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On the finiteness of the discrete spectrum of the Dirac operator
Vol. 57 (2006)
No. 3
REPORTS ON MATHEMATICAL PHYSICS
ON THE FINITENESS OF THE DISCRETE SPECTRUM OF THE DIRAC OPERATOR P. A. COJUHARI Department of Applied Mathematics, AGH University of Science and Technology, A1. Mickiewicza 30, 30-059 Cracow, Poland (e-mail: cojuhari @uci.agh.edu.pl) (Received June 6, 2005 - - Revised February 27, 2006)
New conditions for the finiteness of the discrete spectrum in the spectral gap of the perturbed Dirac operator are established. Keywords: Dirac operators, spectral theory, relatively compact perturbation. Mathematics Subject Classification: Primary 35P05, 47F05; Secondary 47A55, 47A75.
1.
Introduction The present paper is concerned with a spectral problem for the Dirac operator 3 H = ZotjDj
+ d o + Q,
(1.1)
j=l
where Dj = i ~axj, x = ( x l , x 2 ,
x3)~3,
and % ( j = 1 , 2 , 3 )
and or0 are the Dirac
matrices, i.e. 4 × 4 Hermitian matrices which satisfy the anticommutative relations
otjOlk + OtkOtj = 26jk 14 Here ~jk 14 is the In the 0, 1, 2, 3)
(j, k = 0, 1, 2, 3).
denotes the Kronecker symbol (rjk = 1 if j = k and 3jk = 0 if j ~ k), 4-dimensional unit matrix. standard representation [1, 8, 13] (see also [15]) the matrices a j (j = are chosen as follows: O~j =
/° o-j
for j = 1, 2, 3; 0
or0 =
°/ -12
where
oil , [333]
,
(1.2)
334
p.A. COJUHARI
are the Pauli matrices (12 denotes the 2 × 2 identity matrix). Q is considered as a perturbation of the free Dirac operator (which describes the free electron in relativistic quantum mechanics) 3
H0 = )__~otjDj +elo
(1.3)
j=l and represents the operator of multiplication by a given 4 x 4 Hermitian matrix-valued function Q(x), x E R 3. In accordance with our interests we assume that the entries of Q(x) are measurable functions from the space Lec(I~3). The operators H and H0 are considered in the space L2(R3; C 4) with their maximal domains of definition. Namely, it is considered that the domain of H0 is the Sobolev space WI(N3; C a ) and, due to the fact that the operator Q is supposed to be bounded, the perturbed operator H is defined on the same domain W1(]~3; C 4) as well. Then, the Dirac operators H0 and H are self-adjoint, the spectrum of the unperturbed operator /4o is only absolutely continuous and covers the set or(H0)= ( - e c , - 1 ] U [1, +oc). In addition, due to Weyl type theorems, if the entries of the matrix-valued function Q(x) vanish at infinity rapidly enough, the continuous spectrum of the perturbed Dirac operator H = H0 ÷ Q is identical with that of /4o and the perturbation Q can excite in the spectral gap ( - 1 , 1) a nontrivial set of eigenvalues with their possible points of accumulation only at the end points - 1 or 1. Our aim is to establish conditions for the finiteness of the created discrete spectrum in the gap ( - 1 , 1). The first results concerning this problem were established by M. S. Birman in [3] (see also [2]). However, the problem was treated mostly for the case of an electrical potential, i.e. while Q(x) is of the form Q(x) = q(x)I4. It is proved in [3] that if the potential q(x) belongs to the class Q1, that means that
p
f:
Iq(r, co)[dr
) O,
as p > ec ((p, co) designates the spherical coordinates of the point x), then the spectrum of the Dirac operator H in the interval ( - 1 , 1) is finite (in the sense that it consists only of a finite number of eigenvalues of finite multiplicity). As it has been shown in [12] (using the same methods developed by M. S. Birman in [3]), this kind of conditions can be considered in a sense to be exact. In our case, the perturbation Q is generated by a Hermitian matrix-valued function Q(x) of a general structure. By this the results of the present paper are extended in particular for the case of the operator corresponding to a Dirac particle in an electromagnetic field. The Dirac operator for this case is typically written in the physics literature (see, for instance, [8, 9, 15]) as 3 Hu = E olj (Dj - A j (x))u ÷ OtoU ÷ q (x)u, j=l
u
C W 1 (I~3;
ca),
(1.4)
ON THE FINITENESS OF THE DISCRETE SPECTRUM OF THE DIRAC OPERATOR
335
where A(x) = ( A l ( x ) , A z ( x ) , A3(x)) (the vector potential) and q(x) (the scalar potential) are given functions on ]R3. Moreover, the case of a scalar potential of the form Q ( x ) = m(x)o~o, where m(x) is a real-valued function of x 6 IR3, i.e. the case of the perturbed operator of the form 3
H
=
EotjDj
+Or0 +m(x)oto,
j=l
is covered as well. As usual 1 + m ( x ) is interpreted like an x-dependent rest mass of the particle (see [14], and references therein). It turns out that the general case gives some new effects (in this respect see Theorem 2.1 below). In particular, the conditions on the magnetic field look to be weaker than those required from the electric field. For instance, in the case of the Dirac operator (1.4) under the conditions IAj(x)[ < C(1 + Ixl) -1-a
(j = 1, 2, 3),
Iq(x)l < C(1 + Ixl) -2-*, for some ~ > 0, the discrete spectrum of the operator H is finite. The setting of the problem is based on the technique of perturbation theory. The abstract results obtained in the framework of those developed in [6] (and for related results see also [3, 5, 11]) play a crucial role in establishing our main results. For convenience we cite the corresponding abstract result. Let 7-(, "7-/1, ~t-~2 be Hilbert spaces. C(~-{l, 7-/2) (resp. /3(7-/1, J"~2)) stands for the set of all densely defined and closed (resp. bounded) linear operators from 7-/1 into 7-/2. We write C(7-/) for C(7-(, 7-(), and similarly define /3(7-/). The domain of an operator A is denoted by Dom(A). In the case of a self-adjoint operator A we say that an open interval A of real axis is a gap in the spectrum of A (or, simply, a spectral gap of A) if A C p(A). THEOREM 1.1. Let A and B be linear operators in 7-[ such that the following conditions are satisfied: (1) A is a self-adjoint operator, and the interval A = (a, b) is a spectral gap
of A. (2))~ = a is not an eigenvalue of A. (3) The operator B is represented as B = S*TS, where S ~ C(~, ~1), Dom(S) D Dora(A) and T ~ /3(7-/1). (4) The operator ] A - all-l/2S* is densely defined and admits a bounded extension ~S*. If B = S*T~S is a compact operator in ~ , then the operator A + B has a self-adjoint extension A1, the spectrum of A1 on A is only discrete and a is not an accumulation point for the set cr(A]) (3 A.
