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Kane model, dynamical algebra and coherent states
Physica A 152 (1988) 451-458 North-Holland, Amsterdam
KANE MODEL,
DYNAMICAL
ALGEBRA
K.A. Shemakha Astrophysical
Observatory,
Revised
Dynamical Weyl algebra are derived.
AND COHERENT
STATES
RUSTAMOV
Academy
of Sciences of the Azerbaijan
Received 3 November manuscript received
1987 5 April
SSR, USSR
1988
invariance algebra is developed for the Kane model. On the basis of the Heisenbergof invariants explicit expressions for interband coherent states and Green functions
1. Introduction Methods for investigation of algebraic and group properties of theoretical and mathematical physics equations have formed in recent years an independent scientific direction with its internal problems and established principles, which have led to important qualitative results in many fields of applications’-6). In quantum physics such investigations are interesting because they allow for a deeper insight into the nature of the studied phenomena by finding successive ways of development of the basic statements of the theoretical models and deriving some of their exact solutions1-5 ). Besides that, these investigations are also useful because they allow for a unified way in considering such traditional problems as the construction of a complete set of system states with its energy spectrum, as well as Green functions in different representations, matrix elements, probabilities of various transitions3’6), etc. The present paper deals with the study of the algebraic properties of the Kane mode17) describing the energy spectrum of electric charge carriers in a large class of solids, and, therefore, frequently used as a first step for the calculation of the physical characteristics of crystals8-13). Possible ways to obtain a nontrivial result within the given formulation of a problem were prompted by works on relations under certain conditions between the Kane model, the classical equations of theoretical physics and representations of rotational group O(3) (see e.g. refs. 8,13). 03784371/88/$03.50 0 (North-Holland Physics
Elsevier Science Publishers Publishing Division)
B.V.
452
K.A. RUSTAMOV
Later on in this paper we will isolate states generating algebra from the algebra of transformations allowed by the Kane model, and on the basis of this the interband coherent states and Green functions will be obtained. It should be noted that the investigation of algebraic properties of analytical models and the possibilities of further applications of the obtained results are not yet sufficiently employed in solid state physics’“) as compared to other disciplines of theoretical physics’-6 ). Hence, the presented ideas can be very interesting and may also lead to simple implementations for other models of solid state physics.
2. Algebraic Let
analysis
us consider
i?i x
=
of the Kane model the
Schrodinger
c?cp ,
0
and
cp=
C,(k t> C,(k t> C,k t) C,(kl t> C,(k t) Cc@,t>
with
the
Kane
Hamiltonian
(1)
where
-%
equation
exp(ik-
r)
KANE MODEL,
DYNAMICAL
ALGEBRA
AND COHERENT
STATES
453
Here k = (k,, k,, k3) and r = (x,, x2, xg) are 3-dimensional vectors, the functions Cj(k, t) exp(ik * r), j = 1, . . . ,6, are the expansion coefficients of the wave function of a particle in Bloch amplitudes ukcO(r); sg, B are constant parameters and d, = al&,, a, = a/ax, + ialax, 13). For the simplicity of further calculations and in accordance diagonalize i.e.
the symbol15)
fi(P)fi(P)fi+(P) where
E?(P)
of the operator
= A,(P)apV P = (PO, P,,
and
(a,, a+, a_), respectively, R,(E;/~ + 2B2P2/3)“‘,
&
,
P_) is
I? by the unitary
/_L,v = 1, . . . ,6,
- P,P” p:VF
VTP-2vY
0
i?(P) = P-
DO-
R,=
v=1,2,
0,
v=3,4,
1
o=(2,1A;E)1’2, g
)
-P+ 2vF
fiP+
-pop:
2vT
Pp
(3)
P=(2A5A;a)1’2> 8
P2 = Pi + P: .
So, the study of algebraic the solution of this problem (ifi alat - A,(P))lk;,, (ifi a/at
0
v=5,6,
1,
P: = P,P_
0
2e
-1,
(2)
a BP_ A,v%
0
oop
i?,
are the symbols of operators fi and a = a Kronecker symbol, h,(P) = - EJ~ -
-aBP, A,%% Oa
with ref. 5 we
transformation
- A,(P))!&,
properties for three
of the system of eqs. (1) is reduced decoupled equations:
to
= 0,
(4’)
= 0,
(4”)
K.A.
