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NEW YORK UNIVERSITY

COURANT LMSTITUTE - LIBRARY

IMM-NYU 329

XEW YORK INIVHRSITY

COURANT INSTITUTF. OF

MATHEMATICAL SCIENCES

Some Absolutely Continuous Operators I

P. A. Rejto

â€¢r

\ -

o

r

^

PREPARED UNDER

CONTRACT NO. NONR 285(46)

OFFICE OF NAVAL RESEARCH

IMM-NYU 329

September, 1964

NEW YORK UNIVERSITY

COURANT INSTITUTE OF MATHEMATICAL SCIENCES

SOME ABSOLUTELY CONTINUOUS OPERATORS I

P. A. Rejto

This work represents results obtained at the Courant

Institute of Mathematical Sciences, New York University, under

the sponsorship of the Office of Naval Research, Contract No.

NONR-285(46).

Reproduction In whole or In part is permitted for any

purpose of the United States Government.

â– 301:R12Q j:iOITA^iHHTAr.l W JTJTITBV!! T,

I GnOTAHaiO aUOUKITKOO YJaTUJCjSSA 2M0a

Acknowledgement

The author Is indebted to Professor K, 0. Friedrichs

for his valuable criticism.

.:iiiilolil^io a.Ld.:^t .i.iiv o.trf 'lo'-.

ABSTRACT

In this report abstract conditions are formulated v/hich

assure that the part of an operator over a given interval is

absolutely continuous Â» Tliese conditions are verified for the

2

operators - â€” 5- + q(x) acting on the Intervals (- Â«, + ] â€¢

For convenience set

A, = A + P ,

1 o

A and A, being strictly self adjoint on the same domain. For brevity

o 1

we call A and /-> perturbed and unperturbed operators respectively;

P is the perturbation. Distinguishing the corresponding resolvents by

subscripts, for non real z we have the elementary relation,

R, ( 2) - R^( z) = R, ( z) PR ( z) ,

1 o 1 o

which is called the second resolvent equation. In case ( 1 - PR ( z) )

admits an inverse this equation allows one to express R { z) in terms of

R ( z) and P ,

o

(2.2) R^(z) =R^{z)(l - PR^(z))"^ .

Next we assume that the unperturbed resolvent has property [ *J , and

ask for conditions on the family of operators ( 1 - PR ( C + iÂ« ) ) which

ensure that the perturbed resolvent also has property [ *] . First one is

tempted to say tliat all that one needs is the strong convergence of this

sequence on S , arguing that the product of a strongly and a weakly

convergent sequence of

\ .t

n

yM^'-t' ''^-'^ .cdziuob sm;

\'a

. o I

:J pfiii

:t08 vX'isirio,/:'.

a ...V ;;

^.q ^A On3 _^

- (!

D

.noiittC'iJj^-isq ^u?

'.:i07 aor: ".'Oi ^a:

' .â– *

, 1 bll:

Bsnv^zh

.K- SH'.

2.5 5

operators converges weakly. Then according to (2o2) the per-

turbed resolvent would converge weakly. ?Iowever this arguunent

does not apply to our operators since v/e do not have convergence

on the entire space. Another pathology concerning vjeak conver-

gence of operators on a dense set Is that the space of bounded

operators Is not complete with reference to this convergence. In

order to exclude such pathologies vje need some additional condi-

tions and definitions.

Consider an abstract Banach space B and a form F on

B "x B which is conjugate linear in its first and linear in its

second argument, in short, Hermitian. Let < f", Fg > denote

the value of F for the pair (f^g). The complex conjugate

sign above f does not imply that the conjugate of an element

of an abstract Banach space is defined; it merely emphasizes that

F is conjugate linear in its first argument. In analogy to the

notion of the norm of an operator we define the norm of the form

F by setting

(2.5) |||| = sup ^ . I '^^^H .

f,S llfllB-llsllB

In analogy to weak convergence of a family of operators, v;e say

that the family of forms F(e) converges v;eakly to the form

F(0) if for every f and g in B

(2.4) lim < f, F(e)g > = < f,F(0)g> .

sono'ii'i'-'Viico 9Y,Â£irf d-orr ob sv; ,^icqo nuo oct "c'lq'-.r.G :)on

â€¢^â€¢â€¢....â– â– vr..;o "'-.c^s;; ;;: 'â– ;.', -grBqe; C':'I:vno erl3 no

â€¢x/^i'OL' iBaoJt;! .'.5jb.G 0;:to3 Ibo^er, sv 8e'.:"o.Co.l"'j:j';: :io^? ebislnxo â– 'â– d' â€¢â€¢

2A 6

Note that if P(e) Is a weak Cauchy sequence and the

correspondlnG norms |||| , remain bounded independently

of e then F(e) converges to some form, say F(0). Let A

be a bounded operator on B and define, FA , the product of the

form F v;ith the operator A , to be the form determined by

(2.5) < f , FAg > = < f, FlAg) > .

