Refined comparison theorems for the Dirac equation with spin and pseudo--spin symmetry in $d$ dimensions

0.3 MB·

by  Richard L. Hall

CUQM - 155
Refined comparison theorems for the Dirac equation with spin and pseudo–spin
symmetry in d dimensions.
Richard L. Hall1,∗ and Petr Zorin1,†
1Department of Mathematics and Statistics, Concordia University,
1455 de Maisonneuve Boulevard West, Montr´eal, Qu´ebec, Canada H3G 1M8
6
The classic comparison theorem of quantum mechanics states that if two potentials are ordered
1
then the corresponding energy eigenvalues are similarly ordered, that is to say if Va Vb, then
0 ≤
Ea Eb. Such theorems have recently been established for relativistic problems even though
2 ≤
the discrete spectra are not easily characterized variationally. In this paper we improve on the

n basic comparison theorem for the Dirac equation with spin and pseudo–spin symmetry in d 1
a dimensions. The graphs of two comparison potentials may now cross each other in a prescri≥bed
J
manner implying that the energy values are still ordered. The refined comparison theorems are

5 valid for the ground state in one dimension and for thebottom of an angular momentum subspace
1 in d > 1 dimensions. For instance in a simplest case in one dimension, the condition Va Vb is
] replaced byUa ≤Ub, where Ui(x)=R0xVi(t)dt,x∈[0, ∞),and i=a or b. ≤
h
p PACSnumbers: 03.65.Pm,03.65.Ge,36.20.Kd.
Keywords: Dirac equation, ground state, spin symmetry, pseudo–spin symmetry, comparison theorems,
-
h refinedcomparisontheorems.
t
a
m
[

I. INTRODUCTION
1
v
2 Spin and pseudo–spin symmetry were first introduced in [1, 2] more than forty years ago. Spin symmetry occurs
2 in the spectrum of a meson [3–6]. Pseudo–spin symmetry helps explain the spectra of deformed nuclei [7] and
0 superdeformation [8], which occurs in the spectra of certain nuclei [9]. Spin symmetry helps in the design of nuclear
4 shell–model schemes [10–12], and is used to explain certain identical bands [13–15]. Exact spin symmetry in the
0
Dirac equation occurs when the difference between the scalar S and vector V potentials is equal to a constant, i.e.
.
1 S V =c [4]. While exact pseudo–spin symmetry exists when the sum of scalar S and vector V potentials is equal
1
0 to−a constant, i.e. S +V = c [16, 17]. Here we consider potentials of equal magnitude, so that S = V , and the
2
6 | | | |
constants c and c are zero.
1 1 2
Under spin or pseudo–spin symmetries a relativistic system of Dirac coupled equations can be written as a single
:
v Schr¨odinger–likeequation. Thenonecanusemethodswhichweredevelopedtosolvenon–relativisticequationsexactly
i or approximately, such as factorization and path–integralmethods [18–22], the Nikiforov–Uvarovmethod [23], shape
X
invariance [24, 25], asymptotic iteration method [26–30], supersymmetric quantum mechanics [31], and so on. For
r
a instance,theDiracequationwassolvedfortheMorsepotential[32–36],the harmonic–oscillatorpotential[37–39],the
pseudoharmonic potential [40], the Po¨schl–Teller potential [41–44], the Woods–Saxon potential [45, 46], the Eckart
potential [47, 48], the Coulomband the Hartmann potentials [49], the Hyperbolic potentials and the Coulombtensor
interaction [50, 51], the Rosen–Morse potential [52], the Hulth´en potential [53–55], the Hulth´en potential including
the Coulomb–like tensor potential [56], the v tanh2(r/d) potential [57], the Coulomb–like tensor potential [58], the
0
modified Hylleraas potential [59], the Manning–Rosen and the generalized Manning–Rosen potentials [60–64], and
others. The pointisthatthere aremany knownexactsolutionsthatcanbe usedforcomparisonswithnew potentials
found in given problems.
