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A generalization of the Birman trace theorem
JOURNAL
OF FUNCTIONAL
ANALYSIS
A Generalization
28, 182-l 86 (1978)
of the Birman
Trace Theorem
D. B. PEARSON Deportment
of Applied Mathematics,
University
of Hull, Hull HU6 712x, England
Communicated by Tosio Kato Received August
17, 1976; revised November
3, 1976
A simple and direct proof is presented of a version of the Birman trace theorem which is sufficiently general to apply to potential scattering with absorption at local singularities.
1. INTRODUCTION The proof by Kato [I ; 2, Chap. X] and Rosenblum [3] of the existence of generalized wave operators L&(2?, , Hr) for trace class perturbations has subsequently been considerably extended, culminating in the very general results of Belopolskii and Birman [4] which deal with scattering between a pair of Hilbert spaces. These results give sufficient conditions for the existence on M,.,.(H,) of s-liml+fm eiHztJeciHlt, where HI , H, are two self-adjoint operators and J is a mapping between the respective Hilbert spaces in which HI , H, act. Although there are many important applications to potential scattering and elsewhere, the development of a more general scattering theory for real singular absorptive potentials [5, 61 requires a reformulation of the trace theorem. (Theorem 4.2 of [4] is insufficient since the compactness condition B would rule out the possibility of absorption and would imply unitarity of the scattering operator.) We here present a generalization of previous results of the present author [7], deriving a new version of the trace theorem which is independent of any compactness requirement (other than the trace condition itself) and which is applicable to scattering by absorptive potentials (see [5, 61 for applications). For further applications to acoustic scattering, where also the invariance of the wave operators is used, see [8].
2. PROOF OF THE TRACE
THEOREM
In a Hilbert space Z’, D(H) denotes the domain of a self-adjoint operator H, Ma.c.(H) the subspace of absolute continuity, and E,.,.(H) the projection
182 0022-1236/78/02824182$02.00/0
Copyright0 1978by Academic Press,Inc. All riphtsof retxoduction in anvformreserved.
BIFUvIAN
TRACE
183
THEOREM
onto M, .C(H). The spectral projection (of H) corresponding to the interval (-co, A) is denoted by E(X). LEMMA. (Let HI and H, be self-adjoint operators acting in Xl and Hz, respectively.) Supp ose A E gl(Zl , &J (trace class operators from J& to &$) oun dde op era t ors f rom yi”z to Xl). For any B E a(Zl, andTE3?(&Zz,#l)(b &‘$ define
F,,(B) Then for any 4 E Ma.C.(Hl)
=
1” dt eiHltBeWiHlt.
satisfying
d ess, sup a (#J, E,(h) 6) = m2 < 03, where (E,(h)} is the spectral family
(2)
of HI , we have, with a > 0,
I(C, FOa(eiH1tTAeCH1s) +)I
< (2~r)“~ m /I T I/ * (11A Ij#”
(lm 111A 11’2eCiH1’$ II2d,)lia.
s
(3)
Here 1A 1 = (A*A)‘12, and /j A /I1 denotes the trace norm of A. The integral orz the right-hand side of (3) is convergent and bounded by 2am2 11A II1 . Proof. We can write / A / = x:n a, j #n)(+n 1,where {C,} is an orthonormal sequence of eigenvectors of I A ) corresponding to nonzero (positive) eigenvalues a, . We also have A = Cn a, I #,)(& j, where 3, = (A+,)/11 A& 11and the sequence {&} is orthonormal. Now I(#, Foa(eiH1’ TAepiHIS) 4) 1 =
c a, Oadr(q3, eiH1(T+t)T&J(+, I 11 s
< (1 a, l’ n
, e-iHl(r+s)$) )
(4)
dr I($, e’H1(‘+t)T#,J[2)1’2
(1 a, l’ n
dr I
The second factor on the r.h.s. is just
(1adrll I A I
112 e-iH1(r+~)+
,,2)1’2,
0
and is bounded by
(Sswll14
l/2
e--iH,r+ 112 &)l;l.
184
D.
B.