336 2.
P. A. COJUHAR1 Main results
Let H be the Dirac operator defined by (1.1) in which the matrices a j (j = 0, 1, 2, 3, ) are chosen as in (1.2). It will be convenient to divide the perturbation matrix Q(x) into 2 x 2 blocks as Q ( x ) = [Qjk(X)]~,k= 1. The first block Qll(X) is formed by elements qjk(x) (j, k = 1, 2), the block Q:2(x) by elements qjk(x) (j, k = 3,4), Q12(x) and Q21(x), where Q12(x) : Q~l(x), from the remaining elements. In what follows we assume that the elements of the perturbation matrix Q(x) are L~-functions and qjk(X) > 0 as Ixl > oc. We denote by L" I the standard matrix norm. Our main results are contained in the following theorem. THEOREM 2.1. Let H be a perturbed Dirac operator defined by (1.1). Under the conditions IQu(x)[ =
O(Ixl-2-~),
IQj~(x)l = O([x[ -1-~)
Q22(x)l = (j ¢:k;
O(Ixl-~),
j , k = 1,2),
(2.1)
for some ~ > 0 as Ix[ ~ oc, the spectrum of H on the spectral gap ( - 1 , 1) is only discrete and 1 is not a point of accumulation of or(H) fq ( - 1 , 1). If instead of (2.1) one requires that ]QII(X)[ :
O(Ixl- ),
IQj~(x)l = O(Ix1-1-~)
lQ22(x)[ = O([xl-Z-~), (j •k;
j , k = 1,2),
(2.2)
for some 8 > 0 as Ix l ~ oo, then the spectrum of H in the spectral gap ( - 1 , 1) is also discrete for which - 1 is not the point of accumulation. In both cases each eigenvalue is of finite multiplicity. The conditions (2.1) and (2.2) taken together guarantee the finiteness of the spectrum on the interval ( - 1 , 1). COROLLARY 2.2.
if Iajj(x)l-- O([x[-2-6), for some ~ > 0 when Ixl is finite.
IOjk(x)[ = O([x[ -1-8)
(j # k ;
j,k=
1,2),
> oo, then the spectrum of H on the interval ( - 1 , 1)
In the particular case of the Dirac operator defined by formula (1.4) the following result is true. THEOREM 2.3.
If [Aj(x)l <_ C(1 + Ixl) -l-~,
Iq(x)l _< C(1 + Ixl) -z-a,
for some ~ > 0 and for some positive constant C, then the spectrum of the Dirac operator defined by (1.4) on the interval ( - 1 , 1) is finite.
ON THE FINITENESS OF THE DISCRETE SPECTRUM OF THE DIRAC OPERATOR
3.
337
Proofs of the main results
It is sufficient to prove the assertion concerning the endpoint X = 1. The arguments are similar for the other case of the endpoint ;v = - 1 . The proof will base on the abstract results presented by Theorem 1.1 from Introduction. In our case A is equal to the free Dirac operator /4o and in the capacity of B is taken Q. As it was already mentioned, the unperturbed operator /40 is self-adjoint and has a gap ( - 1 , 1) in its spectrum. Moreover, ), = 1 is not an eigenvalue of H0. So, the conditions (l) and (2) of Theorem 1.1 are satisfied automatically. Next, the perturbed operator Q will be factorised in such a way that the conditions (3) and (4) of Theorem 1.1 will be also fulfilled. To this end, we describe first the structure of the inverse operator of H 0 - I (which is unbounded, of course). H0 represents a matrix differential operator (of the dimension 4 x 4) of order 1. The symbol of H0 is the following matrix-valued function ho(~) =
-
a(~)
-12
where
a(~) = a ~
+ a2~2 + a3~3,
= (~, ~2, ~3) s R 3.
Note that by applying the Fourier transformation to the elements of the space L2(~3;C4), /4o is transformed (in the momentum space) into a multiplication operator by the matrix-valued function h0(~). The Fourier transformation is defined by the formula ff(~) =
(Fu)(~)
--
(27r)3/~
' f
ei(X'~)u(x)dx
(U E
L2(]~3)),
in which Ix, ~) denotes the scalar product of the elements x, ~ ~ R 3 (here and in what follows f := fR3). It is well known that eigenvalues of the matrix h0(~) are the following )Vl(~) = )v2(~) =
r(~),
)v3(~) =
X4(~) = - r ( ~ ) ,
where r ( ~ ) = (1 + [~[2)1/2. The corresponding normalized eigenelements are given by wj(~) = a(~)uj(~) (j --- 1, 2, 3, 4), where a(~) = (½(1 + r(~)-l))½, and
ul(~) =t (e21, (1 + r(~))-la(8)e~), u3(~) =t ( - ( 1 + r(8))-la(~)e~, e~),
u2(~) =t (e 2, (1 + r(~))-~a(~)e2), u4(~ ) =t ( - ( 1 -~- r(~))-la(~)e
2, e2).