Concerning such products we have a simple but important proposi-

tion: if the fam.ily of forms FCe) is bounded in norm and

convernes weakly to F(0) and if the family of operators A(e)

convers;es stronp;ly to A(0) , then the family of forms F(e)A(e)

conver?;es weakly to the form F(0)A(0). Tnis an immediate con-

sequence of definitions (2.4) and (2.5) if we remember that for

the family of operators ^A(e) i the principle of imiform bound-

edness holds.

After this digression on abstract Hermitian forms v;e

return to the question of finding conditions v/hich ensure that

property [*] holds for the perturbed resolvent. Recall that

we already assumed that [*] holds for the unperturbed resol-

vent and we consider the set S entering it. Then roughly

speaking v;e shall require the existence of a norm such that: if

B is the completion of S with reference to this norm then one

can set

and

F(e) = R^(C+le)

A(Â£) = 'i-PR^((^+ie))"^,

'.T.-r ; .^ ...

iwuriCwO/. n-^ol edcf srf

< -J

( .: > â– â€¢! n-;I J

. -^ (riA)': ,M

â– â– *â– : cfr- on:; 101}- iyis-Ci/j [-^rrVS 'io e-r5.:?:olo e;lci llii^^ g:; r .jri'j'' Â«iK'xon-S

K no 'io:f.y^Lrp hebrvjcd b sd i rial â€¢^I'XJiIxfrrxB ,A \u hanS-jy . sJ i:,b

/li^ol Gilo :ji'r'o svioqqjjs bas

K it

H n -) .-; (H - a) no (y;'\:i;

X-

2.6 8

converp;G strongly on B as Â£ converc:es to from above, and

this Is uniform In C.

V/e see from the previous proposition and from the second resol-

vent equation (2.2) that If the perturbation problem A^,? ,

satisfies Condition l-I^C^^^Cg] then property [*] holds for

the perturbed resolvent, R,(i;Â±le). Thus these conditions assure

the absolute continuity of the part of A^ over the Interval

In the Appendix v/e shall verify this condition for the

Sturm-Llouvllle operator -D + M(q) , provided that the

potential q dies out at infinity. In doing this, v/e shall need

detailed information on the perturbed resolvent, in spite of the

fact that it does not enter the condition explicitly. For this

reason v;e introduce two more conditions, vjhich are more restric-

tive then this one. Nevertheless for a large class of perturba-

tion problems we can verify them without reference to the per-

turbed resolvent. They read as follows:

Condition 2 .[^j^,^g]

For every ^ in this Interval the two families of operators

determined by PR (^Â±ie) on B , conver,Q;e in norm to compact

operators, as e converp;es to from above, and this is

uniform in ^ .

Condition ^ .fCj^^C^I

For every t, in this interval the nullspacec of 1-PR ({;Â±)

on B are trivial.

bnr:- â– iTOrfi C

â– ~I.-.'.:3'-i b:i'joe-6 sr'rJ irro'il -Mis:, n - ^'

^ -^o

/\ j'j aoi jjjdnuvi'ing

'â– xol a/Mo/! [â– *] \j'i3C/ri'] noilf [^^ < |-'*] -^ - . - ' â– c^llyJ-Cs^.

!'rtn->

r

o

tirlo diii':d- jc.jj.i:v/.wcr ^ ^i.;;.! -r "V.;.. - ;;

2.7 9

If this condition is satisfied at a given point C only-

then v/e refer to it as Condition J.C- Similarly v/e shall speak

of Condition l-C and 2.^.