PEARSON
where
(Since .!&(Hr) commutes with e-iHl(r+s), we may here assume without loss of generality & E M,.C.(HI).) H ence by Plancherel’s theorem we have
I-02m dy I(&
, ewiH1(r+s)$)lz< 2m2 fm dh $ (4, , E&I) &,) = ~~2, -m
so that certainly
IA I1’2 eeiHlr$ /j2dr < IdmII
27rm2c a, = 2m2(1 A II1. n
Similarly, for the first factor on the r.h.s. of (4) we have the bound (27mt2I] T II211A 11r)1/2,from which (3) follows. It is important to note that the bound is uniform in a. THEOREM. Let HI and H, be self-adjoint operators acting in .& and Z2, respectively. Let J E 9(Sl , Z2) satisfy JD(H,) C D(H,), and suppose that the closure A of H, J - JH, (de$ned on D(H,)) is in ~l(~l , X2). Then, if w(t) = e%tJe-iHlt, the strong limits Q&J2
>HI; I) = ;iiz 44 LAW
exist on the entire Hilbert space. Proof.
Using the formula (see [2, X, (Kg)]) Y - eiHlaYePla
= -iFO,([Hl
in the case Y = (w(t))* w(s), we have (w(t))* w(s) - eiH1”(w(t))* w(s) eCCHla
, Y]),
BIRMAN
TRACE
185
THEOREM
Subtracting the same identity with s + t we have (w(t))* (w(t) - w(s)) - P+(w(t))* = ;F,a{e’w(A*]
(w(t) - w(s)) ewiHla
_ ]*A) e-iw _
_
etHlt(A*eiH2(S-t)]
]*eiHa(s-t)A)
e-iH,s)*
(5) Now take matrix elements with 4 E A&&HI) satisfying condition (2) of the lemma. We may use the lemma (that is, (3) together with the corresponding bound for (9,Fo,(eiH1SA*T*e-“Hlt)~), to estimate (+,FO,{.)+) on the r.h.s. of Eq. (5). These estimates give lim s,ttm($,Foa{~}+) = 0, the convergence is compact, being uniform in a. Moreover, w(t) - w(s) = i s,”dr e%r/P%r so that, for fixed s and t, s;&n
(w(t)
- w(s)) e--y
= 0.
Hence, from Eq. (5) we have
(Suppose that, for s, t > N(E), and for all a, I<& (W(t)>* (w(t) -
w(s)) 4) -
(9, eiHl”(w(t))*
(w(t) -
w(s)) emiH1”+)I < E.
Fix s, t with s, t > N(E); then the bound remains satisfied in the limit as a -+ co, so that I(+, (w(t))*(w(Q - w(s))$)I < e.) From Eq. (5) with s +-+ t we similarly deduce that jgm (4, (w(s))* W)
- 4s)) 4) = 0,
and combining the two results we have
Hence s-lim,,, w(tW exists. Since the C$satisfying the conditions of the lemma are dense in &&&HI), we have proved the existence on 9’ of Q-(H, , Hr ; J). The existence of Q+(H, , HI ; J) follows similarly, and this completes the proof of the theorem. Remark 1. For given J (bounded), A = Hz J - JH, is most conveniently defined by the bilinear form on D(H,) @ D(H,). Thus A(f, g) = (H,f, Jg) - (J*f, H,g), and this form defines a bounded linear operator A if and only if the domain property JD(H,) C D(H,) holds. Hence if A is defined by the bilinear form, the domain property in the statement of the theorem is superfluous, since A is assumed to be of trace class, and therefore certainly bounded.