Here by e~ and e 2 are denoted the vectors: e~ =t (1, 0), e~ =t (0, 1).
338
p.A. COJUHARI
Next, we consider the matrix
U(~)= [
a(~)I2 -b(~)a(~)
b(~)g(~)], a(~)I2 _]
where b (~) = a ($) (1 + r (~)) -1, composed from the coordinates of the eigenelements coj(~) (j = 1,2, 3,4). U(~) brings the matrix h0(~) to the diagonal form, i.e.
U(~)ho(~)U(~)* = a0r(~).
(3.1)
Now, let S(~) = [d1(~)I20
d2(~)I20 1 U(~),
where d l ( ~ ) = ( r ( ~ ) - 1) 1/2 and d 2 ( ~ ) = ( r ( ~ ) + 1) 1/2, and denote by S = S(D) a pseudodifferential operator corresponding to its symbol S(~). The operator S is defined in the space L2(]~3; C 4) b y
(Su)(x)
-
-
(2~)3/2
e-i(x'~)S(~)~ff(~)d~,
x E R 3,
on the domain D(S)
= {u ~ L2(IR3; C 4) I S(~)~'(~) E L2(~3; C4)}.
Similarly, we define the operators U = U(D), dj(D) (j = 1, 2), and denote R=
0
d2(D)I2 Obviously, S = RU and u ~ D(S) if and only if fi" E L2,w(~3; C4), where L2,w(R3; C 4) stands for the space weighted by w(~) = (1 + }~12)1/4, i.e. the space of all functions f E L2(]~3; C 4) such that wf c L2(R3; C4). Note that F*L2,w(R3; C 4) ----- W21/2(~3; C 4) D W21(R3; C 4) (the Fourier transformation in the space L2(~.3; C 4) is again denoted by F). Thus, we see that D(S) D Dora(H0)
(= wl(]~3; C4)).
It follows from (3.1) that
Ho - I = U*Ro~oRU, and since IRaoRIl/2-= R, we see immediately that IH0 - I[ t/2 = U*RU. Then, for each v E Dora(R) we obtain Sv* := IHo - II-1/2S*v = U*R-1U U*Rv -= U'v,
ON THE FINITENESS OF THE DISCRETE SPECTRUM OF THE DIRAC OPERATOR
i.e. the operator ] H o (= U*). Now, we write
339
II-I/2S* is densely defined and admits a bounded extension Q = S*[S*-1QS-1]S,
and denote To = S*-IQS -1. It turns out that the operator To, due to the hypothesis on the elements of the matrix Q(x), is densely defined (in fact, by arguments of Remark 4.2 To is defined on the range Ran(S) of the operator S), and as it will be proved, it admits a compact extension T on L2(R3; C4). In order to show this fact, denote by a(D), b(D) and a(D) pseudodifferential operators with symbols a(~), b(~) and a(~), respectively (in fact, a(D) is a 2× 2 matrix differential operator) and observe that the pseudodifferential operators a(D) and b(D)a(D) are bounded (because of their symbols a(~) and b(~)a(~) constitute bounded functions on ~3). Let Q = UQU*. It is clear that Q ~ /3(L2(~3; C4)) (U = U(D2 is a unitary 2 operator). It will be convenient to divide Q into 2 x 2 blocks as Q = [Qjk]j,k=l. The operator To is obtained from Q by multiplication concomitantly from the right and from the left by the diagonal operator
d2(D)-lh As a result, the block C)11 is multiplied from the left and the right by dl(D) -1, the block Q12 ~ multiplied from the left by dl(D) -1 and from the right by dz(D) -1, the block Q21--from the left by d2(D) -1 and the right by dl(D) -1, and finaly the block Q22 is multiplied from both sides by dz(D) -1. Thus, after all of these calculation, we conclude that To admits a compact extension to the whole space if a similar property is required from each of the following operators:
dI(D)-la(D)Qlla(D)dl(D) -1, dl(D)-IQllb(D)a(D)d2(D) -l , dz(D)-lb(D)a(D)Q11b(D)a(D)d2(O) -1, dl(D)-la(D)Q12b(D)a(O)dl(D) -1, dl(D)-lQlza(D)d2(D) -1, d2(D)-lb(D)a(D)Qlzb(D)o(D)d1(D) -1, dz(D)-lb(D)cr(D)Qlza(D)dz(D) -1, d1(D)-lb(D)a(D)Qzzb(D)a(D)dl(D) -1, dl(D)-lb(D)a(D)Qz2a(D)d2(D) -1, d2(D)-1a(D)Qz2a(D)d2(D) -1. Note that the blocks of the operator T are represented as sums of such operators taken together with their adjoint operators. Since a(D), b(D)a(D), b(D)a(D)dl (D) -1 and d2(D) -1 are bounded operators, it is clear that the latter is guaranteed if in turn the operators
dl(D)-llqjk[ 1/2 (j, k = 1, 2), d2(D)-l[qjk[ 1/2 (j, k = 3, 4), dl(D)-llqjkl (j = l, 2; k = 3, 4; or j = 3 , 4 ; k = l , 2 ) , or, equivalently, the integral operators with the kernels
[~[-llqjk(x)ll/Zeilx'~)
(j, k --- 1, 2),
340
P.A. COJUHARI (r(~) + 1)-l/21qjt(x)lUZei(X'~) I~l-Xlqjk(x)le ~
(j, k = 3, 4),
(j = 1 , 2 ; k = 3,4 or j = 3 , 4 ; k = 1,2),
(3.2)
are compact in the space L2(I[~.3). The compactness criteria of such type integral operators are well known [4] (see also Lemma from [6], p. 45). By virtue of the results obtained in [4], conditions (2.1) guarantee compactness of the integral operators with kernels g~en b y (3.2). Thus, conditions (2.1) imply also the compactness of To. Therefore B = S*TS is a compact operator in the space L2(•3; C 4) and so all assumptions of Theorem 1.1 are satisfied. The proof is complete. [] 4.