454
RUSTAMOV
(4"')
(ih alat - A5(P))YJ5,6 = 0, where
P is the Fourier transformation operator. It is to be noted that we will refer to the transformation the equation &Y,
allowed
by
alay, . . .)f(y> = 0
as the algebra the relation: &Y,
algebra
consisting
of operators
o,( y, alay,
alay,. . .>&,(Y, a/ah. . MY)
. . .) (n = 1, . . .) satisfying
= ~Y>&Y,
a&,
. . .)f(y) ,
(5)
where g(y) is an arbitrary function of the independent variable y lm6). Then, in accordance with the general theory proved in ref. 5, as a consequence of the two-fold degenerate eigenvalues Al(P) = A,(P), h,(P) = A,(P), h,(P) = A,(P), we conclude that the system of equations (inalat-h,(P)$y)~=O, allows for a twelve-dimensional
j,b,~=l,..., algebra
(6)
6,
of pure-matrix
transformations:
where {e} is a four-dimensional algebra of group GL(2), realized by the 2 X 2 Pauli matrices aj (j = 1,2,3) of the algebra of the group SU(2) and by the unit matrix a,, . Then, it is not difficult to ascertain that each of the three equations (4) allows for a three-dimensional algebra of transformations, too:
, j=1,2,3.
(8)
KANE MODEL,
In addition, transformation ties in order
DYNAMICAL
ALGEBRA
AND COHERENT
STATES
as was shown in ref. 16, one can perform the inverse of equations (4) and relate the study of their algebraic to carry out a similar
(
a*/at* +
p1
a/at + i j=l
p3
analysis
455
Fourier proper-
of equation
avax; +p* f(r, >
t) = 0,
(9)
where p, (j = 1,2,3) are some constants, and, then, it is easy to prove that each equation (4) allows for transformations from the ten-dimensional Poincare algebra {fi,} (m = 1, . . . , lo), where
b, =aiat,
hj+,
=alaxj,
ljj+7=~:talaxj-~jaIat,
Ijj+4 =xjaiax,-xk alaxj, R;=3n
-2iB*
(10) R:,
k,j=L2,3.
Here, as in (8) and (9), the constants RY for (4’), (4”) and (4”‘) take on the values of - 1,O and + 1, respectively. So, the Kane model (1) allows for a rather wide set of transformations, i.e.
where 0 is the operator of unitary transformation (3), fi is the Fourieroperator”). The study of the structure characteristics of the set (11) is of evident interest. And, for our purposes, it should be noted that from the elements of the set (ll), we can get the following linear combinations:
ai
=
+( _& _ 2iR;i2tpj ($ +T)-“‘)
+pi,
I
aj(R,)
=
+( $ _ 2iR;i2tpi( $ +?$!?-“‘)
_ pi,
(12)
I
satisfying
the commutation
relations:
(13) where
I^ is a unit operator,
[a, b] = ab - ba.
K.A. RUSTAMOV
456
3. Interband
coherent states and Green functions
So, the algebraic structure of the set of operators (12) and r^ determines the Heisenberg-Weyl algebra. According to the Stone-von Neumann theorem’7XL8 ) in the space of the states of the investigated dynamical system there are no invariant subspaces of these operators (they form a complete system). By transformations, whose generators are the above mentioned operators, from only one state of a dynamical system all other states can be constructed. Conforming to the standard terminology, the operators A:(RY) and Ai will be called the creation and annihilation operators. As soon as, depending on the value of the parameter R,,the
~i(R,)lr*l,Ry) = a,I’y>R,)
(14)
2
where the CX~ are constant complex numbers. It is clear that the Ia, R,) satisfy the Schrodinger equation (6), too. Often it proves to be convenient to construct functions la, R,) by means of a unitary shift operator &a) 6,14),
D(a) = fi
exp(a,A:
- aTA,>
= ,t
exp{ z]
exp(O!jAT)
exp(-c_uqA,)
,
j=l
(15) by applying Aj(R,)lO,
it to the basic state Ry) = 0,
10, Ry
j = 1,2,3
)
determined
by
(16)
Namely,
(a, R,,) = &)(O, R,) .
