A NNALES DE L ’I. H. P., SECTION A
R. A. W EDER
Spectral properties of one-body relativistic spin-zero hamiltonians
Annales de l’I. H. P., section A, tome 20, n
o2 (1974), p. 211-220
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211
Spectral properties of one-body relativistic
spin-zero hamiltonians (1)
R. A. WEDER
(2)
Instituut voor Theoretische en Wiskundige Fysica
Katholieke Universiteit Leuven (Belgium)
(3)
Poincaré, Vol. XX, n° 2, 1974,
Section A :
Physique ’ théorique. ’
ABSTRACT. 2014 We
study
thespectral properties
of the relativisticspin-
zero Hamiltonian H =
p2 + ,u2
+ V of aspinless particle, by
an exten-sion of the method of
Aguilar-Combes,
for a class of interactionsincluding
V =-~~0~
1.Absence of
singular-continuous spectrum
isproved, together
with theexistence of an
absolutely-continuous spectrum [,u, oo).
InRB{ /1 }
thepoint
spectrum
consists of finite-dimensionaleigenvalues
which are bounded.Properties
of resonances areinvestigated.
RESUME. 2014 Nous etudions les
proprietes
de l’Hamiltonien relativiste despin
zero : H =p2 + ~c2
+ V d’uneparticule
sansspin, grace
a uneextension de la methode
d’Aguilar-Combes,
pour la classe d’interactionscomprenant :
V = - 0~3
1.L’absence de
spectre singulierement
continu estprouvee,
en memetemps
que 1’existence d’un spectre absolument continu[/1, oo).
DansRB { /1 }
Ie
spectre ponctuel
est formee de valeurs propres de dimension finiequi
sont bornees.
Les
proprietes
des resonances sontanalysees.
e)
Work supported by : A. G. C. D. (Belgium).(2)
On leave of absence from Universidad de Buenos Aires (Argentina).e)
Postal address: Instituut voor Theoretische en Wiskundige Natuurkunde Celestij-nenlaan 200 D, B-3030, Heverlee (Belgium).
Annales de l’Institut Henri Poincaré - Section A - Vol. XX, n° 2 - 1974
212 R. A. WEDER
INTRODUCTION
Recently [1] [2], spectral properties
ofSchrodinger operators
with inter- actionssatisfying analyticity
conditions withrespect
to the dilatation group were studied.In this work we
investigate
thespectral properties
of the relativisticspin-zero
Hamiltonian H =~/p~ + /~
+ V of aspinless particle by
anextension of the method of
[7],
for a class of interactionsincluding
V
= - ~’~ 0 ~
1.In Section I we
study
theanalyticity properties,
with respect to the dilatation group, of the free HamiltonianHo
=~/p~ + /~.
In Section II we define the class of
interactions,
we areconsidering,
and prove that the interactions V = -
gr-a,
0~3
1 are allowed.In Section III we show that the
singular-continuous spectrum
isempty together
with the existence of anabsolutely-continuous spectrum [~, oo).
We also show that the
point spectrum
consists of a bounded set of finite- dimensionaleigenvalues
differentfrom ,u (accumulating,
at most, at~),
and
possibly
aneigenvalue
at ~.Properties
of resonances are also investi-gated.
We stress the difference with the non-relativistic case,
namely
theessential-spectrum
of theanalytic
extention of the Hamiltonian is not astraight-line (see fig. 1 )
but apart
of anhyperbola starting
at ,u. The basicnew technical results for the relativistic case are contained in lemmas
1,
2and 3. The method of the
proof
of lemma 4 and theorem 1 were taken from[1] [2]
and[6].
For the definitions ofvector-valued,
andoperator-
valuedanalytic functions,
and for the classification of thespectrum
we referto T. Kato
[3].
In this paper we use the same notation as in
[2].
I. THE FREE HAMILTONIAN
Let,
the space of wavefunctions of aspinless particle,
be the Hilbert space J~ ==22(1R3)
ofsquare-integrable
functions in1F~3.
Let ,u, theparticle
mass, be a
strictly positive
constant, and letM(p)
=p2 + ,u2.