Remarks
1. In [4] there are established conditions under which an operator of the form b ( X ) a ( D ) (X denotes the family of operators of multiplication by coordinates xj (j = 1, 2, 3) in R 3) belongs to the von Neumann-Schatten class of compact operators /3p. In concordance with these results it is possible to calculate the ideal-norm I[" lip of the operator T defined as in the proof of Theorem 2.1 by factorizing the perturbation operator Q as Q = S*TS. Note that the operator S is chosen such that the densely defined operator S ( H - l ) - l S * has a bound extension of which norm (of the space Lz(R3; C4)) is equal to 1. Thus, the abstract results from [7] concerning estimation of the number of eigenvalues excited by a perturbation in a given spectral gap can be applied. On the problems connected with the estimation of the discrete spectrum for Dirac type operators together with some asymptotic formulae related with those obtained by M. Klaus in [10] the author hopes to undertake discussions in a separate publication. 2. In the proof of Theorem 2.1 it was substantially used the following fact. Let be a Hilbert space, and let A and B denote densely defined operators in 7-/. The operator A is assumed to be self-adjoint and A > 0. In addition, 0 6 cr(A)\cyp(A), i.e. A is invertible but the inverse A -1 represents an unbounded operator. It is affirmed that if A - 1 B is densely defined and possesses a bounded extension, then the domains of the operators A - I B and B coincide. Indeed, let ~. denote an arbitrary point on the negative semi-axis, and denote R0~; A ) = ( A - ~.I) -l . One has R()~; A ) B u - A - 1 B u = ~R(L; A)A 1Bu for each u from the domain of A-1B, i.e. u ~ D ( A - 1 B ) . Since D ( A - 1 B ) = 7-[, it follows [R(~.; A)B] - [A-1B] = ~.R(~.; A)[A-1B], where IT] denotes complete extension of a densely defined and bounded operator T. Since )~R()~; A) converges strongly to 0, it follows that R(~.; A ) B u > [A-IB]u for each u ~ D(B). On the other hand, due to the relation AR(£; A ) B : B + )~R(X; A)B, one obtains AR(~.; A ) B u > Bu for each u ~ D(B). Consequently, since A is a closed operator, one can conclude that [A-1B]u ~ D(A) and A[A-1B]u = Bu for
ON THE FINITENESS OF THE DISCRETE SPECTRUM OF THE DIRAC OPERATOR
341
each u ~ D(B). Therefore, Bu belongs to the range of A and A-I(Bu) = [ A - 1 B ] u , and so D ( B ) C D(A-1B). Since the opposite inclusion is evident our assertion follows. REFERENCES [1] H. A. Bethe and E. E. Salpeter: Quantum Mechanics of One- and Two- Electron Atoms, Academic Press, New York 1959. [2] M. S. Birman: On the spectrum of Schr6dinger and Dirac operators, Dokl. Acad. Nauk 129, no. 2, (1959), 239-241. [3] M. S. Birman, On the spectrum of singular boundary-value problems, Mat. Sb. 55, no. 2, (1961), 125-174. [4] M. S. Birman, G. E. Karadzhov snd M. Z. Solomyak: Boundedness conditions and spectrum estimates for the operators b(X)a(D) and their analogs, Adv. Soviet. Math. 7 (1991), 85-106. [5] P. A. Cojuhari: On the finiteness of the discrete spectrum of some pseudodifferential operators, Mat. Issled. 80 (1985), 85-97. [6] P. A. Cojuhari: On the finiteness of the discrete spectrum of some matrix pseudodifferential operators, Vyssh. Uchebn. Zaved. Mat. 1 (1989), 42-50. [7] P. A. Cojuhari: Estimates of the number of perturbed eigenvalues, Operator Theory, Operator Algebras and Related Topics in Proc. of OT 17 Conf. (2000), 97-111. [8] V. A. Fock: Introduction to Quantum Mechanics, Nauka, Moscow 1976. [9] W. Greiner: Relativistic Quantum Mechanics: Wave Equations, Springer, Berlin-Heidelberg-New York 1994. [10] M. Klaus: On the point spectrum of Dirac operators, Helv. Phys. Acta. 53 (1980), 453--462. [11] R. Konno and S. T. Kuroda: On the finiteness of perturbed eigenvalues, J. Fac. Sci. Univ. To~,o Sect. IA Math. 13 (1966), 55~53. [12] O. I. Kurbenin: The discrete spectra of the Dirac and Pauli operators, Topics in Mathematical Physics, vol. 3, Spectral Theory (1969), 43-52. [13] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (nonrelativistic theoo'), Pergamon Press. Oxford, and Addison-Wesley, Reading, MA 1965. [14] C. Pladdy: Asymptotics of the resolvent of the Dirac operator with a scalar short-range potential, Analysis 21 (2001), 79-97. [15] B. Thaller: The Dirac Equation, Springer, Berlin-Heidelberg-New York 1992.