'v/e claim that Conditions 2-5 imply Condition I. For,

according to Condition 2, PR (^4-) is compact and so the

Fredholm alternative applies to the operator 1-PR (C+)Â« Tlius

from Condition 5 v;e can conclude that 1-PR^(C+) admits a boimd-

ed Inverse defined on all of B. From this in turn, using the

other part of Condition 2 v;e conclude that for small enough e

the operators 1-PR (C+le) admit Inverses; moreover

d-PR (^+le)) converges in the B-operator norm to

d-PR (C+))" . Thus Conditions 2 and p imply Condition I. In

conclusion let us remark that if v;e replace norm convergence by-

strong convergence in Condition 2 this implication may possibly

be false.

Summarizing our results we arrive at the follov/lng:

Lemma 2Â»1

Suppose that the perturbation problem A .P , satisfies condi -

tions l-l,[^-^,^rj or. Conditions 1-2-3. [^3_. ^2^ â€¢ "Jl^en the part

of the perturbed operator A + P over the Interval [^ ,4^] is

absolutely continuous *

In the applications that we consider we can verify the first

two of these conditions directly. In establishing Condition

5Â«[C-i>Coi * hov/ever, it is convenient to proceed indirectly.

I.e., to assume that it is violated and to derive a false con-

clusion from this. Such a conclusion is stated in the next

section.

â€¢ : \ â– - â–

â€¢â€¢)ifi-c;3 Ii^dE ov. â– .';I':tj;I.i.;:lC .^.o ',o ':.â– .:. b..r>0 5\: cfx ....

. -â– . S '.jaa

.^:iC'i Â»I -:o.: - ;.j:iinc-0 v;Iqi;..L c-=:- ;Dr!C..7X!:iKX ;ri:

Â£..i'j ./:, ;;nr; JCJ^'-irivOu j.i; (^^i) HI ^G no.:;'!;:. â– . '- . ^.-

o::j cjfi-f^^^-' t: '^-v^ fi - . strict ;:;â€¢. \L'd Â«o 'lo il^- no 'jenlloij St.. .

' o

^- f .'

r

V;Id'l:roq -^iEr; r;oJ;>: -1 '

iooB'j:-^";o ^shj xC JoenlrrnOoOij ul CI .â– â– ' ii no (.-^J;-^) ..,.n ;.i':^o'i

sonoil :Ii

k "'.^

;T . ,P > r.M.

< (3)B(.>X-^)^^fi ,

. '3 !.. >J J-

- (3}

3.5

12

lim [im < f , R^(C-le)Cl-PR (^+)'^S. >] = < f , ^J^-)^ =^ â€¢

e-^ +o â– - Â° o E 4

On the other hand according to (3 ..6) the left member equals zero,

thus

(5.8) Im < f , RqIC - )^ > = .

How from this relation v;e can easily derive conclusion

(5,4). For, according to Condition 1,^, for an arbitrary func-

tion f in B,

(5.9) < f , R (t;-)f > = lim ff^,R^(C-lÂ£)0 ,

o

e.n

n' o

n

the limit on the ri:;ht being a double limit. Next set

Sq(z) = (z-M)

-1

and observe that according to (5*1)

R (z) = V S (z)V .

o o

Hence

Im(f^ , R^(C-lB)f^) = (Vf^,

S^it^-le) - S(C+ie)]vf^) .

From this fact and from the assumption that V maps vectors in

B n H into continuous functions v;e can conclude

lim (Vf ,

Â£-^ +o

S^(C-iÂ£) - SQ(C+ie)

Vf^) = 7r|5^Vf^r ,

if we use a theorem of Fatou. Since V is a bounded map of

B H into C , this yields

-((-:â€¢:

â€¢ Ciij'': alfij.;p9 nsditiSv/ Jlsl G^i-y (^^a-^) o;/ y.nJ:&icoos 'xTBrJ lerf'o

:^ < ^(.^^ii^hl t T

o

I:U

^â€¢li

f.I 1 nolo

('^,-i(..f:-'^)^.n,^;:;') .nij; â€¢ < â– '-J^^h . i > i'i.O

as;;: ^rs'^t .ctlrffJ:! â€¢-/.[c^u^^b r :-r:.i:i'0 d'ri,:j:';i er.i no JJ:n;;.I 8ri

â€¢- I .V ;

o

(.r.;.'j o:;' â– \n-':b'-.oooa J^-il:i- 6VTÂ©scfo Ijks

\rf

U:) !T

sonsH

O J ;^ Jk O a

oev :.

1 ' v; J.