The comparison theorem of quantum mechanics states that if the comparison potentials are ordered then the
corresponding energy eigenvalues are ordered as well, i.e. if V V then E E [65–71], thus the graphs of the
a b a b
≤ ≤
comparison potentials are not allowed to cross over each other. The comparison theorem was also established for
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
Refined comparison theorems for the Dirac equation with spin and pseudo–spin symmetry in d dimensions. 2
the Dirac equation under the spin and pseudo–spin symmetry [72]. Similarly to the non–relativistic case [73], here
we refine the comparison theorem for the Dirac equation under the spin and pseudo–spin symmetry by establishing
conditions under which the potentials can intersect and still preserve the ordering of eigenvalues. In the simplest
x
one–dimensionalcase,the conditionV V is replacedby U U , where U (x)= V (t)dt, x [0, ), and i=a
a ≤ b a ≤ b i 0 i ∈ ∞
or b. R
Thepaperisorganizedinthefollowingway: westartwiththeDiracequationinonedimensionandderivetheusual
comparisontheorem (section II. A.). Then in section II. B. we establishsome generalrelations between the potential
V, the energy E, and mass of the particle m. In section II. C. we refine the comparison theorem, using necessary
monotonebehaviourofthe wavefunctions. Finally,wedemonstratehowto applythe refinedcomparisontheoremsin
practice, often by taking advantage of the corollaries with specially designed simplified sufficient conditions (section
II. D.). Following a similar path we then consider the family of d > 1 dimensional cases. In order to simplify
the statements and proofs of the theorems, we shall usually combine the formulation of the spin–symmetric and
pseudo–spin-symmetric cases by the use of a parameter s= 1.
±
II. THE ONE–DIMENSIONAL CASE d=1.
A. The Dirac equation
The Dirac equation in one dimension is given by [74]:
∂
σ (E V)σ +m+S ψ =0,
1 3
(cid:18) ∂x − − (cid:19)
in natural units ~ = c = 1, m is the mass of the particle, and σ and σ are Pauli matrices. The potentials V and
1 3
S are monotone even functions such that the energy eigenvalue E exists. Both potentials are bounded at the origin,
ϕ
that is to say V(0) and S(0) are finite. By taking the two–component Dirac spinor as ψ = 1 the above matrix
(cid:18)ϕ2 (cid:19)
equation can be decomposed into the following system of first–order linear differential equations [75, 76]:
ϕ′ = (E+m V +S)ϕ , (1a)
1 − − 2
(cid:26) ϕ′ = (E m V S)ϕ , (1b)
2 − − − 1
where the prime denotes the derivative with respect to x. For bound states, ϕ and ϕ satisfy the normalization
1 2
′
condition
∞
(ϕ ,ϕ )+(ϕ ,ϕ )= (ϕ2+ϕ2)dx=1.
1 1 2 2 Z 1 2
−∞
We now compare two problems with potentials V and S , i = a or b, and corresponding energies E and E for
i i a b
which the system (1a)–(1b) becomes respectively
ϕ′ = (E +m V +S )ϕ , (2a)
1a − a − a a 2a
(cid:26) ϕ′ = (E m V S )ϕ , (2b)
2a a− − a− a 1a
and
ϕ′ = (E +m V +S )ϕ , (3a)
1b − b − b b 2b
(cid:26) ϕ′ = (E m V S )ϕ . (3b)
2b b− − b− b 1b
Let us consider the following combination of the above equations:
(2a)ϕ (2b)ϕ (3a)ϕ +(3b)ϕ ,
2b 1b 2a 1a
− −
which, after some simplifications, becomes
(ϕ ϕ )′ (ϕ ϕ )′ =(ϕ ϕ +ϕ ϕ )(E E V +V ) (ϕ ϕ ϕ ϕ )(S S ).