(17)
KANE MODEL,
DYNAMICAL
ALGEBRA
AND COHERENT
STATES
457
Using expressions (12) of the creation and the annihilation operators for the normalized interband coherent states we get: Ia, R,)
= ( $)“4
exp( !!T$(
$ + F,“’
+ p2
(18) It is not difficult to ascertain that the functions (18) possess all basic properties of coherent states6’14). In particular, the standard conditions of orthogonality and completeness are satisfied for them: z
1 (&R,,IP)(P~cx,R,.)~‘P=,~~~~~(-~-~+cx~~~), -CC
57
-3
‘?(a, R,)(a,
R,(P’)
d2a = 6(P-
(19)
P’),
(20)
where 6(P - P’) = II:=, 8(Pj - Pi) is Dirac’s 6-function, d2a = d(Re CX) x d(Im u). As the functions (18) form a complete system of functions, the interband Green function G(P’, t’; P, t), being the probability amplitude of the transition from the state (P, t) to the state (P’, t’) of the system described by eqs. (4’), (4”), (4”‘), can be calculated from s G(P’, t’; P, t) = Y3
I --T
RvIP, t) d2a .
(P’, t’(a, RJa,
(21)
From
1% R,)
= 10, R,)
exp{i:
((y’ ,‘I’
- zajpj)]
(22)
,
j=l
taking into account the orthogonality
condition of coherent
states, we get
G(P’, t’; P, t) = exp
3 (P, + Pi)2
+ P2 + PI2 - c j=l
2
I
6(P - P’) .
(23)
K.A.
458
Note that the calculation states
is equivalent
to taking
of Green
RUSTAMOV
functions
the Feynman
by the integration
integral
over coherent
along the classical
paths6).
References
1) L.V.
Ovsyannikov, Gruppovie Svoystva Differentsialnikh Uravneniy SO AN SSSR, Novosibirsk, 1962); Gruppovoy Analys Differentsialnikh Uravneniy (Nauka, Moscow, 1978). Fiziki (NGU, 2) N. Kh. Ibragimov, Gruppi Li v Nekotorikh Voprosakh Mathematicheskoy Novosibirsk, 1972); Gruppi Preobrasovaniy v Mathematicheskoy Fizike (Nauka, Moscow, 1983). Theory of Group Representations and Applications (PWN, 3) A.O. Barut and R. Raczka, Warszawa, 1977). and separation of variables, Encyclopedia of Mathematics and its 4) W. Miller Jr., Symmetry Applications, vol. 4, Gian-Carlo Rota, ed. (Addison-Wesley, Reading, MA, 1977). metodi v mathematicheskoy fizike, VI. Fushich, ed., 5) V.I. Fushich, in: Theoretico-gruppovie IM AN SSSR. Kiev, 1978); in: Theoretico-gruppovie issledovaniya v mathematicheskoy fizike, V.I. Fushich, ed., IM AN SSSR, Kiev, 1981). Dynamicheskie Symmetrii i Kogerentnie Sostoyaniya Kvan6) I.A. Malkin and V.I. Manko, tovikh System (Nauka, Moscow, 1979). 7) A.O. Kane, J. Phys. Chem. Sol. 1 (1957) 249. 8) L.V. Keldysh, JETF 45 no. 2/8 (1963). i Deformatsionniye Effekti v Poluprovodnikakh 9) G.L. Bir and G.E. Pikus, Symmetriya (Nauka, Moscow, 1972). Kineticheskiye Effekti v Poluprovodnikakh (Nauka, Moscow, 1970). 10) B.M. Askerov, and W. Szymanska, Phys. Stat. Sol. (b) 45 (1971) 415. 11) W. Zawadski Phys. Stat. Sol. (b) 87 (1978) 447. 12) M.A. Mekhtiev and S.M. Seid-Rzaeva, and A.M. Babaev, Izv. VUZov SSSR, Fizika, no.1, 1988; preprint no. 435, 13) K.A. Rustamov Dipartamento di Fizika, Universita di Roma, La Sapienza, 1985. Comm. Math. Phys. 26 (1972) 222; preprint ITEF-144, Institute of 14) A.M. Perelomov, Theoretical and Experimental Physics, Moscow, 1983. Psevdodifferentsialnie Operatori i Spektralnaya Teoriya (Nauka, Moscow, 15) M.A. Shubin, 1978). and A.M. Babaev, preprint no. 6, Naucho-Proizvodstvennoe Obyedinenie 16) K.A. Rustamov Kosmicheskikh Issledovaniy (Baku, 1986). 17) M. Stone, Proc. Nat. Acad. Sci. USA 16 (1930) 172. Math. Ann. 104 (1931) 570. 18) J. von Neumann,
KANE MODEL,
DYNAMICAL
ALGEBRA
K.A. Shemakha Astrophysical
Observatory,
Revised
Dynamical Weyl algebra are derived.