3bJjl:

5.^ 15

(3.9) llm [lm(fj^,R^(C-lE)f^)] = Trl&^Vfl^

Finally combining relations (3.8), (3.9) and (3.10 v;e obtain

(3.^), v;hich establishes the lemma.

'NMV.oi-vf - I (^^-^Isx-'^ . r i':.0

4.1 14

2

4. The operator p + q(x) on (-~, + oo)

Let D be the set of tv;ice continuously dlfferentlable

functions on [-oo, + oo] with bounded support. We define the

u. perturbed operator to be the D-closure of the formal operator

- 2__ and denote it by -D , For the perturbation we take the

operator defined by

(4.1) M(q)f(x) = q(x)f(x) .

2

As is well known the operator -D is strictly self adjoint.

Hence if the function q is bounded so is the operator Il(q)

and we see that -D + M(q) is also strictly self adjoint and

\ie take it to be the perturbed operator.

VJe formulate additional conditio- s on the function q which

o

assure that the perturbation problem -D , M(q) , satisfies the

abstract conditions of Section 2. Specifically v;e assume that

q is a continuous function for v.'hlch

(4.2) 11m q(x) = .

Ixl-i CO

Moreover, q is integrable v;ith respect to a weight function,

(4.3) / |q(x) lw(x)dx < 00 ,

>-' -co

such that w(x) ^ 1 and

(4.4) lim w(x) = w ,

|xM w

Define the Banach space B to be the closure of D with refer-

ence to the norm

n

X.-l'

( 00 -i , ^ii- ) cfo { .: ] .

-'xJb

2,.

ftr.'c* sr-lBct ew nold'sd'Awo'isc: si'.i lOu

V J-

â€¢ d.ixo'cbB llscj â€¢'rlc^ci'r:; - ;.;.r. "'G- '-loci-.s-xsao ario m.'ca^' IXs'.: ':â– .â– : r.A

.'iocf.n'ioo;o iiS'a'X0;J*"r3q ?ri.:f sd od" j.!: sx.sj Â£

;0'yj

'(-)pi ^^

X -â€¢ \:uu

.^0^)

-^Ui

-''ZZj.Q'l iiuXi

4.2 15

CO

(4.5) llfMB= suplf(x)| + / |f(x)|v;(x)doc .

Note that assumptions (4t3) and (4.4) imply that

(4.6) Iklig < ~ .

o

We maintain that the perturbation problem -D , W(q) satisfies

the conditions of Section 2. More specifically v;e maintain the

following:

Sub theorem 4.1

Suppose that the function q satisfies conditions ( 4 . 3 ) * (4.4)

and that the bounded interval [^, ,Cp] does not contain the

o

poin t or a point eigenvalue of the operator -D + M(q) in

Lp. Then the perturbation problem -D , M(q) satisfies condi -

tions 1-2-3' [C-] jCpl with reference to the Banach space B,

The proof of this subtheorem makes essential use of the

representation of the unperturbed resolventt, Specifically

setting

(4.7) k(u) = e^'^' ,

v;e have

(4.8) R (z)(x,y) = ik(/i-(x-y)) ,

\/z being defined in the plane cut along the positive real axis

and having positive imaginary part,

a) Condition 1 . [ C â€¢â– > C ^ 3 Â» Let [Ci^^o^ ^Â® ^ closed and bounded

interval not containing the point 0. First observe that for

:d:>(::h'! (:â€¢)â€¢:! \ -i- !(>â€¢

f ;i li

^farvj '\

rS.qril (f^.!-) tit^ (â– .%â– !') ?^noi:oqfnf;C2S d-j3f{J ^:>oVI

II I I

lip!!

:;: â– \ :.o1

j,.4^ :r{5

f :i }1\

i \

i-

/-^Â«

kU.

Â« {i:;):I

â€¢3V

)(s)^H

I

':' -J â€¢_

4.3 16

t, In [?, , ?, ] the kemol of the unperturbed resolvent converges pointwise.

Specifically if on the positive real axis

six = llm \/t, + ie ,

â‚¬-*+0

then according to (4. 7) ,

(4.9) lim R (^ + iO(x,y) = â€” k(\T(x - y) ) .

Since this convergence is uniform in any bounded (x, y) region, we see

that setting

(4.10) RCi+)(x,v) = â€”k{^{x-Y)),

for any pair of functions ( f, g) In D ,

(4.11) lim = .