1a 2b 2a 1b 1a 1b 2a 2b b a b a 1a 1b 2a 2b b a
− − − − − −
Refined comparison theorems for the Dirac equation with spin and pseudo–spin symmetry in d dimensions. 3
Integrating the left side of the above expression by parts from 0 to , and using the boundary conditions, we find
∞[(ϕ ϕ )′ (ϕ ϕ )′]dx=0. We then integrate the right side t∞o obtain
0 1a 2b − 2a 1b
R
∞ ∞
(E E ) (ϕ ϕ +ϕ ϕ )dx= [(S +V S V )ϕ ϕ +(S V S +V )ϕ ϕ ]dx. (4)
b a 1a 1b 2a 2b b b a a 1a 1b a a b b 2a 2b
− Z Z − − − −
0 0
We canmergethe spin andpseudo–spinsymmetric cases (as wasdone in [72]) by introducing the parameters, which
is equal to 1 if S =V and 1 if S = V, so S =sV. Then the above expression becomes
− −
∞ ∞
(E E ) (ϕ ϕ +ϕ ϕ )dx=2 (V V )ϕ ϕ dx, (5)
b a 1a 1b 2a 2b b a qa qb
− Z Z −
0 0
where q = 1 if s = 1 and q = 2 if s = 1. Expression (5) yields spectral ordering if the comparison potentials are
−
orderedandtheintegrandshaveconstantsigns,i.e. E E ifV V . Thisisequivalenttothecomparisontheorem
a b a b
≤ ≤
[72]whichwasderivedbyHallandYe¸silta¸susingmonotonicitypropertiesandisvalidalsoforexitedstates. However,
the potentials are not allowed to crossover. In the present paper we refine this theorem by letting the potentials
intersect each other in a suitable controlled manner and still imply spectral ordering.
B. Classes of potentials
By differentiation and substitution, system (1a)–(1b) in the case S = sV can be written as a Schr¨odinger–like
equation
ϕ′′+2V(E+sm)ϕ=(E2 m2)ϕ, (6)
− −
where ϕ = ϕ if s = 1 and ϕ = ϕ if s = 1. The radial function ϕ is normalizable but not necessarily normalized.
1 2
−
In any case, and the above eigenequation determines, the eigenvalue E. By using the spectral properties of the
Schr¨odingeroperator[77],we proposeto considertwosubclassesofpotentials: (i)V isfinite forlarge x andwithout
| |
loss of generality we choose the energy scale so that lim V =0; and (ii) V is unbounded for large x and without
|x|→∞ | |
loss of generality we choose a coordinate system so that V(0)=0. Analysing (6) and (1a)–(1b) for the S =sV case
we can finally state the three classes of potentials and corresponding relationship between energy E and mass m:
(i) V is finite near infinity, sV(0)<0, and
(1) sV 0 and lim V =0. This implies m<E <m;
≤ |x|→∞ −
(ii) V is unbounded near infinity, V(0)=0, and
(2) sV 0 and lim V = s . This implies sE > m
≥ |x|→∞ ∞
or
(3) sV 0 and lim V = s . This implies sE < m.
≤ |x|→∞ − ∞ −
For instance, consider s = 1 case. Then it follows from (6) that if V 0 and E m > 0 then E2 m2 < 0.
Inequality E m>0 leads to−E >m>0, but E2 m2 <0 leads to E < ≤m<0, whi−ch is a contradictio−n. Then if
V 0 and E− m<0 we should have E2 m2 >0−. Both inequalities E −m<0 and E2 m2 >0 lead to E < m.