AND COHERENT
STATES
RUSTAMOV
Academy
of Sciences of the Azerbaijan
Received 3 November manuscript received
1987 5 April
SSR, USSR
1988
invariance algebra is developed for the Kane model. On the basis of the Heisenbergof invariants explicit expressions for interband coherent states and Green functions
1. Introduction Methods for investigation of algebraic and group properties of theoretical and mathematical physics equations have formed in recent years an independent scientific direction with its internal problems and established principles, which have led to important qualitative results in many fields of applications’-6). In quantum physics such investigations are interesting because they allow for a deeper insight into the nature of the studied phenomena by finding successive ways of development of the basic statements of the theoretical models and deriving some of their exact solutions1-5 ). Besides that, these investigations are also useful because they allow for a unified way in considering such traditional problems as the construction of a complete set of system states with its energy spectrum, as well as Green functions in different representations, matrix elements, probabilities of various transitions3’6), etc. The present paper deals with the study of the algebraic properties of the Kane mode17) describing the energy spectrum of electric charge carriers in a large class of solids, and, therefore, frequently used as a first step for the calculation of the physical characteristics of crystals8-13). Possible ways to obtain a nontrivial result within the given formulation of a problem were prompted by works on relations under certain conditions between the Kane model, the classical equations of theoretical physics and representations of rotational group O(3) (see e.g. refs. 8,13). 03784371/88/$03.50 0 (North-Holland Physics
Elsevier Science Publishers Publishing Division)
B.V.
452
K.A. RUSTAMOV
Later on in this paper we will isolate states generating algebra from the algebra of transformations allowed by the Kane model, and on the basis of this the interband coherent states and Green functions will be obtained. It should be noted that the investigation of algebraic properties of analytical models and the possibilities of further applications of the obtained results are not yet sufficiently employed in solid state physics’“) as compared to other disciplines of theoretical physics’-6 ). Hence, the presented ideas can be very interesting and may also lead to simple implementations for other models of solid state physics.
2. Algebraic Let
analysis
us consider
i?i x
=
of the Kane model the
Schrodinger
c?cp ,
0
and
cp=
C,(k t> C,(k t> C,k t) C,(kl t> C,(k t) Cc@,t>
with
the
Kane
Hamiltonian
(1)
where
-%
equation
exp(ik-
r)
KANE MODEL,
DYNAMICAL
ALGEBRA
AND COHERENT
STATES
453
Here k = (k,, k,, k3) and r = (x,, x2, xg) are 3-dimensional vectors, the functions Cj(k, t) exp(ik * r), j = 1, . . . ,6, are the expansion coefficients of the wave function of a particle in Bloch amplitudes ukcO(r); sg, B are constant parameters and d, = al&,, a, = a/ax, + ialax, 13). For the simplicity of further calculations and in accordance diagonalize i.e.
the symbol15)
fi(P)fi(P)fi+(P) where
E?(P)
of the operator
= A,(P)apV P = (PO, P,,
and
(a,, a+, a_), respectively, R,(E;/~ + 2B2P2/3)“‘,
&
,
P_) is
I? by the unitary
/_L,v = 1, . . . ,6,
- P,P” p:VF
VTP-2vY
0
i?(P) = P-
DO-
R,=
v=1,2,
0,
v=3,4,
1
o=(2,1A;E)1’2, g
)
-P+ 2vF
fiP+
-pop:
2vT
Pp
(3)
P=(2A5A;a)1’2> 8
P2 = Pi + P: .
So, the study of algebraic the solution of this problem (ifi alat - A,(P))lk;,, (ifi a/at
0
v=5,6,
1,
P: = P,P_
0
2e
-1,
(2)
a BP_ A,v%
0
oop
i?,
are the symbols of operators fi and a = a Kronecker symbol, h,(P) = - EJ~ -
-aBP, A,%% Oa
with ref. 5 we
transformation
- A,(P))!&,
properties for three
of the system of eqs. (1) is reduced decoupled equations:
to
= 0,
(4’)
= 0,
(4”)
K.A.