Note that R ( C+) is a possibly unbounded Hermitian fonTi, defined by the

COURANT LMSTITUTE - LIBRARY

IMM-NYU 329

XEW YORK INIVHRSITY

COURANT INSTITUTF. OF

MATHEMATICAL SCIENCES

Some Absolutely Continuous Operators I

P. A. Rejto

â€¢r

\ -

o

r

^

PREPARED UNDER

CONTRACT NO. NONR 285(46)

OFFICE OF NAVAL RESEARCH

IMM-NYU 329

September, 1964

NEW YORK UNIVERSITY

COURANT INSTITUTE OF MATHEMATICAL SCIENCES

SOME ABSOLUTELY CONTINUOUS OPERATORS I

P. A. Rejto

This work represents results obtained at the Courant

Institute of Mathematical Sciences, New York University, under

the sponsorship of the Office of Naval Research, Contract No.

NONR-285(46).

Reproduction In whole or In part is permitted for any

purpose of the United States Government.

â– 301:R12Q j:iOITA^iHHTAr.l W JTJTITBV!! T,

I GnOTAHaiO aUOUKITKOO YJaTUJCjSSA 2M0a

Acknowledgement

The author Is indebted to Professor K, 0. Friedrichs

for his valuable criticism.

.:iiiilolil^io a.Ld.:^t .i.iiv o.trf 'lo'-.

ABSTRACT

In this report abstract conditions are formulated v/hich

assure that the part of an operator over a given interval is

absolutely continuous Â» Tliese conditions are verified for the

2

operators - â€” 5- + q(x) acting on the Intervals (- Â«, + ] â€¢

For convenience set

A, = A + P ,

1 o

A and A, being strictly self adjoint on the same domain. For brevity

o 1

we call A and /-> perturbed and unperturbed operators respectively;

P is the perturbation. Distinguishing the corresponding resolvents by

subscripts, for non real z we have the elementary relation,

R, ( 2) - R^( z) = R, ( z) PR ( z) ,

1 o 1 o

which is called the second resolvent equation. In case ( 1 - PR ( z) )

admits an inverse this equation allows one to express R { z) in terms of

R ( z) and P ,

o

(2.2) R^(z) =R^{z)(l - PR^(z))"^ .

Next we assume that the unperturbed resolvent has property [ *J , and

ask for conditions on the family of operators ( 1 - PR ( C + iÂ« ) ) which

ensure that the perturbed resolvent also has property [ *] . First one is

tempted to say tliat all that one needs is the strong convergence of this

sequence on S , arguing that the product of a strongly and a weakly

convergent sequence of

\ .t

n

yM^'-t' ''^-'^ .cdziuob sm;

\'a

. o I

:J pfiii

:t08 vX'isirio,/:'.

a ...V ;;

^.q ^A On3 _^

- (!

D

.noiittC'iJj^-isq ^u?

'.:i07 aor: ".'Oi ^a:

' .â– *

, 1 bll:

Bsnv^zh

.K- SH'.

2.5 5

operators converges weakly. Then according to (2o2) the per-

turbed resolvent would converge weakly. ?Iowever this arguunent

does not apply to our operators since v/e do not have convergence

on the entire space. Another pathology concerning vjeak conver-

gence of operators on a dense set Is that the space of bounded

operators Is not complete with reference to this convergence. In

order to exclude such pathologies vje need some additional condi-

tions and definitions.

Consider an abstract Banach space B and a form F on

B "x B which is conjugate linear in its first and linear in its

second argument, in short, Hermitian. Let < f", Fg > denote

the value of F for the pair (f^g). The complex conjugate

sign above f does not imply that the conjugate of an element

of an abstract Banach space is defined; it merely emphasizes that

F is conjugate linear in its first argument. In analogy to the

notion of the norm of an operator we define the norm of the form

F by setting

(2.5) |||| = sup ^ . I '^^^H .

f,S llfllB-llsllB

In analogy to weak convergence of a family of operators, v;e say

that the family of forms F(e) converges v;eakly to the form

F(0) if for every f and g in B

(2.4) lim < f, F(e)g > = < f,F(0)g> .

sono'ii'i'-'Viico 9Y,Â£irf d-orr ob sv; ,^icqo nuo oct "c'lq'-.r.G :)on

â€¢^â€¢â€¢....â– â– vr..;o "'-.c^s;; ;;: 'â– ;.', -grBqe; C':'I:vno erl3 no

â€¢x/^i'OL' iBaoJt;! .'.5jb.G 0;:to3 Ibo^er, sv 8e'.:"o.Co.l"'j:j';: :io^? ebislnxo â– 'â– d' â€¢â€¢

2A 6

Note that if P(e) Is a weak Cauchy sequence and the

correspondlnG norms |||| , remain bounded independently

of e then F(e) converges to some form, say F(0). Let A

be a bounded operator on B and define, FA , the product of the

form F v;ith the operator A , to be the form determined by

(2.5) < f , FAg > = < f, FlAg) > .