≤ − − − − −
Now we assume that lim V =0, then system (1a)–(1b) asymptotically becomes
x→∞
ϕ′ = (E+m)ϕ , (7a)
1 − 2
(cid:26) ϕ′ = (E m)ϕ . (7b)
2 − 1
If ϕ 0 before vanishing, then ϕ′ 0 and, using E < m, above system yields ϕ 0 and ϕ′ 0 near infinity,
1 ≥ 1 ≤ − 2 ≤ 2 ≤
which is the contradiction. Assumption lim V = leads to
x→∞ −∞
ϕ′ =2Vϕ , (8a)
1 2
(cid:26) ϕ′ =(E m)ϕ . (8b)
2 − 1
Refined comparison theorems for the Dirac equation with spin and pseudo–spin symmetry in d dimensions. 4
Now if ϕ 0 and ϕ′ 0 we have ϕ 0 and ϕ′ 0, which means that ϕ approaches zero with positive sign.
1 ≥ 1 ≤ 2 ≥ 2 ≤ 2
FinallyweconcludethatifS = V andV 0thenE < mand lim V = ,whichcorrespondsto(2). Following
− ≤ − x→∞ −∞
thesamepath,thecases=1andtheremainingclassesofpotentialandcorrespondinginequalitiescanbeestablished.
C. Refined comparison theorems
Supposethat ϕ (x), ϕ (x) isasolutionoftheDiraccoupledequations(1a)–(1b). SincethepotentialV isaneven
1 2
{ }
function, itfollowsfrom(1a)–(1b)that ϕ ( x), ϕ ( x) and ϕ ( x), ϕ ( x) arealsosolutionsof(1a)–(1b).
1 2 1 2
{ − − − } {− − − }
Thusϕ andϕ havedefiniteandoppositeparities,i.e. ifϕ iseventhenϕ isoddandviceversa. Therefore,because
1 2 1 2
of the symmetry of the wave functions, we shall consider only the positive half axis x 0.
≥
Now we prove the lemma which characterizes the behaviour of the one dimensional Dirac wave functions in the
ground state:
Lemma 1: In the ground state the upper ϕ and lower ϕ components of the Dirac spinor are monotone in the spin
1 2
and pseudo–spin symmetric cases respectively.
Proof: In the s= 1 case equation (1b) becomes
−
ϕ′ =(E m)ϕ . (9)
2 − 1
Since in the groundstate ϕ hasconstantsign, the function ϕ′ has constantsignaswell,whichresultends the proof.
1 2
The case s=1, for which the roles of ϕ and ϕ are interchanged, can be similarly proved.
1 2
(cid:3)
For example, consider the s= 1 case with potential V satisfying (2). We are looking for the ground state. Thus
without loss of generality, we pu−t ϕ 0 on [0, ). Then equation (9) yields ϕ′ 0, so ϕ has to be even and
nonnegative. Consequentlyϕ isodd1,s≥oϕ′ mustch∞angeitssignfrompositivetoneg2a≤tive. Inor2dertoguaranteesuch
1 1
behaviour of ϕ , the potential V has to be smaller then E+m near the originand then dominate the term E+m at
1
infinity: this is true since V(0)=0 and lim V = .
|x|→∞ −∞
Now we refine the basic comparisontheorem which follows from relation (5).
Theorem 1: The potential V belongs to one of the classes (1)–(3) and has area, S =sV, and
x
g(x)= (V (t) V (t))dt, x [0, ). (10)
b a
Z − ∈ ∞
0
Then if g 0, the eigenvalues are ordered, i.e. E E .
a b
≥ ≤
Proof: We prove the theorem for the pseudo–spin symmetric case, i.e. s = 1; for the other case the proof is
−
essentially the same. We integrate the right side of (5) by parts to obtain
∞ ∞
2 (V V )ϕ ϕ dx=ϕ ϕ g ∞ 2 g(ϕ ϕ )′dx,
Z b− a 2a 2b 2a 2b |0 − Z 2a 2b
0 0
where g is defined by (10). Since g(0)=0 and lim ϕ =0, relation (5) becomes
2
x→∞
∞ ∞
(E E ) (ϕ ϕ +ϕ ϕ )dx= 2 g(ϕ ϕ )′dx,
b a 1a 1b 2a 2b 2a 2b
− Z − Z
0 0
According to Lemma 1, ϕ is monotone and, since it is also square integrable,it follows that the functions ϕ and ϕ′
2 2 2
have different signs in the ground state, i.e. if ϕ 0 then ϕ′ 0 on [0, ) and vice versa. Thus the derivative of
the product satisfies (ϕ ϕ )′ 0. Finally, if g 2 ≥0, it follows2f≤rom the ab∞ove expression that E E .