We definethe free Hamiltonian in momentum space
by
on the
domain, !?Ø(Ho),
of all T in ~f such that isagain
in ~f.We
take,
of course, the square root withpositive sign. Ho
is apositive,
and
selfadjoint operator [3].
Annales de l’Institut Henri Poincaré - Section A
213
SPECTRAL PROPERTIES OF ONE-BODY RELATIVISTIC SPIN-ZERO HAMILTONIANS
Let
U(8),
8 E!R,
be thestrongly-continuous unitary representation
on -~ of the dilatation group defined
by
Thus, we have
where
LEMMA 1. The
family
of operatorsHo(0),
Oe!?,
can be extended toan
analytic family
in thestrip
of thecomplex-plane
Proof :
- We can writewith, c/>(E>, p),
the argument; and a modulusp(O, p)
> 0 for allpe[0, oo)
and all 0 E
S 1t.
There exist tworeal, positive,
and bounded functions ofO(M 1 (O),
andM2(0398))
such thatThus
That is to say
By
a trivial argument, which weomit,
we can show thatThen,
we have thefollowing
estimationVol. XX, n° 2 - 1974.
214 R. A. WEDER
Then,
as03C9(0398, p)
is ananalytic
function of 8 for all 8 ES 7[’ by
the Lebes-2
gue’s
dominated convergence theorem we havein the
strong topology
in J~.LEMMA 2. The
spectrum
of denotedo-(Ho(0)),
is a continuouscurve,
starting
j at , and otending
iasymptoticaly
tothe straightline
’(see fig. 1).
a) The spectrum of
Hotter
b ) The spectrum
FiG.l.
2014 It is an immediate consequence of the definition of the
spectrum
thatAnnales de Henri Poincare - Section A
215 SPECTRAL PROPERTIES OF ONE-BODY RELATIVISTIC SPIN-ZERO HAMILTONIANS
where
p(0, p)
and~(0, p)
arerespectively
the modulus and theargument
of +p2.
For 0 Im 0
2 , 03C6(0398, p)
is astrictly decreasing
functionof p
and03C1(0398, p)
is astrictly increasing
functionof p,
bounded below for7T
o ~
1m8 ~ - ;
and is aconvex-function,
with a minimum at4
As
(~ ~~
+/12)
lies in the second-sheet for p 5~ 0 and 0 1m e -,we have 2
thus,
the spectrum ofHo(0) is,
for0
Im 0-,
a continuous curve .starting
andtending asymptoticaly
to thestraight-line
The
proof for - 03C0 2
2 Im0398
0 is similar.Q.
E. D.Remark. - The spectrum of
Ho(8)
isindependent
ofRe(O)
becauseHo(81)
and areunitary equivalents
for ImOi
1 = Im02.
II THE CLASS
OF DILATATION ANALYTIC INTERACTIONS
We define a dilatation
analytic
interaction[1]
as asymmetric
andHo-compact operator [3]
Vhaving
thefollowing property:
thefamily
of operators
has an
Ho-compact analytic
continuation in an open connected domain 0 of thecomplex-plane (4).
We
consider,
now, the total Hamiltonianwhere V is a dilatation
analytic
interaction in thestrip Sa,
0 a-.
,(4) It is clear that the analyticity domain 0 of V(0) can always be extended to a complex strip Sa = {Z~C| |Im Z| a}, a > 0 [1].
Vol. XX, n° 2 - 1974.
216 R. A. WEDER
As V is
symmetric
andHo-compact,
H isselfadjoint,
boundedbelow,
and
!Ø(H) = !Ø(Ho) [3].
By
lemma1,
and the definition of dilatationanalytic interactions,
thefamily
ofoperators H(8),
defined for eby
has an extension to an
analytic
andselfadjoint family [3],
within the
strip Sb,
where b = min(~ - j .
LEMMA 3. The
multiplication operator (denoted V) by
the function- 0
{3 1, where g
is a constant, is dilatationanalytic
in theentire
complex-plane (5).
Proof.
- V is asymmetric operator [3];
andwhere
which has an
analytic
extension to the entirecomplex-plane.