Concerning such products we have a simple but important proposi-

tion: if the fam.ily of forms FCe) is bounded in norm and

convernes weakly to F(0) and if the family of operators A(e)

convers;es stronp;ly to A(0) , then the family of forms F(e)A(e)

conver?;es weakly to the form F(0)A(0). Tnis an immediate con-

sequence of definitions (2.4) and (2.5) if we remember that for

the family of operators ^A(e) i the principle of imiform bound-

edness holds.

After this digression on abstract Hermitian forms v;e

return to the question of finding conditions v/hich ensure that

property [*] holds for the perturbed resolvent. Recall that

we already assumed that [*] holds for the unperturbed resol-

vent and we consider the set S entering it. Then roughly

speaking v;e shall require the existence of a norm such that: if

B is the completion of S with reference to this norm then one

can set

and

F(e) = R^(C+le)

A(Â£) = 'i-PR^((^+ie))"^,

'.T.-r ; .^ ...

iwuriCwO/. n-^ol edcf srf

< -J

( .: > â– â€¢! n-;I J

. -^ (riA)': ,M

â– â– *â– : cfr- on:; 101}- iyis-Ci/j [-^rrVS 'io e-r5.:?:olo e;lci llii^^ g:; r .jri'j'' Â«iK'xon-S

K no 'io:f.y^Lrp hebrvjcd b sd i rial â€¢^I'XJiIxfrrxB ,A \u hanS-jy . sJ i:,b

/li^ol Gilo :ji'r'o svioqqjjs bas

K it

H n -) .-; (H - a) no (y;'\:i;

X-

2.6 8

converp;G strongly on B as Â£ converc:es to from above, and

this Is uniform In C.

V/e see from the previous proposition and from the second resol-

vent equation (2.2) that If the perturbation problem A^,? ,

satisfies Condition l-I^C^^^Cg] then property [*] holds for

the perturbed resolvent, R,(i;Â±le). Thus these conditions assure

the absolute continuity of the part of A^ over the Interval

In the Appendix v/e shall verify this condition for the

Sturm-Llouvllle operator -D + M(q) , provided that the

potential q dies out at infinity. In doing this, v/e shall need

detailed information on the perturbed resolvent, in spite of the

fact that it does not enter the condition explicitly. For this

reason v;e introduce two more conditions, vjhich are more restric-

tive then this one. Nevertheless for a large class of perturba-

tion problems we can verify them without reference to the per-

turbed resolvent. They read as follows:

Condition 2 .[^j^,^g]

For every ^ in this Interval the two families of operators

determined by PR (^Â±ie) on B , conver,Q;e in norm to compact

operators, as e converp;es to from above, and this is

uniform in ^ .

Condition ^ .fCj^^C^I

For every t, in this interval the nullspacec of 1-PR ({;Â±)

on B are trivial.

bnr:- â– iTOrfi C

â– ~I.-.'.:3'-i b:i'joe-6 sr'rJ irro'il -Mis:, n - ^'

^ -^o

/\ j'j aoi jjjdnuvi'ing

'â– xol a/Mo/! [â– *] \j'i3C/ri'] noilf [^^ < |-'*] -^ - . - ' â– c^llyJ-Cs^.

!'rtn->

r

o

tirlo diii':d- jc.jj.i:v/.wcr ^ ^i.;;.! -r "V.;.. - ;;

2.7 9

If this condition is satisfied at a given point C only-

then v/e refer to it as Condition J.C- Similarly v/e shall speak

of Condition l-C and 2.^.