2a 2b a b
≤ ≥ ≤ (cid:3)
If we know more details of the interlacing relations of the comparison potentials, we can state a corollary of the
above theorem which is easier to apply on practice:
Corollary 1: Let the comparison potentials belong to one of the classes (1)–(3). If the potentials cross over once,
say at x , V V for x [0, x ], and
1 a b 1
≤ ∈
∞
g( )= (V V )dx,
b a
∞ Z −
0
Refined comparison theorems for the Dirac equation with spin and pseudo–spin symmetry in d dimensions. 5
or if the potentials cross over twice, say at x and x , x <x , V V for x [0, x ], and
1 2 1 2 a b 1
≤ ∈
x2
g(x )= (V V )dx.
2 b a
Z −
0
Then if g( ) 0 and g(x ) 0 it follows that g(x) 0 and the eigenvalues are ordered, i.e. E E .
2 a b
∞ ≥ ≥ ≥ ≤
We can extend Corollary 1 to the case of n intersections, n=1, 2, 3, ..., say at points x , x , x , .... As before
1 2 3
we suppose that V V on the first interval x [0, x ]. Then we assume that the sequence xi+1 V V dx,
a ≤ b ∈ 1 xi | b − a|
i = 1, 2, 3, ..., n, of absolute areas is nonincreasing (if n is odd then xn V V dx ∞ RV V dx ), this
xn−1| b − a| ≥ xn | b − a|
leads to g 0 on x [0, ) thus, according to the first theorem, E ER. R
a b
≥ ∈ ∞ ≤
Now we state and give proof of the second refined comparisontheorem. Where the difference V V is multiplied
b a
−
by upper ϕ or lower ϕ component of the Dirac spinor.
1 2
Theorem 2: The potential V belongs to one of the classes (1)–(3) and has ϕ –weighted area, S =sV, and
l
x
p(x)= (V (t) V (t))ϕ (t)dt, x [0, ). (11)
b a l
Z − | | ∈ ∞
0
Then if p 0, the eigenvalues are ordered, i.e. E E , where ϕ =ϕ if s=1 and ϕ =ϕ if s= 1, i=a or b.
a b l 1i l 2i
≥ ≤ −
Proof: We prove the theorem for the spin symmetric case and assume that the upper component of the Dirac
spinor is known,so s=1 and ϕ =ϕ ; for the other casethe proofis essentially the same. The rightside of(5) after
l 1i
integration by parts becomes
∞ ∞
2 (V V )ϕ ϕ dx=ϕ p∞ 2 p(ϕ )′dx,
Z b− a 1a 1b 1b |0 − Z 1b
0 0
where p is defined by (11) for ϕ = ϕ . The expression ϕ p∞ = 0, because p(0) = 0 and lim ϕ = 0. Then
1i 1a 1b |0 x→∞ 1
relation (5) takes the form
∞ ∞
(E E ) (ϕ ϕ +ϕ ϕ )dx= 2 p(ϕ )′dx.
b a 1a 1b 2a 2b 1b
− Z − Z
0 0
Functions ϕ andϕ′ havedifferent signs thus p(ϕ )′ 0 andwe conclude E E , whichinequality establishes the
1 1 1b ≤ a ≤ b
theorem.