No. 3
REPORTS ON MATHEMATICAL PHYSICS
ON THE FINITENESS OF THE DISCRETE SPECTRUM OF THE DIRAC OPERATOR P. A. COJUHARI Department of Applied Mathematics, AGH University of Science and Technology, A1. Mickiewicza 30, 30-059 Cracow, Poland (e-mail: cojuhari @uci.agh.edu.pl) (Received June 6, 2005 - - Revised February 27, 2006)
New conditions for the finiteness of the discrete spectrum in the spectral gap of the perturbed Dirac operator are established. Keywords: Dirac operators, spectral theory, relatively compact perturbation. Mathematics Subject Classification: Primary 35P05, 47F05; Secondary 47A55, 47A75.
1.
Introduction The present paper is concerned with a spectral problem for the Dirac operator 3 H = ZotjDj
+ d o + Q,
(1.1)
j=l
where Dj = i ~axj, x = ( x l , x 2 ,
x3)~3,
and % ( j = 1 , 2 , 3 )
and or0 are the Dirac
matrices, i.e. 4 × 4 Hermitian matrices which satisfy the anticommutative relations
otjOlk + OtkOtj = 26jk 14 Here ~jk 14 is the In the 0, 1, 2, 3)
(j, k = 0, 1, 2, 3).
denotes the Kronecker symbol (rjk = 1 if j = k and 3jk = 0 if j ~ k), 4-dimensional unit matrix. standard representation [1, 8, 13] (see also [15]) the matrices a j (j = are chosen as follows: O~j =
/° o-j
for j = 1, 2, 3; 0
or0 =
°/ -12
where
oil , [333]
,
(1.2)
334
p.A. COJUHARI
are the Pauli matrices (12 denotes the 2 × 2 identity matrix). Q is considered as a perturbation of the free Dirac operator (which describes the free electron in relativistic quantum mechanics) 3
H0 = )__~otjDj +elo
(1.3)
j=l and represents the operator of multiplication by a given 4 x 4 Hermitian matrix-valued function Q(x), x E R 3. In accordance with our interests we assume that the entries of Q(x) are measurable functions from the space Lec(I~3). The operators H and H0 are considered in the space L2(R3; C 4) with their maximal domains of definition. Namely, it is considered that the domain of H0 is the Sobolev space WI(N3; C a ) and, due to the fact that the operator Q is supposed to be bounded, the perturbed operator H is defined on the same domain W1(]~3; C 4) as well. Then, the Dirac operators H0 and H are self-adjoint, the spectrum of the unperturbed operator /4o is only absolutely continuous and covers the set or(H0)= ( - e c , - 1 ] U [1, +oc). In addition, due to Weyl type theorems, if the entries of the matrix-valued function Q(x) vanish at infinity rapidly enough, the continuous spectrum of the perturbed Dirac operator H = H0 ÷ Q is identical with that of /4o and the perturbation Q can excite in the spectral gap ( - 1 , 1) a nontrivial set of eigenvalues with their possible points of accumulation only at the end points - 1 or 1. Our aim is to establish conditions for the finiteness of the created discrete spectrum in the gap ( - 1 , 1). The first results concerning this problem were established by M. S. Birman in [3] (see also [2]). However, the problem was treated mostly for the case of an electrical potential, i.e. while Q(x) is of the form Q(x) = q(x)I4. It is proved in [3] that if the potential q(x) belongs to the class Q1, that means that
p
f:
Iq(r, co)[dr
) O,
as p > ec ((p, co) designates the spherical coordinates of the point x), then the spectrum of the Dirac operator H in the interval ( - 1 , 1) is finite (in the sense that it consists only of a finite number of eigenvalues of finite multiplicity). As it has been shown in [12] (using the same methods developed by M. S. Birman in [3]), this kind of conditions can be considered in a sense to be exact. In our case, the perturbation Q is generated by a Hermitian matrix-valued function Q(x) of a general structure. By this the results of the present paper are extended in particular for the case of the operator corresponding to a Dirac particle in an electromagnetic field. The Dirac operator for this case is typically written in the physics literature (see, for instance, [8, 9, 15]) as 3 Hu = E olj (Dj - A j (x))u ÷ OtoU ÷ q (x)u, j=l
u
C W 1 (I~3;
ca),
(1.4)
ON THE FINITENESS OF THE DISCRETE SPECTRUM OF THE DIRAC OPERATOR
335
where A(x) = ( A l ( x ) , A z ( x ) , A3(x)) (the vector potential) and q(x) (the scalar potential) are given functions on ]R3. Moreover, the case of a scalar potential of the form Q ( x ) = m(x)o~o, where m(x) is a real-valued function of x 6 IR3, i.e. the case of the perturbed operator of the form 3
H
=
EotjDj
+Or0 +m(x)oto,
j=l
is covered as well. As usual 1 + m ( x ) is interpreted like an x-dependent rest mass of the particle (see [14], and references therein). It turns out that the general case gives some new effects (in this respect see Theorem 2.1 below). In particular, the conditions on the magnetic field look to be weaker than those required from the electric field. For instance, in the case of the Dirac operator (1.4) under the conditions IAj(x)[ < C(1 + Ixl) -1-a
(j = 1, 2, 3),
Iq(x)l < C(1 + Ixl) -2-*, for some ~ > 0, the discrete spectrum of the operator H is finite. The setting of the problem is based on the technique of perturbation theory. The abstract results obtained in the framework of those developed in [6] (and for related results see also [3, 5, 11]) play a crucial role in establishing our main results. For convenience we cite the corresponding abstract result. Let 7-(, "7-/1, ~t-~2 be Hilbert spaces. C(~-{l, 7-/2) (resp. /3(7-/1, J"~2)) stands for the set of all densely defined and closed (resp. bounded) linear operators from 7-/1 into 7-/2. We write C(7-/) for C(7-(, 7-(), and similarly define /3(7-/). The domain of an operator A is denoted by Dom(A). In the case of a self-adjoint operator A we say that an open interval A of real axis is a gap in the spectrum of A (or, simply, a spectral gap of A) if A C p(A). THEOREM 1.1. Let A and B be linear operators in 7-[ such that the following conditions are satisfied: (1) A is a self-adjoint operator, and the interval A = (a, b) is a spectral gap
of A. (2))~ = a is not an eigenvalue of A. (3) The operator B is represented as B = S*TS, where S ~ C(~, ~1), Dom(S) D Dora(A) and T ~ /3(7-/1). (4) The operator ] A - all-l/2S* is densely defined and admits a bounded extension ~S*. If B = S*T~S is a compact operator in ~ , then the operator A + B has a self-adjoint extension A1, the spectrum of A1 on A is only discrete and a is not an accumulation point for the set cr(A]) (3 A.