'v/e claim that Conditions 2-5 imply Condition I. For,

according to Condition 2, PR (^4-) is compact and so the

Fredholm alternative applies to the operator 1-PR (C+)Â« Tlius

from Condition 5 v;e can conclude that 1-PR^(C+) admits a boimd-

ed Inverse defined on all of B. From this in turn, using the

other part of Condition 2 v;e conclude that for small enough e

the operators 1-PR (C+le) admit Inverses; moreover

d-PR (^+le)) converges in the B-operator norm to

d-PR (C+))" . Thus Conditions 2 and p imply Condition I. In

conclusion let us remark that if v;e replace norm convergence by-

strong convergence in Condition 2 this implication may possibly

be false.

Summarizing our results we arrive at the follov/lng:

Lemma 2Â»1

Suppose that the perturbation problem A .P , satisfies condi -

tions l-l,[^-^,^rj or. Conditions 1-2-3. [^3_. ^2^ â€¢ "Jl^en the part

of the perturbed operator A + P over the Interval [^ ,4^] is

absolutely continuous *

In the applications that we consider we can verify the first

two of these conditions directly. In establishing Condition

5Â«[C-i>Coi * hov/ever, it is convenient to proceed indirectly.

I.e., to assume that it is violated and to derive a false con-

clusion from this. Such a conclusion is stated in the next

section.

â€¢ : \ â– - â–

â€¢â€¢)ifi-c;3 Ii^dE ov. â– .';I':tj;I.i.;:lC .^.o ',o ':.â– .:. b..r>0 5\: cfx ....

. -â– . S '.jaa

.^:iC'i Â»I -:o.: - ;.j:iinc-0 v;Iqi;..L c-=:- ;Dr!C..7X!:iKX ;ri:

Â£..i'j ./:, ;;nr; JCJ^'-irivOu j.i; (^^i) HI ^G no.:;'!;:. â– . '- . ^.-

o::j cjfi-f^^^-' t: '^-v^ fi - . strict ;:;â€¢. \L'd Â«o 'lo il^- no 'jenlloij St.. .

' o

^- f .'

r

V;Id'l:roq -^iEr; r;oJ;>: -1 '

iooB'j:-^";o ^shj xC JoenlrrnOoOij ul CI .â– â– ' ii no (.-^J;-^) ..,.n ;.i':^o'i

sonoil :Ii

k "'.^

;T . ,P > r.M.

< (3)B(.>X-^)^^fi ,

. '3 !.. >J J-

- (3}

3.5

12

lim [im < f , R^(C-le)Cl-PR (^+)'^S. >] = < f , ^J^-)^ =^ â€¢

e-^ +o â– - Â° o E 4

On the other hand according to (3 ..6) the left member equals zero,

thus

(5.8) Im < f , RqIC - )^ > = .

How from this relation v;e can easily derive conclusion

(5,4). For, according to Condition 1,^, for an arbitrary func-

tion f in B,

(5.9) < f , R (t;-)f > = lim ff^,R^(C-lÂ£)0 ,

o

e.n

n' o

n

the limit on the ri:;ht being a double limit. Next set

Sq(z) = (z-M)

-1

and observe that according to (5*1)

R (z) = V S (z)V .

o o

Hence

Im(f^ , R^(C-lB)f^) = (Vf^,

S^it^-le) - S(C+ie)]vf^) .

From this fact and from the assumption that V maps vectors in

B n H into continuous functions v;e can conclude

lim (Vf ,

Â£-^ +o

S^(C-iÂ£) - SQ(C+ie)

Vf^) = 7r|5^Vf^r ,

if we use a theorem of Fatou. Since V is a bounded map of

B H into C , this yields

-((-:â€¢:

â€¢ Ciij'': alfij.;p9 nsditiSv/ Jlsl G^i-y (^^a-^) o;/ y.nJ:&icoos 'xTBrJ lerf'o

:^ < ^(.^^ii^hl t T

o

I:U

^â€¢li

f.I 1 nolo

('^,-i(..f:-'^)^.n,^;:;') .nij; â€¢ < â– '-J^^h . i > i'i.O

as;;: ^rs'^t .ctlrffJ:! â€¢-/.[c^u^^b r :-r:.i:i'0 d'ri,:j:';i er.i no JJ:n;;.I 8ri

â€¢- I .V ;

o

(.r.;.'j o:;' â– \n-':b'-.oooa J^-il:i- 6VTÂ©scfo Ijks

\rf

U:) !T

sonsH

O J ;^ Jk O a

oev :.

1 ' v; J.