(cid:3)
The wave functions vanish at infinity, thus the potential difference might be bigger in the second theorem than in
the first one and still lead to E E . As before we can formulate simpler sufficient condition for spectral ordering
a b
≤
if more detailed potential behaviour is known:
Corollary 2: Let the comparison potentials belong to one of the classes (1)–(3). If the potentials cross over once,
say at x , V V for x [0, x ], and
1 a b 1
≤ ∈
∞
p( )= (V V )ϕ dx,
b a l
∞ Z − | |
0
or if the potentials cross over twice, say at x and x , x <x , V V for x [0, x ], and
1 2 1 2 a b 1
≤ ∈
x2
p(x )= (V V )ϕ dx.
2 b a l
Z − | |
0
Then if p( ) 0 or p(x ) 0 it follows that p(x) 0 and the eigenvalies are ordered, i.e. E E , where ϕ =ϕ
2 a b l 1i
∞ ≥ ≥ ≥ ≤
if s=1 and ϕ =ϕ if s= 1, i=a or b.
l 2i
−
Corollary 2 can also be generalized for the case of n intersections: if V V on x [0, x ] and the sequence
a b 1
xi+1 (V V )ϕ dx, i = 1, 2, 3, ..., n and ϕ = ϕ if s = 1 and ϕ = ϕ ≤if s = 1,∈i = a or b, is nonincreasing
xi | b− a l| l 1i l 2i −
R(and, if n is odd, xn (V V )ϕ dx ∞ (V V )ϕ dx), then p 0 on x [0, ), so, according to Theorem
xn−1| b− a l| ≥ xn | b− a l| ≥ ∈ ∞
2, we conclude E R E . R
a b
≤
Refined comparison theorems for the Dirac equation with spin and pseudo–spin symmetry in d dimensions. 6
6
5
4
3
2
1
0 0.5 1 1.5 2 2.5 3 3.5
x
FIG. 1: Potential Va dashed lines and Vb full line.
D. An Example
In this section as an example we consider the extension of Corollary 1 to the case of n intersections in the spin
symmetriccase. WetaketheharmonicoscillatorV andamodifiedharmonicoscillatorV asourcomparisonpotentials:
a b
sin(x3+β)
V =ax2 and V =bx2 1+ .
a b (cid:18) x3+β (cid:19)
Both comparison potentials satisfy (2) for s = 1. If a = b the substitution z = x3 +β transforms the integral (10)
into
∞ b ∞ sinz
(V V )dt= dz.
b a
Z − 3Z z
0 β
Choosing β =1.64, and calculating numerical values, we find that the first area is bigger then the second one:
π sinz 2π sinz
| |dz =0.43810> | |dz =0.43379.
Z z Z z
β π
The sinz is a periodic function, thus sinx = siny where x [(k 1)π, kπ] and y =x+π, k =3, 4, 5, ..., then
| | | | ∈ −
it is clear that
kπ sinz (k+1)π sinz
| |dz > | |dz.
Z z Z z
(k−1)π kπ
Therefore
∞
(V V )dt 0,
b a
Z − ≥
0
because successive positive and negative areas of the integrand do not increase in absolute value. Thus g >0 and by
Theorem1wehaveE E . Thispredictionisverifiedbyaccuratenumericalcalculations: fora=b=0.5,β =1.64,
a b
≤
and m = 1.2 the comparison potentials intersect at infinitely many points (see Figure 1) and numerical eigenvalues
are E =1.77935 E =1.85470.