Let us definethe
following operator
where(n
is a entirepositive number)
in the domain~(Vn(O)),
of all ’P EJf,
suchthat
[( - g(O)r - ~)n~’(r)]
isagain
in ~f.Then is
Ho-compact by
a theorem of[4].
e)
Clearly, V is defined as a multiplication operator in configuration-space that is tosay
on the domain, !Z’(V); of all BII in ~ such that ( - is again in ’P(r) is the Fourier
transform of the wave-function in momentum space
~’(p).
In momentum space V is anintegral operator.
(6) ~3 +«~~3)
is the Banach space of complex valued functions on [R3, such thatAnnales de l’Institut Henri Poincaré - Section A
SPECTRAL PROPERTIES OF ONE-BODY RELATIVISTIC SPIN-ZERO HAMILTONIANS 217
But as
a ~ n v v i .... ’ . ~ ’
Q. E. D.
We will now
study
thespectrum
of theoperators H(0),
0 E andthen make the transition to real 0. We note that the remark
following
lemma 2 is also valid for
H(0).
III.
SPECTRAL
PROPERTIES OF H =Ho
+ VLEMMA 4. The
spectrum
ofH(0)
with 0 Im8
b consists of:essential
spectrum : 6e(H(O))
=7(Ho(0)).
Real bound state
energies (o~(H(0))):
a bounded set ofisolated,
finite-dimensional, real-eigenvalues, independent
of8, with
as theonly possible
accumulation
point.
Non-real resonance
energies (~(H(0))/~(H(0))):
a bounded set ofnon-real
isolated,
finite-dimensionaleigenvalues,
contained in the sectorof the
complex-plane
boundedby (J1, oo)
and~(H(0)).
Theonly possible
accumulation
point
is ~c. Agiven
resonance energy isindependent
of 0 aslong
as itbelongs
to~(H(0))B~(H(e)).
For |03C6| > 1m 0! I
there existC(4))
> 0 such that for0
p 00.where
~,o
is the minimum of thespectrum (which
isindependent
of0).
Proof 2014 By
the second resolventequation [3J
for all Z E such that
exists.
But,
asV(0)
isH0(0398)-compact and ~V(0398)(H0(0398)
+ 1 forreal a and a > K >
0,
this holds for all Z Eexcept for,
at most, a set S of isolatedpoints [3]..
Let,
for ~, ES, P~
be theprojection operator
definedby [3]
where r is a circle
separating ~,
fromcr(H(0)) 2014 {/L}.
The first
integrand
isholomorphic
in Z =,1"
and the second compact;hence
P Â
is a compactoperator.
Vol. XX, n° 2 - 1974.
218 R. A. WEDER
Then, ~,
is an isolatedfinite-dimensional eigenvalue
ofH(8) [3]. By exchanging
the roles ofHo(8)
andH(8)
we can prove thatTake us
~,o
in the set~d(H(0o))
of isolated finite-dimensionaleigen- values
ofH(8o).
AsH(8)
is (inanalytic family
with spectrum constant for Im 8 constant,~,o
E(jd(H(é))
for 8 in aneighborhood
of80 [3].
Thusthe isolated finite-dimensional
eigenvalues
ofH(8)
can accumulateonly
at ~u; and the real bound-state
energies
areindependent
of8,
and the non-real resonance
energies
areindependent
of 8 aslong
asthey belong
to~(H(0))B~(H(0)),
and are contained in the sectorof
thecomplex-plane
bounded
by (J1, oo)
and(
The fact that the set
6d(H(O))
is bounded(and
thevalidity
of the estima-tion for the resolvent
given above)
can be proven in the same lines asin
[2],
then we will omit theproof
here.,
Q.
E. D.THEOREM 1. The
point spectrum
of H consists of a bounded set of finite-dimensionaleigenvalues
different from ~c(which
areprecisely
thereal
eigenvalues
ofH(8)
ImO ~ 0,
different from~c) accumulating
atmost at ,u, and
possibly
aneigenvalue
at ~c. Theprojection operators P(0, A),
e ESb,
on theeigenspace
ofH(8) corresponding
to a fixed-real-eigenvalue
03BB differentfrom
form aselfadjoint analytic family
inSb.