454
RUSTAMOV
(4"')
(ih alat - A5(P))YJ5,6 = 0, where
P is the Fourier transformation operator. It is to be noted that we will refer to the transformation the equation &Y,
allowed
by
alay, . . .)f(y> = 0
as the algebra the relation: &Y,
algebra
consisting
of operators
o,( y, alay,
alay,. . .>&,(Y, a/ah. . MY)
. . .) (n = 1, . . .) satisfying
= ~Y>&Y,
a&,
. . .)f(y) ,
(5)
where g(y) is an arbitrary function of the independent variable y lm6). Then, in accordance with the general theory proved in ref. 5, as a consequence of the two-fold degenerate eigenvalues Al(P) = A,(P), h,(P) = A,(P), h,(P) = A,(P), we conclude that the system of equations (inalat-h,(P)$y)~=O, allows for a twelve-dimensional
j,b,~=l,..., algebra
(6)
6,
of pure-matrix
transformations:
where {e} is a four-dimensional algebra of group GL(2), realized by the 2 X 2 Pauli matrices aj (j = 1,2,3) of the algebra of the group SU(2) and by the unit matrix a,, . Then, it is not difficult to ascertain that each of the three equations (4) allows for a three-dimensional algebra of transformations, too:
, j=1,2,3.
(8)
KANE MODEL,
In addition, transformation ties in order
DYNAMICAL
ALGEBRA
AND COHERENT
STATES
as was shown in ref. 16, one can perform the inverse of equations (4) and relate the study of their algebraic to carry out a similar
(
a*/at* +
p1
a/at + i j=l
p3
analysis
455
Fourier proper-
of equation
avax; +p* f(r, >
t) = 0,
(9)
where p, (j = 1,2,3) are some constants, and, then, it is easy to prove that each equation (4) allows for transformations from the ten-dimensional Poincare algebra {fi,} (m = 1, . . . , lo), where
b, =aiat,
hj+,
=alaxj,
ljj+7=~:talaxj-~jaIat,
Ijj+4 =xjaiax,-xk alaxj, R;=3n
-2iB*
(10) R:,
k,j=L2,3.
Here, as in (8) and (9), the constants RY for (4’), (4”) and (4”‘) take on the values of - 1,O and + 1, respectively. So, the Kane model (1) allows for a rather wide set of transformations, i.e.
where 0 is the operator of unitary transformation (3), fi is the Fourieroperator”). The study of the structure characteristics of the set (11) is of evident interest. And, for our purposes, it should be noted that from the elements of the set (ll), we can get the following linear combinations:
ai
=
+( _& _ 2iR;i2tpj ($ +T)-“‘)
+pi,
I
aj(R,)
=
+( $ _ 2iR;i2tpi( $ +?$!?-“‘)
_ pi,
(12)
I
satisfying
the commutation
relations:
(13) where
I^ is a unit operator,
[a, b] = ab - ba.
K.A. RUSTAMOV
456
3. Interband
coherent states and Green functions
So, the algebraic structure of the set of operators (12) and r^ determines the Heisenberg-Weyl algebra. According to the Stone-von Neumann theorem’7XL8 ) in the space of the states of the investigated dynamical system there are no invariant subspaces of these operators (they form a complete system). By transformations, whose generators are the above mentioned operators, from only one state of a dynamical system all other states can be constructed. Conforming to the standard terminology, the operators A:(RY) and Ai will be called the creation and annihilation operators. As soon as, depending on the value of the parameter R,,the
~i(R,)lr*l,Ry) = a,I’y>R,)
(14)
2
where the CX~ are constant complex numbers. It is clear that the Ia, R,) satisfy the Schrodinger equation (6), too. Often it proves to be convenient to construct functions la, R,) by means of a unitary shift operator &a) 6,14),
D(a) = fi
exp(a,A:
- aTA,>
= ,t
exp{ z]
exp(O!jAT)
exp(-c_uqA,)
,
j=l
(15) by applying Aj(R,)lO,
it to the basic state Ry) = 0,
10, Ry
j = 1,2,3
)
determined
by
(16)
Namely,
(a, R,,) = &)(O, R,) .