186
D. B. PEARSON
Remark 2. Given a bounded operator j, H,f - yHl bounded bilinear form on (range E,(Q) @ (range &(A,)), are bounded Bore1 sets and E,(d) is a spectral projection Provided A = H,E&l,)] - ]HIE,(d,) E aI(Z’), the applied with J = E,(i3,) @l(A,) to deduce convergence
will always define a where A, (k = 1, 2) of Hk . theorem may be on M,.,.(H,) of
If, for fixed A, , the Bore1 sets A, such that A E al(#) cover R apart from a set of measure zero, one may conclude the existence of s-lim,,*, E&lz) ezHztle-rHlt. The existence of Q*(H, , HI ; J) does not follow in general even if the trace conditions hold for all pairs d, , A, of bounded Bore1 sets; a further condition on the domain of Hz is required, such as range (I) C D(H,). (If 2: EL~([W~) is a short-range absorptive potential, HI = --d + v’, and H, = -A, Q*(H, , HI ,I) does not exist on the whole of AI&.~.(H~), even though s-lim,,+, E&l,) ezHzfe-ZHltEa.c. (HI) does exist. This is due to the presence of states for which the kinetic energy tends to infinity.) Remark 3. Under the conditions of the theorem, the invariance of the wave operators holds; that is for suitable functions F we have &(H, , HI ; J) = QJF(H2), F(HJ, J). SeeP, PP. 540 ff.1;the P roof, which uses the same estimates as for the proof of the theorem, follows closely the proof of Theorem 4.7 in [2].
ACKNOWLEDGMENTS The author is indebted to both J. Ginibre and T. Kato for many helpful suggestions and comments, and for considerable simplifications to the original proof. He also wishes to thank P. Deift for his many helpful comments on the invariance property of the wave operators.
REFERENCES 1. T. KATO, Perturbation of continuous spectra by trace class operators, PYOC. Japan Acad. 33 (1957), 26&264. 2. T. KATO, “Perturbation Theory for Linear Operators,” Interscience, New York, 1963. 3. M. ROSFZNBLUM,Perturbation of the continuous spectrum and unitary equivalence, Pacijic /. Math. 7 (1957), 997-1010. 4. A. BELOPOLSKII AND M. BIRMAN, The existence of wave operators in scattering theory for pairs of spaces, Math. USSR-Izw. 2 (1968), 1117-t 130. 5. D. B. PEARSON, General theory of potential scattering with absorption at local singularities, Hek. Phys. Acta 48 (1975), 639-653. 6. M. COMBESCURE AND J. GINIBFLE, “Scattering and Local Absorption for the SchrGdinger Operator,” Universite de Paris Sud, Preprint. 7. D. B. PEARSON, Conditions for the existence of the generalised wave operators, J. Math. Phys. 13 (1972), 149&1499. 8. P. DEW, Ph.D. Thesis, Princeton University.
OF FUNCTIONAL
ANALYSIS
A Generalization
28, 182-l 86 (1978)
of the Birman
Trace Theorem
D. B. PEARSON Deportment
of Applied Mathematics,
University
of Hull, Hull HU6 712x, England
Communicated by Tosio Kato Received August
17, 1976; revised November
3, 1976
A simple and direct proof is presented of a version of the Birman trace theorem which is sufficiently general to apply to potential scattering with absorption at local singularities.
1. INTRODUCTION The proof by Kato [I ; 2, Chap. X] and Rosenblum [3] of the existence of generalized wave operators L&(2?, , Hr) for trace class perturbations has subsequently been considerably extended, culminating in the very general results of Belopolskii and Birman [4] which deal with scattering between a pair of Hilbert spaces. These results give sufficient conditions for the existence on M,.,.(H,) of s-liml+fm eiHztJeciHlt, where HI , H, are two self-adjoint operators and J is a mapping between the respective Hilbert spaces in which HI , H, act. Although there are many important applications to potential scattering and elsewhere, the development of a more general scattering theory for real singular absorptive potentials [5, 61 requires a reformulation of the trace theorem. (Theorem 4.2 of [4] is insufficient since the compactness condition B would rule out the possibility of absorption and would imply unitarity of the scattering operator.) We here present a generalization of previous results of the present author [7], deriving a new version of the trace theorem which is independent of any compactness requirement (other than the trace condition itself) and which is applicable to scattering by absorptive potentials (see [5, 61 for applications). For further applications to acoustic scattering, where also the invariance of the wave operators is used, see [8].
2. PROOF OF THE TRACE
THEOREM
In a Hilbert space Z’, D(H) denotes the domain of a self-adjoint operator H, Ma.c.(H) the subspace of absolute continuity, and E,.,.(H) the projection
182 0022-1236/78/02824182$02.00/0
Copyright0 1978by Academic Press,Inc. All riphtsof retxoduction in anvformreserved.