336 2.
P. A. COJUHAR1 Main results
Let H be the Dirac operator defined by (1.1) in which the matrices a j (j = 0, 1, 2, 3, ) are chosen as in (1.2). It will be convenient to divide the perturbation matrix Q(x) into 2 x 2 blocks as Q ( x ) = [Qjk(X)]~,k= 1. The first block Qll(X) is formed by elements qjk(x) (j, k = 1, 2), the block Q:2(x) by elements qjk(x) (j, k = 3,4), Q12(x) and Q21(x), where Q12(x) : Q~l(x), from the remaining elements. In what follows we assume that the elements of the perturbation matrix Q(x) are L~-functions and qjk(X) > 0 as Ixl > oc. We denote by L" I the standard matrix norm. Our main results are contained in the following theorem. THEOREM 2.1. Let H be a perturbed Dirac operator defined by (1.1). Under the conditions IQu(x)[ =
O(Ixl-2-~),
IQj~(x)l = O([x[ -1-~)
Q22(x)l = (j ¢:k;
O(Ixl-~),
j , k = 1,2),
(2.1)
for some ~ > 0 as Ix[ ~ oc, the spectrum of H on the spectral gap ( - 1 , 1) is only discrete and 1 is not a point of accumulation of or(H) fq ( - 1 , 1). If instead of (2.1) one requires that ]QII(X)[ :
O(Ixl- ),
IQj~(x)l = O(Ix1-1-~)
lQ22(x)[ = O([xl-Z-~), (j •k;
j , k = 1,2),
(2.2)
for some 8 > 0 as Ix l ~ oo, then the spectrum of H in the spectral gap ( - 1 , 1) is also discrete for which - 1 is not the point of accumulation. In both cases each eigenvalue is of finite multiplicity. The conditions (2.1) and (2.2) taken together guarantee the finiteness of the spectrum on the interval ( - 1 , 1). COROLLARY 2.2.
if Iajj(x)l-- O([x[-2-6), for some ~ > 0 when Ixl is finite.
IOjk(x)[ = O([x[ -1-8)
(j # k ;
j,k=
1,2),
> oo, then the spectrum of H on the interval ( - 1 , 1)
In the particular case of the Dirac operator defined by formula (1.4) the following result is true. THEOREM 2.3.
If [Aj(x)l <_ C(1 + Ixl) -l-~,
Iq(x)l _< C(1 + Ixl) -z-a,
for some ~ > 0 and for some positive constant C, then the spectrum of the Dirac operator defined by (1.4) on the interval ( - 1 , 1) is finite.
ON THE FINITENESS OF THE DISCRETE SPECTRUM OF THE DIRAC OPERATOR
3.
337
Proofs of the main results
It is sufficient to prove the assertion concerning the endpoint X = 1. The arguments are similar for the other case of the endpoint ;v = - 1 . The proof will base on the abstract results presented by Theorem 1.1 from Introduction. In our case A is equal to the free Dirac operator /4o and in the capacity of B is taken Q. As it was already mentioned, the unperturbed operator /40 is self-adjoint and has a gap ( - 1 , 1) in its spectrum. Moreover, ), = 1 is not an eigenvalue of H0. So, the conditions (l) and (2) of Theorem 1.1 are satisfied automatically. Next, the perturbed operator Q will be factorised in such a way that the conditions (3) and (4) of Theorem 1.1 will be also fulfilled. To this end, we describe first the structure of the inverse operator of H 0 - I (which is unbounded, of course). H0 represents a matrix differential operator (of the dimension 4 x 4) of order 1. The symbol of H0 is the following matrix-valued function ho(~) =
-
a(~)
-12
where
a(~) = a ~
+ a2~2 + a3~3,
= (~, ~2, ~3) s R 3.
Note that by applying the Fourier transformation to the elements of the space L2(~3;C4), /4o is transformed (in the momentum space) into a multiplication operator by the matrix-valued function h0(~). The Fourier transformation is defined by the formula ff(~) =
(Fu)(~)
--
(27r)3/~
' f
ei(X'~)u(x)dx
(U E
L2(]~3)),
in which Ix, ~) denotes the scalar product of the elements x, ~ ~ R 3 (here and in what follows f := fR3). It is well known that eigenvalues of the matrix h0(~) are the following )Vl(~) = )v2(~) =
r(~),
)v3(~) =
X4(~) = - r ( ~ ) ,
where r ( ~ ) = (1 + [~[2)1/2. The corresponding normalized eigenelements are given by wj(~) = a(~)uj(~) (j --- 1, 2, 3, 4), where a(~) = (½(1 + r(~)-l))½, and
ul(~) =t (e21, (1 + r(~))-la(8)e~), u3(~) =t ( - ( 1 + r(8))-la(~)e~, e~),
u2(~) =t (e 2, (1 + r(~))-~a(~)e2), u4(~ ) =t ( - ( 1 -~- r(~))-la(~)e
2, e2).