3bJjl:

5.^ 15

(3.9) llm [lm(fj^,R^(C-lE)f^)] = Trl&^Vfl^

Finally combining relations (3.8), (3.9) and (3.10 v;e obtain

(3.^), v;hich establishes the lemma.

'NMV.oi-vf - I (^^-^Isx-'^ . r i':.0

4.1 14

2

4. The operator p + q(x) on (-~, + oo)

Let D be the set of tv;ice continuously dlfferentlable

functions on [-oo, + oo] with bounded support. We define the

u. perturbed operator to be the D-closure of the formal operator

- 2__ and denote it by -D , For the perturbation we take the

operator defined by

(4.1) M(q)f(x) = q(x)f(x) .

2

As is well known the operator -D is strictly self adjoint.

Hence if the function q is bounded so is the operator Il(q)

and we see that -D + M(q) is also strictly self adjoint and

\ie take it to be the perturbed operator.

VJe formulate additional conditio- s on the function q which

o

assure that the perturbation problem -D , M(q) , satisfies the

abstract conditions of Section 2. Specifically v;e assume that

q is a continuous function for v.'hlch

(4.2) 11m q(x) = .

Ixl-i CO

Moreover, q is integrable v;ith respect to a weight function,

(4.3) / |q(x) lw(x)dx < 00 ,

>-' -co

such that w(x) ^ 1 and

(4.4) lim w(x) = w ,

|xM w

Define the Banach space B to be the closure of D with refer-

ence to the norm

n

X.-l'

( 00 -i , ^ii- ) cfo { .: ] .

-'xJb

2,.

ftr.'c* sr-lBct ew nold'sd'Awo'isc: si'.i lOu

V J-

â€¢ d.ixo'cbB llscj â€¢'rlc^ci'r:; - ;.;.r. "'G- '-loci-.s-xsao ario m.'ca^' IXs'.: ':â– .â– : r.A

.'iocf.n'ioo;o iiS'a'X0;J*"r3q ?ri.:f sd od" j.!: sx.sj Â£

;0'yj

'(-)pi ^^

X -â€¢ \:uu

.^0^)

-^Ui

-''ZZj.Q'l iiuXi

4.2 15

CO

(4.5) llfMB= suplf(x)| + / |f(x)|v;(x)doc .

Note that assumptions (4t3) and (4.4) imply that

(4.6) Iklig < ~ .

o

We maintain that the perturbation problem -D , W(q) satisfies

the conditions of Section 2. More specifically v;e maintain the

following:

Sub theorem 4.1

Suppose that the function q satisfies conditions ( 4 . 3 ) * (4.4)

and that the bounded interval [^, ,Cp] does not contain the

o

poin t or a point eigenvalue of the operator -D + M(q) in

Lp. Then the perturbation problem -D , M(q) satisfies condi -

tions 1-2-3' [C-] jCpl with reference to the Banach space B,

The proof of this subtheorem makes essential use of the

representation of the unperturbed resolventt, Specifically

setting

(4.7) k(u) = e^'^' ,

v;e have

(4.8) R (z)(x,y) = ik(/i-(x-y)) ,

\/z being defined in the plane cut along the positive real axis

and having positive imaginary part,

a) Condition 1 . [ C â€¢â– > C ^ 3 Â» Let [Ci^^o^ ^Â® ^ closed and bounded

interval not containing the point 0. First observe that for

:d:>(::h'! (:â€¢)â€¢:! \ -i- !(>â€¢

f ;i li

^farvj '\

rS.qril (f^.!-) tit^ (â– .%â– !') ?^noi:oqfnf;C2S d-j3f{J ^:>oVI

II I I

lip!!

:;: â– \ :.o1

j,.4^ :r{5

f :i }1\

i \

i-

/-^Â«

kU.

Â« {i:;):I

â€¢3V

)(s)^H

I

':' -J â€¢_

4.3 16

t, In [?, , ?, ] the kemol of the unperturbed resolvent converges pointwise.

Specifically if on the positive real axis

six = llm \/t, + ie ,

â‚¬-*+0

then according to (4. 7) ,

(4.9) lim R (^ + iO(x,y) = â€” k(\T(x - y) ) .

Since this convergence is uniform in any bounded (x, y) region, we see

that setting

(4.10) RCi+)(x,v) = â€”k{^{x-Y)),

for any pair of functions ( f, g) In D ,

(4.11) lim = .

Note that R ( C+) is a possibly unbounded Hermitian fonTi, defined by the