BIFUvIAN
TRACE
183
THEOREM
onto M, .C(H). The spectral projection (of H) corresponding to the interval (-co, A) is denoted by E(X). LEMMA. (Let HI and H, be self-adjoint operators acting in Xl and Hz, respectively.) Supp ose A E gl(Zl , &J (trace class operators from J& to &$) oun dde op era t ors f rom yi”z to Xl). For any B E a(Zl, andTE3?(&Zz,#l)(b &‘$ define
F,,(B) Then for any 4 E Ma.C.(Hl)
=
1” dt eiHltBeWiHlt.
satisfying
d ess, sup a (#J, E,(h) 6) = m2 < 03, where (E,(h)} is the spectral family
(2)
of HI , we have, with a > 0,
I(C, FOa(eiH1tTAeCH1s) +)I
< (2~r)“~ m /I T I/ * (11A Ij#”
(lm 111A 11’2eCiH1’$ II2d,)lia.
s
(3)
Here 1A 1 = (A*A)‘12, and /j A /I1 denotes the trace norm of A. The integral orz the right-hand side of (3) is convergent and bounded by 2am2 11A II1 . Proof. We can write / A / = x:n a, j #n)(+n 1,where {C,} is an orthonormal sequence of eigenvectors of I A ) corresponding to nonzero (positive) eigenvalues a, . We also have A = Cn a, I #,)(& j, where 3, = (A+,)/11 A& 11and the sequence {&} is orthonormal. Now I(#, Foa(eiH1’ TAepiHIS) 4) 1 =
c a, Oadr(q3, eiH1(T+t)T&J(+, I 11 s
< (1 a, l’ n
, e-iHl(r+s)$) )
(4)
dr I($, e’H1(‘+t)T#,J[2)1’2
(1 a, l’ n
dr I
The second factor on the r.h.s. is just
(1adrll I A I
112 e-iH1(r+~)+
,,2)1’2,
0
and is bounded by
(Sswll14
l/2
e--iH,r+ 112 &)l;l.
184
D.
B.
PEARSON
where
(Since .!&(Hr) commutes with e-iHl(r+s), we may here assume without loss of generality & E M,.C.(HI).) H ence by Plancherel’s theorem we have
I-02m dy I(&
, ewiH1(r+s)$)lz< 2m2 fm dh $ (4, , E&I) &,) = ~~2, -m
so that certainly
IA I1’2 eeiHlr$ /j2dr < IdmII
27rm2c a, = 2m2(1 A II1. n
Similarly, for the first factor on the r.h.s. of (4) we have the bound (27mt2I] T II211A 11r)1/2,from which (3) follows. It is important to note that the bound is uniform in a. THEOREM. Let HI and H, be self-adjoint operators acting in .& and Z2, respectively. Let J E 9(Sl , Z2) satisfy JD(H,) C D(H,), and suppose that the closure A of H, J - JH, (de$ned on D(H,)) is in ~l(~l , X2). Then, if w(t) = e%tJe-iHlt, the strong limits Q&J2
>HI; I) = ;iiz 44 LAW
exist on the entire Hilbert space. Proof.
Using the formula (see [2, X, (Kg)]) Y - eiHlaYePla
= -iFO,([Hl
in the case Y = (w(t))* w(s), we have (w(t))* w(s) - eiH1”(w(t))* w(s) eCCHla
, Y]),
BIRMAN
TRACE
185
THEOREM
Subtracting the same identity with s + t we have (w(t))* (w(t) - w(s)) - P+(w(t))* = ;F,a{e’w(A*]
(w(t) - w(s)) ewiHla
_ ]*A) e-iw _
_
etHlt(A*eiH2(S-t)]
]*eiHa(s-t)A)
e-iH,s)*
(5) Now take matrix elements with 4 E A&&HI) satisfying condition (2) of the lemma. We may use the lemma (that is, (3) together with the corresponding bound for (9,Fo,(eiH1SA*T*e-“Hlt)~), to estimate (+,FO,{.)+) on the r.h.s. of Eq. (5). These estimates give lim s,ttm($,Foa{~}+) = 0, the convergence is compact, being uniform in a. Moreover, w(t) - w(s) = i s,”dr e%r/P%r so that, for fixed s and t, s;&n
(w(t)
- w(s)) e--y
= 0.
