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The variational principle for the scattering phase
as as as as as as as as Volume 76A, number .5,6
PHYSICS LETTERS
14 April 198C
THE VARIATIONAL PRINCIPLE FOR THE SCATTERING PHASE a a
J. PlAN and C.S. SHARMA Department of Mathematics, Birkbeck College, London WCJE 7HX, England
a Received 3 January 1980
The proof of Kato’s lemma for Schwinger’s variational principle for the scattering phase is extended to a semi-bounded symmetric transformation on a complex Hubert space by using the new calculus on complex Banach spaces.
The new calculus on Banach spaces developed by Sharma and Rebelo [1] provides the foundations for a rigorous treatment of the calculus of variations in a complex Hilbert space (cf. refs. [2,3]). We use this calculus to refine and extend the result concerning Kato’s lemma proved earlier by Sharma and Rebelo [4] and we also give the correct sufficiency conditions (cf. ref. [5]). We first prove the following proposition: Proposition]. Let f be a function from a complex Banach space x to a complex Banach space y and let f be twice semi-differentiable on x. A sufficient condition for a stationary point u E x off to be minimum is that f2s(h,h)~clIhlI2 V/i Ex, for some c E R~. U
Proof The proof foilows directly from the next proposition. Proposition 2 (Taylor’s formula with remainder). Let f be as in proposition 1. Then there exists an r E R~such that for all h E x with Il/i II
Proof Letg: [0,1] -*ybedefinedbyg(t)=f(v th). Then g is semi-differentiable and g~(t)= g’(t) =f~÷rh(h). Hence
~ ‘
~
~
~v ~
‘
where R’ -÷0as it/ill -÷0.The proof follows by cornputing (1) with the help of (2), one can then find an r E R~such that for Il/ill
where a E C and in order that a stationary value of the functional is minimum it is necessary that (hA/i) + c(h,fXf, h> ~s 0 (4) for some c E R and Vh E H. Furthermore a sufficient condition for the stationary value to be a minimum is that J(u)<~Jllf12 where ~iis the infimum of the spectrum ofA. Proof It is easily verified that F is an open set in H and that the functionalJ is semi-differentiable. Cornputing its first semi-derivative at v yields
+
1
g(1)_g(0)__ff~5+~~(h).
From the definition of f~we have = ÷“~t/i~÷R’ 11th I
(1) 365
Volume 76A, number 5,6
PHYSICS LETTERS
J,~(h) ((v,f)(f, u))[h,(Au—(Kv,Av)/(u,f)) f) +
(Au— ((u,Au)/(u,f))f, h)]
ii
14 April [980
((u,
~A/’~Bv,f)A)Bf,
that is
.
At a stationary point u we must haveJ~= 0, that is
Au =((u,Au)/(u,f))f.
((u,Au)/(u,f))f= af.
A necessary condition forf(u)
ComputingJ~(h,h)gives 2~(h, h) = 2((u,fXf, u))—1 J~ X [(h,Ah) J(u)(h,f)(f, h)].
(h,h)4 —J(u)(Bh,f)A (f,Bh)A that is,
Au
—
The necessary condition for
u to be a minimum fol-
lows from proposition 2 by taking c = —J(u). Furthermore if i~is the infimum of the spectrum ofA, then J~S(h,h) ~ 2(Ku, f)(f, u)) [i~ J(u) If 112] 1/112 —
.