(17)
KANE MODEL,
DYNAMICAL
ALGEBRA
AND COHERENT
STATES
457
Using expressions (12) of the creation and the annihilation operators for the normalized interband coherent states we get: Ia, R,)
= ( $)“4
exp( !!T$(
$ + F,“’
+ p2
(18) It is not difficult to ascertain that the functions (18) possess all basic properties of coherent states6’14). In particular, the standard conditions of orthogonality and completeness are satisfied for them: z
1 (&R,,IP)(P~cx,R,.)~‘P=,~~~~~(-~-~+cx~~~), -CC
57
-3
‘?(a, R,)(a,
R,(P’)
d2a = 6(P-
(19)
P’),
(20)
where 6(P - P’) = II:=, 8(Pj - Pi) is Dirac’s 6-function, d2a = d(Re CX) x d(Im u). As the functions (18) form a complete system of functions, the interband Green function G(P’, t’; P, t), being the probability amplitude of the transition from the state (P, t) to the state (P’, t’) of the system described by eqs. (4’), (4”), (4”‘), can be calculated from s G(P’, t’; P, t) = Y3
I --T
RvIP, t) d2a .
(P’, t’(a, RJa,
(21)
From
1% R,)
= 10, R,)
exp{i:
((y’ ,‘I’
- zajpj)]
(22)
,
j=l
taking into account the orthogonality
condition of coherent
states, we get
G(P’, t’; P, t) = exp
3 (P, + Pi)2
+ P2 + PI2 - c j=l
2
I
6(P - P’) .
(23)
K.A.
458
Note that the calculation states
is equivalent
to taking
of Green
RUSTAMOV
functions
the Feynman
by the integration
integral
over coherent
along the classical
paths6).
References
1) L.V.
Ovsyannikov, Gruppovie Svoystva Differentsialnikh Uravneniy SO AN SSSR, Novosibirsk, 1962); Gruppovoy Analys Differentsialnikh Uravneniy (Nauka, Moscow, 1978). Fiziki (NGU, 2) N. Kh. Ibragimov, Gruppi Li v Nekotorikh Voprosakh Mathematicheskoy Novosibirsk, 1972); Gruppi Preobrasovaniy v Mathematicheskoy Fizike (Nauka, Moscow, 1983). Theory of Group Representations and Applications (PWN, 3) A.O. Barut and R. Raczka, Warszawa, 1977). and separation of variables, Encyclopedia of Mathematics and its 4) W. Miller Jr., Symmetry Applications, vol. 4, Gian-Carlo Rota, ed. (Addison-Wesley, Reading, MA, 1977). metodi v mathematicheskoy fizike, VI. Fushich, ed., 5) V.I. Fushich, in: Theoretico-gruppovie IM AN SSSR. Kiev, 1978); in: Theoretico-gruppovie issledovaniya v mathematicheskoy fizike, V.I. Fushich, ed., IM AN SSSR, Kiev, 1981). Dynamicheskie Symmetrii i Kogerentnie Sostoyaniya Kvan6) I.A. Malkin and V.I. Manko, tovikh System (Nauka, Moscow, 1979). 7) A.O. Kane, J. Phys. Chem. Sol. 1 (1957) 249. 8) L.V. Keldysh, JETF 45 no. 2/8 (1963). i Deformatsionniye Effekti v Poluprovodnikakh 9) G.L. Bir and G.E. Pikus, Symmetriya (Nauka, Moscow, 1972). Kineticheskiye Effekti v Poluprovodnikakh (Nauka, Moscow, 1970). 10) B.M. Askerov, and W. Szymanska, Phys. Stat. Sol. (b) 45 (1971) 415. 11) W. Zawadski Phys. Stat. Sol. (b) 87 (1978) 447. 12) M.A. Mekhtiev and S.M. Seid-Rzaeva, and A.M. Babaev, Izv. VUZov SSSR, Fizika, no.1, 1988; preprint no. 435, 13) K.A. Rustamov Dipartamento di Fizika, Universita di Roma, La Sapienza, 1985. Comm. Math. Phys. 26 (1972) 222; preprint ITEF-144, Institute of 14) A.M. Perelomov, Theoretical and Experimental Physics, Moscow, 1983. Psevdodifferentsialnie Operatori i Spektralnaya Teoriya (Nauka, Moscow, 15) M.A. Shubin, 1978). and A.M. Babaev, preprint no. 6, Naucho-Proizvodstvennoe Obyedinenie 16) K.A. Rustamov Kosmicheskikh Issledovaniy (Baku, 1986). 17) M. Stone, Proc. Nat. Acad. Sci. USA 16 (1930) 172. Math. Ann. 104 (1931) 570. 18) J. von Neumann,