Hence, from Eq. (5) we have
(Suppose that, for s, t > N(E), and for all a, I<& (W(t)>* (w(t) -
w(s)) 4) -
(9, eiHl”(w(t))*
(w(t) -
w(s)) emiH1”+)I < E.
Fix s, t with s, t > N(E); then the bound remains satisfied in the limit as a -+ co, so that I(+, (w(t))*(w(Q - w(s))$)I < e.) From Eq. (5) with s +-+ t we similarly deduce that jgm (4, (w(s))* W)
- 4s)) 4) = 0,
and combining the two results we have
Hence s-lim,,, w(tW exists. Since the C$satisfying the conditions of the lemma are dense in &&&HI), we have proved the existence on 9’ of Q-(H, , Hr ; J). The existence of Q+(H, , HI ; J) follows similarly, and this completes the proof of the theorem. Remark 1. For given J (bounded), A = Hz J - JH, is most conveniently defined by the bilinear form on D(H,) @ D(H,). Thus A(f, g) = (H,f, Jg) - (J*f, H,g), and this form defines a bounded linear operator A if and only if the domain property JD(H,) C D(H,) holds. Hence if A is defined by the bilinear form, the domain property in the statement of the theorem is superfluous, since A is assumed to be of trace class, and therefore certainly bounded.
186
D. B. PEARSON
Remark 2. Given a bounded operator j, H,f - yHl bounded bilinear form on (range E,(Q) @ (range &(A,)), are bounded Bore1 sets and E,(d) is a spectral projection Provided A = H,E&l,)] - ]HIE,(d,) E aI(Z’), the applied with J = E,(i3,) @l(A,) to deduce convergence
will always define a where A, (k = 1, 2) of Hk . theorem may be on M,.,.(H,) of
If, for fixed A, , the Bore1 sets A, such that A E al(#) cover R apart from a set of measure zero, one may conclude the existence of s-lim,,*, E&lz) ezHztle-rHlt. The existence of Q*(H, , HI ; J) does not follow in general even if the trace conditions hold for all pairs d, , A, of bounded Bore1 sets; a further condition on the domain of Hz is required, such as range (I) C D(H,). (If 2: EL~([W~) is a short-range absorptive potential, HI = --d + v’, and H, = -A, Q*(H, , HI ,I) does not exist on the whole of AI&.~.(H~), even though s-lim,,+, E&l,) ezHzfe-ZHltEa.c. (HI) does exist. This is due to the presence of states for which the kinetic energy tends to infinity.) Remark 3. Under the conditions of the theorem, the invariance of the wave operators holds; that is for suitable functions F we have &(H, , HI ; J) = QJF(H2), F(HJ, J). SeeP, PP. 540 ff.1;the P roof, which uses the same estimates as for the proof of the theorem, follows closely the proof of Theorem 4.7 in [2].
ACKNOWLEDGMENTS The author is indebted to both J. Ginibre and T. Kato for many helpful suggestions and comments, and for considerable simplifications to the original proof. He also wishes to thank P. Deift for his many helpful comments on the invariance property of the wave operators.
REFERENCES 1. T. KATO, Perturbation of continuous spectra by trace class operators, PYOC. Japan Acad. 33 (1957), 26&264. 2. T. KATO, “Perturbation Theory for Linear Operators,” Interscience, New York, 1963. 3. M. ROSFZNBLUM,Perturbation of the continuous spectrum and unitary equivalence, Pacijic /. Math. 7 (1957), 997-1010. 4. A. BELOPOLSKII AND M. BIRMAN, The existence of wave operators in scattering theory for pairs of spaces, Math. USSR-Izw. 2 (1968), 1117-t 130. 5. D. B. PEARSON, General theory of potential scattering with absorption at local singularities, Hek. Phys. Acta 48 (1975), 639-653. 6. M. COMBESCURE AND J. GINIBFLE, “Scattering and Local Absorption for the SchrGdinger Operator,” Universite de Paris Sud, Preprint. 7. D. B. PEARSON, Conditions for the existence of the generalised wave operators, J. Math. Phys. 13 (1972), 149&1499. 8. P. DEW, Ph.D. Thesis, Princeton University.