The sufficient condition in the lemma then follows from proposition 1. This proof does not hold if A is unbounded, since in that case the functional J is not semi-differentiable, Let us consider first the case where A is a symmetric transformation such that (Au, u) ~ ~(u, u), r~E R~. Riesz and Nagy [6] have shown that DA, the domain ofA, can be completed to form a Hubert space with inner product )~defined by Ku, A)A = Ku, Au). Furthermore, the transformationB: H (DA ~‘A) defined by (f, h) = (f, Bh ~A with f E (DA, )A) and h E H is self-adjoint and has an inverse B which is an extension of the transformation A. Let fbe a fixed element ofDA. The sets F = fVEDA: (v,f) ~ 0} and F’ = fu E DA: KBu,f)A ~ 0} are identical and F’ is open in (DA >A)’ The functional J of lemma 1, viewed as a functional from F’ C (DA, ~A) to C
can be rewritten as
J(u)=Ku,v)A/(Bv,.f)A (f,Bu)A
to
be minimum is then
~‘
K/i, Ah) J(u)(h,fXf, h) ~ 0. A sufficient condition for J(u) to be a minimum then becomes
J(u) <(Bf, f)1 ~ r~/If 112 (where lf1I2 (f, f)). We have thus proved:
(5)
Lemma 2. Let H be a Hubert space with inner product K ), let A be a semi-bounded positive symi-netric transformation and let f be a fixed element in DA , the domain of A. Let F = {u Eli: (v,f) ~ 0}. Let J: F C be as in lemma 1. Then the conclusions of lemma I about the stationary value off continue to be valid. ,
-~
The lemma will hold also for a semi-bounded symmetric transformation A such that (Au, u) > ~(u, u) where r~<0. This can be seen by considering A’ = A r~Iand viewing J as a functional from a subset of DA to C. We further remark that when A is bounded below all stationary points off satisfying the inequality (5) yield a minimum also in the original space H. References [lj CS. Sharma and I. Rebelo, Int. J. Theor. Phys. 13 (1975) 323.
.
J is semi-differentiable. Computing its first and second semi-derivatives, we find that at a stationary point u
[2] G. Fonte, Nuovo Cimento 49B (1979) [31 S. Pian, submitted for publication.
[41 C.S. Sharma and I.
200.
Rebelo, Phys. Lett. 44A (1973) 29.
[51 S.B.W. Mare, Phys. Lett. 53A (1975) 8. [61 F. Riesz and B.Sz. Nagy, Functional analysis (Ungar, New York, 1955).
a a a a a a a a a a a a a as as as
366 as as as as as as as as as
PHYSICS LETTERS
14 April 198C
THE VARIATIONAL PRINCIPLE FOR THE SCATTERING PHASE a a
J. PlAN and C.S. SHARMA Department of Mathematics, Birkbeck College, London WCJE 7HX, England
a Received 3 January 1980
The proof of Kato’s lemma for Schwinger’s variational principle for the scattering phase is extended to a semi-bounded symmetric transformation on a complex Hubert space by using the new calculus on complex Banach spaces.
The new calculus on Banach spaces developed by Sharma and Rebelo [1] provides the foundations for a rigorous treatment of the calculus of variations in a complex Hilbert space (cf. refs. [2,3]). We use this calculus to refine and extend the result concerning Kato’s lemma proved earlier by Sharma and Rebelo [4] and we also give the correct sufficiency conditions (cf. ref. [5]). We first prove the following proposition: Proposition]. Let f be a function from a complex Banach space x to a complex Banach space y and let f be twice semi-differentiable on x. A sufficient condition for a stationary point u E x off to be minimum is that f2s(h,h)~clIhlI2 V/i Ex, for some c E R~. U
Proof The proof foilows directly from the next proposition. Proposition 2 (Taylor’s formula with remainder). Let f be as in proposition 1. Then there exists an r E R~such that for all h E x with Il/i II
Proof Letg: [0,1] -*ybedefinedbyg(t)=f(v th). Then g is semi-differentiable and g~(t)= g’(t) =f~÷rh(h). Hence
~ ‘
~
~
~v ~
‘
where R’ -÷0as it/ill -÷0.The proof follows by cornputing (1) with the help of (2), one can then find an r E R~such that for Il/ill
where a E C and in order that a stationary value of the functional is minimum it is necessary that (hA/i) + c(h,fXf, h> ~s 0 (4) for some c E R and Vh E H. Furthermore a sufficient condition for the stationary value to be a minimum is that J(u)<~Jllf12 where ~iis the infimum of the spectrum ofA. Proof It is easily verified that F is an open set in H and that the functionalJ is semi-differentiable. Cornputing its first semi-derivative at v yields
+
1
g(1)_g(0)__ff~5+~~(h).