The
eigenvectors 03A6
of Hcorresponding
to sucheigenvalues (03BB)
are inthe dense set
~b
ofanalytic
vectors[5]
inSb,
and theanalytic
extentions~(0)
are
eigenvectors
ofH(8) corresponding
to ~,. Thesingular-continuous
spectrum is
empty,
i. e.Proo f :
be fixed vectors in the dense set~b ;
andand
’P(e)
theiranalytic
extentions.By
lemma 4 the functionis
analytic
in8,
for fixed Z such that Im Z > 0 and such thatSince
it follows that the
equality
holds for all e withNow we fix 8 with Im 8 >
0; by
lemma4, F~~(0, Z)
ismeromorphic
in Z for
Z ~ 03C3e(H(0398)),
then(I), (H - Z)-103A8)
has ameromorphic
continua-Annales de l’Institut Henri Poincaré - Section A
SPECTRAL PROPERTIES OF ONE-BODY RELATIVISTIC SPIN-ZERO HAMILTONIANS 219
tion from above
(Im
Z >0)
across the line[11, oo)
up to the curveLet us denote
by E(~.)
thespectral family
of H[3],
then we have thatwhere
This
implies (together
with a similar result for Im 0 0 and Im Z0)
that the
eigenvalues
ofH,
different from /1, areprecisely
the realeigenvalues
of
H(0),
ImO ~ 0,
different from and that the realpoles of(H(0) - Z)-1,
Im
0398 ~ 0,
differentfrom ,
aresimple. Then,
thepoint spectrum
of H is bounded andaccumulates,
at most, at ,u.Let, P:t(8, ~),
be theprojection operator [3]
on theeigenspace
ofH(8) corresponding
to aneigenvalue, ~,,
differentfrom /1 (+, - corresponds
to Im 8 > 0 and Im 0 0
respectively).
Setting P(À)
=E). - E~o,
we obtain as above :where we have used the fact that A is a
simple pole.
Then,
the function defined for1>, ’ by
is
analytic
inSb,
whereP(e~)=U(0)P(~)U(-e)
is theprojection operator
on theeigenspace
ofH(8) corresponding
to theeigenvalue
Afor 8 E [R.
Now,
it is not difficult to show[1]
the fact that thefamily P(8, x),
8 Ef~,
has ananalytic
extention inSb
whichequals P+(8, ~.) (resp. P"(0, ~,)
forIm 0 > 0 (rcsp. Im 0 0).
This
implies
that theeigenvalues
ofH,
different from /~ are finite-dimensional.
By
standardarguments [1]
it ispossible
to show that theeigenvectors 03A6
of H
corresponding
to sucheigenvalues, /.,
areanalytic
vectors inSb,
andthat their
analytic
extentions1>(0)
areeigenvectors
ofH(8)
with thesame
eigenvalue
/..Let å =
(a, b),
be an interval in(~c, oo)
which containsno-eigenvalue
of
H,
then for Im 0 > 0[3]
where
Vol. XX, n° 2 - 1974.
220 R. A. WEDER
The
integrand being analytic,
the function(0,
isabsolutely
conti-nuous on
A;
since~~
is dense in Jf we obtain that is to sayand also
ACKNOWLEDGEMENTS
I am very indebted to Professor A.
Verbeure,
whosuggested
thiswork,
for discussions and
help.
It is apleasure
to thank Professors J. P. Antoine and W. Amrein forenlightening
discussions and for their encouragement.We thank the referee for
pointing
out to us the formulation of Lemma 3 in its mostgeneral
form.[1] J. AGUILAR, J. M. COMBES, Comm. Math. Phys., t. 22, 1971, p. 269.
[2] E. BALSLEV, J. M. COMBES, Comm. Math. Phys., t. 22, 1971, p. 280.
[3] T. KATO, Perturbation theory for linear operators. Springer-Verlag, 1966.
[4] W. G. FARIS, Quadratic Forms and Essential Self-Adjointness. Helv. Phys. Acta,
t. 45, 7, 1973, p. 1074.
[5] E. NELSON, Analytic Vectors. Ann. Math., t. 70, 1959, p. 3.
[6] E. BALSLEV, Spectral Theory of Schrödinger Operators of Many-Body Systems. Lecture
notes, Leuven, 1971.
( Version revisee reçue ’ le 22 octobre 1973)
Annales de Henri Section A