Here by e~ and e 2 are denoted the vectors: e~ =t (1, 0), e~ =t (0, 1).
338
p.A. COJUHARI
Next, we consider the matrix
U(~)= [
a(~)I2 -b(~)a(~)
b(~)g(~)], a(~)I2 _]
where b (~) = a ($) (1 + r (~)) -1, composed from the coordinates of the eigenelements coj(~) (j = 1,2, 3,4). U(~) brings the matrix h0(~) to the diagonal form, i.e.
U(~)ho(~)U(~)* = a0r(~).
(3.1)
Now, let S(~) = [d1(~)I20
d2(~)I20 1 U(~),
where d l ( ~ ) = ( r ( ~ ) - 1) 1/2 and d 2 ( ~ ) = ( r ( ~ ) + 1) 1/2, and denote by S = S(D) a pseudodifferential operator corresponding to its symbol S(~). The operator S is defined in the space L2(]~3; C 4) b y
(Su)(x)
-
-
(2~)3/2
e-i(x'~)S(~)~ff(~)d~,
x E R 3,
on the domain D(S)
= {u ~ L2(IR3; C 4) I S(~)~'(~) E L2(~3; C4)}.
Similarly, we define the operators U = U(D), dj(D) (j = 1, 2), and denote R=
0
d2(D)I2 Obviously, S = RU and u ~ D(S) if and only if fi" E L2,w(~3; C4), where L2,w(R3; C 4) stands for the space weighted by w(~) = (1 + }~12)1/4, i.e. the space of all functions f E L2(]~3; C 4) such that wf c L2(R3; C4). Note that F*L2,w(R3; C 4) ----- W21/2(~3; C 4) D W21(R3; C 4) (the Fourier transformation in the space L2(~.3; C 4) is again denoted by F). Thus, we see that D(S) D Dora(H0)
(= wl(]~3; C4)).
It follows from (3.1) that
Ho - I = U*Ro~oRU, and since IRaoRIl/2-= R, we see immediately that IH0 - I[ t/2 = U*RU. Then, for each v E Dora(R) we obtain Sv* := IHo - II-1/2S*v = U*R-1U U*Rv -= U'v,
ON THE FINITENESS OF THE DISCRETE SPECTRUM OF THE DIRAC OPERATOR
i.e. the operator ] H o (= U*). Now, we write
339
II-I/2S* is densely defined and admits a bounded extension Q = S*[S*-1QS-1]S,
and denote To = S*-IQS -1. It turns out that the operator To, due to the hypothesis on the elements of the matrix Q(x), is densely defined (in fact, by arguments of Remark 4.2 To is defined on the range Ran(S) of the operator S), and as it will be proved, it admits a compact extension T on L2(R3; C4). In order to show this fact, denote by a(D), b(D) and a(D) pseudodifferential operators with symbols a(~), b(~) and a(~), respectively (in fact, a(D) is a 2× 2 matrix differential operator) and observe that the pseudodifferential operators a(D) and b(D)a(D) are bounded (because of their symbols a(~) and b(~)a(~) constitute bounded functions on ~3). Let Q = UQU*. It is clear that Q ~ /3(L2(~3; C4)) (U = U(D2 is a unitary 2 operator). It will be convenient to divide Q into 2 x 2 blocks as Q = [Qjk]j,k=l. The operator To is obtained from Q by multiplication concomitantly from the right and from the left by the diagonal operator
d2(D)-lh As a result, the block C)11 is multiplied from the left and the right by dl(D) -1, the block Q12 ~ multiplied from the left by dl(D) -1 and from the right by dz(D) -1, the block Q21--from the left by d2(D) -1 and the right by dl(D) -1, and finaly the block Q22 is multiplied from both sides by dz(D) -1. Thus, after all of these calculation, we conclude that To admits a compact extension to the whole space if a similar property is required from each of the following operators:
dI(D)-la(D)Qlla(D)dl(D) -1, dl(D)-IQllb(D)a(D)d2(D) -l , dz(D)-lb(D)a(D)Q11b(D)a(D)d2(O) -1, dl(D)-la(D)Q12b(D)a(O)dl(D) -1, dl(D)-lQlza(D)d2(D) -1, d2(D)-lb(D)a(D)Qlzb(D)o(D)d1(D) -1, dz(D)-lb(D)cr(D)Qlza(D)dz(D) -1, d1(D)-lb(D)a(D)Qzzb(D)a(D)dl(D) -1, dl(D)-lb(D)a(D)Qz2a(D)d2(D) -1, d2(D)-1a(D)Qz2a(D)d2(D) -1. Note that the blocks of the operator T are represented as sums of such operators taken together with their adjoint operators. Since a(D), b(D)a(D), b(D)a(D)dl (D) -1 and d2(D) -1 are bounded operators, it is clear that the latter is guaranteed if in turn the operators
dl(D)-llqjk[ 1/2 (j, k = 1, 2), d2(D)-l[qjk[ 1/2 (j, k = 3, 4), dl(D)-llqjkl (j = l, 2; k = 3, 4; or j = 3 , 4 ; k = l , 2 ) , or, equivalently, the integral operators with the kernels
[~[-llqjk(x)ll/Zeilx'~)
(j, k --- 1, 2),
340
P.A. COJUHARI (r(~) + 1)-l/21qjt(x)lUZei(X'~) I~l-Xlqjk(x)le ~
(j, k = 3, 4),
(j = 1 , 2 ; k = 3,4 or j = 3 , 4 ; k = 1,2),
(3.2)
are compact in the space L2(I[~.3). The compactness criteria of such type integral operators are well known [4] (see also Lemma from [6], p. 45). By virtue of the results obtained in [4], conditions (2.1) guarantee compactness of the integral operators with kernels g~en b y (3.2). Thus, conditions (2.1) imply also the compactness of To. Therefore B = S*TS is a compact operator in the space L2(•3; C 4) and so all assumptions of Theorem 1.1 are satisfied. The proof is complete. [] 4.