From the definition of f~we have = ÷“~t/i~÷R’ 11th I
(1) 365
Volume 76A, number 5,6
PHYSICS LETTERS
J,~(h) ((v,f)(f, u))[h,(Au—(Kv,Av)/(u,f)) f) +
(Au— ((u,Au)/(u,f))f, h)]
ii
14 April [980
((u,
~A/’~Bv,f)A)Bf,
that is
.
At a stationary point u we must haveJ~= 0, that is
Au =((u,Au)/(u,f))f.
((u,Au)/(u,f))f= af.
A necessary condition forf(u)
ComputingJ~(h,h)gives 2~(h, h) = 2((u,fXf, u))—1 J~ X [(h,Ah) J(u)(h,f)(f, h)].
(h,h)4 —J(u)(Bh,f)A (f,Bh)A that is,
Au
—
The necessary condition for
u to be a minimum fol-
lows from proposition 2 by taking c = —J(u). Furthermore if i~is the infimum of the spectrum ofA, then J~S(h,h) ~ 2(Ku, f)(f, u)) [i~ J(u) If 112] 1/112 —
.
The sufficient condition in the lemma then follows from proposition 1. This proof does not hold if A is unbounded, since in that case the functional J is not semi-differentiable, Let us consider first the case where A is a symmetric transformation such that (Au, u) ~ ~(u, u), r~E R~. Riesz and Nagy [6] have shown that DA, the domain ofA, can be completed to form a Hubert space with inner product )~defined by Ku, A)A = Ku, Au). Furthermore, the transformationB: H (DA ~‘A) defined by (f, h) = (f, Bh ~A with f E (DA, )A) and h E H is self-adjoint and has an inverse B which is an extension of the transformation A. Let fbe a fixed element ofDA. The sets F = fVEDA: (v,f) ~ 0} and F’ = fu E DA: KBu,f)A ~ 0} are identical and F’ is open in (DA >A)’ The functional J of lemma 1, viewed as a functional from F’ C (DA, ~A) to C
can be rewritten as
J(u)=Ku,v)A/(Bv,.f)A (f,Bu)A
to
be minimum is then
~‘
K/i, Ah) J(u)(h,fXf, h) ~ 0. A sufficient condition for J(u) to be a minimum then becomes
J(u) <(Bf, f)1 ~ r~/If 112 (where lf1I2 (f, f)). We have thus proved:
(5)
Lemma 2. Let H be a Hubert space with inner product K ), let A be a semi-bounded positive symi-netric transformation and let f be a fixed element in DA , the domain of A. Let F = {u Eli: (v,f) ~ 0}. Let J: F C be as in lemma 1. Then the conclusions of lemma I about the stationary value off continue to be valid. ,
-~
The lemma will hold also for a semi-bounded symmetric transformation A such that (Au, u) > ~(u, u) where r~<0. This can be seen by considering A’ = A r~Iand viewing J as a functional from a subset of DA to C. We further remark that when A is bounded below all stationary points off satisfying the inequality (5) yield a minimum also in the original space H. References [lj CS. Sharma and I. Rebelo, Int. J. Theor. Phys. 13 (1975) 323.
.
J is semi-differentiable. Computing its first and second semi-derivatives, we find that at a stationary point u
[2] G. Fonte, Nuovo Cimento 49B (1979) [31 S. Pian, submitted for publication.
[41 C.S. Sharma and I.
200.
Rebelo, Phys. Lett. 44A (1973) 29.
[51 S.B.W. Mare, Phys. Lett. 53A (1975) 8. [61 F. Riesz and B.Sz. Nagy, Functional analysis (Ungar, New York, 1955).
a a a a a a a a a a a a a as as as
366 as as as as as as as as as