for Im ( 2 ) # 0 ,

x, x ' ) G ( 2 - - 2 ' ; y , y ' ) =

=~-~fd ,~ 'G( ,V;y ,y ' )G( ,~-- 0

2' ; x, x ' ) = ( , t - - A) -~ .

1 f d 2 ' G ( 2 ' ; x , x ' ) G ( 2 - - 2 , y . y ) = - - i ~ . !

f f

( o + )

= 2i~(x- -x ' ) (y - -y ' ) l fdx' exp [ i V 2 z ix - - x ' l + i V a - - 2' lY - - Y ' I ] =

coTie

1 fo = 16izta(x - x , ) ( y _ y , ) d a s i n a l x - - x ' [ exp[ i%/v2 - -a2 l y - - y ' l ] ,

O

where we have pu t 2 ' = a 2, and 2 = v 2, we can eva lua te the integral (put t ing for con- venience v = e x p [-- izt/2]~) (5) and obtain

G ( 2 ; x , x ' ; y , y ' ) = 16zta[(x - - x ' ) z -k (y _ y , ) 2 ]

H~[ V-~ { (x - -x )" § ( y - - y ' ) ~ } ]

which is jus t the formula g iven by SOMMERFELD (5). We would like to conclude with a brief remark about the possibi l i ty of s tudy ing

the analyt ic proper t ies of the to ta l Green 's funct ion. Suppose for s implic i ty t h a t the two Green 's funct ions G(2; A~) and G(2; A S) have a real spec t rum wi th a cu t s ta r t ing f rom 2 = 0 to -4-cr and a single pole in 2=20 and 2=2~o, respect ively. Then

G(2;AI § A~)= - - (2 ' ;A1)G(2--2 ' ;A2)d2 ' 2izt .~

C

can be wr i t ten as follows (by s imply t ak ing residues) (6) (see Fig. 3):

1 fa(z,; A1)G(2 - - 2 ' ; A2)d2' G(2; A 1 § A 2) = Fl(2o)G(2 -- 20; Az) + 2i-~

(~) ~Ao SOI~I)IERFELD: Partial Di]]erenlial Equations in Physics (New York, 19t9), p. 233. (e) Fl(~o) and F2(,1~)' are the residues of G(~'; AI) and G(2 -- ,1'; .4D respectively in their own poles.

1828 L. BIANCHI a n d L. FAVELLA

So it is easy to see that G(2; A I + A 2) has a pole in )~=,,to-F;t,~ and a cut which starts from rain ().o, ;to) and goes to infinity, since the path C can be deformed to

c u t o f G(.l-.,tr;A 2) (

pole of G (.,tr;A I)

C ~!

)" I pole of G (.,l,-.,tr;A 2) C

,(

C r cu t o f G (ZV;A)j,

Fig. 3.

encircle the spectrum of G ( 2 - - 2 ' ; A2), giving thus

, 1 / G ( 2 ' ; A1)G(,~-- ~'; A2)d2' O(~; A 1 -~ A 2) = i~(),~)Gl(~-- ~o; A1) ~- 2i---~

The authors wish to thank prof. VERDE for stimulating discussions.