Remarks
1. In [4] there are established conditions under which an operator of the form b ( X ) a ( D ) (X denotes the family of operators of multiplication by coordinates xj (j = 1, 2, 3) in R 3) belongs to the von Neumann-Schatten class of compact operators /3p. In concordance with these results it is possible to calculate the ideal-norm I[" lip of the operator T defined as in the proof of Theorem 2.1 by factorizing the perturbation operator Q as Q = S*TS. Note that the operator S is chosen such that the densely defined operator S ( H - l ) - l S * has a bound extension of which norm (of the space Lz(R3; C4)) is equal to 1. Thus, the abstract results from [7] concerning estimation of the number of eigenvalues excited by a perturbation in a given spectral gap can be applied. On the problems connected with the estimation of the discrete spectrum for Dirac type operators together with some asymptotic formulae related with those obtained by M. Klaus in [10] the author hopes to undertake discussions in a separate publication. 2. In the proof of Theorem 2.1 it was substantially used the following fact. Let be a Hilbert space, and let A and B denote densely defined operators in 7-/. The operator A is assumed to be self-adjoint and A > 0. In addition, 0 6 cr(A)\cyp(A), i.e. A is invertible but the inverse A -1 represents an unbounded operator. It is affirmed that if A - 1 B is densely defined and possesses a bounded extension, then the domains of the operators A - I B and B coincide. Indeed, let ~. denote an arbitrary point on the negative semi-axis, and denote R0~; A ) = ( A - ~.I) -l . One has R()~; A ) B u - A - 1 B u = ~R(L; A)A 1Bu for each u from the domain of A-1B, i.e. u ~ D ( A - 1 B ) . Since D ( A - 1 B ) = 7-[, it follows [R(~.; A)B] - [A-1B] = ~.R(~.; A)[A-1B], where IT] denotes complete extension of a densely defined and bounded operator T. Since )~R()~; A) converges strongly to 0, it follows that R(~.; A ) B u > [A-IB]u for each u ~ D(B). On the other hand, due to the relation AR(£; A ) B : B + )~R(X; A)B, one obtains AR(~.; A ) B u > Bu for each u ~ D(B). Consequently, since A is a closed operator, one can conclude that [A-1B]u ~ D(A) and A[A-1B]u = Bu for
ON THE FINITENESS OF THE DISCRETE SPECTRUM OF THE DIRAC OPERATOR
341
each u ~ D(B). Therefore, Bu belongs to the range of A and A-I(Bu) = [ A - 1 B ] u , and so D ( B ) C D(A-1B). Since the opposite inclusion is evident our assertion follows. REFERENCES [1] H. A. Bethe and E. E. Salpeter: Quantum Mechanics of One- and Two- Electron Atoms, Academic Press, New York 1959. [2] M. S. Birman: On the spectrum of Schr6dinger and Dirac operators, Dokl. Acad. Nauk 129, no. 2, (1959), 239-241. [3] M. S. Birman, On the spectrum of singular boundary-value problems, Mat. Sb. 55, no. 2, (1961), 125-174. [4] M. S. Birman, G. E. Karadzhov snd M. Z. Solomyak: Boundedness conditions and spectrum estimates for the operators b(X)a(D) and their analogs, Adv. Soviet. Math. 7 (1991), 85-106. [5] P. A. Cojuhari: On the finiteness of the discrete spectrum of some pseudodifferential operators, Mat. Issled. 80 (1985), 85-97. [6] P. A. Cojuhari: On the finiteness of the discrete spectrum of some matrix pseudodifferential operators, Vyssh. Uchebn. Zaved. Mat. 1 (1989), 42-50. [7] P. A. Cojuhari: Estimates of the number of perturbed eigenvalues, Operator Theory, Operator Algebras and Related Topics in Proc. of OT 17 Conf. (2000), 97-111. [8] V. A. Fock: Introduction to Quantum Mechanics, Nauka, Moscow 1976. [9] W. Greiner: Relativistic Quantum Mechanics: Wave Equations, Springer, Berlin-Heidelberg-New York 1994. [10] M. Klaus: On the point spectrum of Dirac operators, Helv. Phys. Acta. 53 (1980), 453--462. [11] R. Konno and S. T. Kuroda: On the finiteness of perturbed eigenvalues, J. Fac. Sci. Univ. To~,o Sect. IA Math. 13 (1966), 55~53. [12] O. I. Kurbenin: The discrete spectra of the Dirac and Pauli operators, Topics in Mathematical Physics, vol. 3, Spectral Theory (1969), 43-52. [13] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (nonrelativistic theoo'), Pergamon Press. Oxford, and Addison-Wesley, Reading, MA 1965. [14] C. Pladdy: Asymptotics of the resolvent of the Dirac operator with a scalar short-range potential, Analysis 21 (2001), 79-97. [15] B. Thaller: The Dirac Equation, Springer, Berlin-Heidelberg-New York 1992.