a b
≤
Refined comparison theorems for the Dirac equation with spin and pseudo–spin symmetry in d dimensions. 7
III. THE d-DIMENSIONAL CASE
A. The Dirac equation in d dimensions
The Dirac equation in d>1 dimensions is given by [78]
d
∂Ψ
i =HΨ, where H = α p +(m+S)β+V,
s s
∂t
Xs=1
where we use naturalunits ~=c=1, m is the mass of the particle, the functions V and S are spherically symmetric
vectorandscalarpotentials,and α andβ areDiracmatrices,whichsatisfyanti–commutationrelations;theidentity
s
{ }
matrix is implied after the potential V. The above equationcan be written as the following system of two first–order
differential equations [78–81]
k
ψ′ =(m+E+S V)ψ dψ , (12a)
 1 − 2− r 1
 ψ′ =(m E+S+V)ψ + kdψ , (12b)
2 − 1 r 2

whereψ andψ areradialwavefunctions,r = r ,prime′ denotesthederivativewithrespecttor,k =τ j+ d−2 ,
1 2 k k d 2
τ = 1,andj =1/2,3/2,5/2,.... WeassumethatthepotentialsV andS aresuchthatthereisanenergy(cid:0)eigenvalu(cid:1)e
±
E and that equations (12a)–(12b) are the eigenequations for the corresponding pair of radial eigenstates. For d>1,
the wave functions vanish at r =0, and for bound states they obey the normalization condition
∞
(ψ ,ψ )+(ψ ,ψ )= (ψ2+ψ2)dr =1.
1 1 2 2 Z 1 2
0
As in one dimension, we now compare the system (12a)–(12b) for the eigenvalues respectively E and E :
a b
k
ψ′ =(m+E +S V )ψ dψ , (13a)
 1 a a− a 2a− r 1a
 ψ′ =(m E +S +V )ψ + kdψ , (13b)
2 − a a a 1a r 2a

and
k
ψ′ =(m+E +S V )ψ dψ , (14a)
 1 b b− b 2b− r 1b
 ψ′ =(m E +S +V )ψ + kdψ . (14b)
2 − b b b 1b r 2b

Then we form the following combination of the equations: (13a)ψ (13b)ψ (14a)ψ + (14b)ψ , which, after
2b 1b 2a 1a
− −
integration and some simplifications, takes the form
∞ ∞
(E E ) (ψ ψ +ψ ψ )dr = [(V V S +S )ψ ψ +(V V +S S )ψ ψ ]dr. (15)
b a 1a 1b 2a 2b b a a b 1a 1b b a a b 2a 2b
− Z Z − − − −
0 0
By introducingthe parameters,we cancombine the spinandpseudo–spinsymmetriccases,i.e. S =sV where s=1
if S =V and s= 1 if S = V. Then the above expression for the S =sV case becomes
− −
∞ ∞
(E E ) (ψ ψ +ψ ψ )dr =2 (V V )ψ ψ dr, (16)
b a 1a 1b 2a 2b b a qa qb
− Z Z −
0 0
where q = 1 if s = 1 and q = 2 if s = 1. If the wave functions are nodeless, i.e. have constant sign on [0, ),
− ∞
and the potentials are ordered, say V V , then the integrands of (16) have constant sign and E E , which is
a b a b
≤ ≤
equivalent to the usual comparison theorem. We shall refine that theorem later, as in the one-dimensional case. For
example, we may replace V V by the weaker condition rV (t)t−2skddt rV (t)t−2skddt for some cases. We
a ≤ b 0 b ≥ 0 a
shall consider theorems for specific classes of potentials in secRtion C. below. R
Refined comparison theorems for the Dirac equation with spin and pseudo–spin symmetry in d dimensions. 8
Now, if two comparisonscalar potentials S and S are equal but the vector potentials V and V are different, i.e.
a b a b
S =S and V =V , the relation (15) can be rewritten as
a b a b
6
∞ ∞
(E E ) (ψ ψ +ψ ψ )dr = (V V )(ψ ψ +ψ ψ )dr. (17)
b a 1a 1b 2a 2b b a 1a 1b 2a 2b
− Z Z −
0 0
Then the following comparison theorem immediately follows:
Theorem 3: If S =S and V V , then E E .
a b a b a b
≤ ≤
s
As an example we consider the Coulomb potential S = S = , with s = 0.7. For the vector potentials we
a b
−r
α 4β
choosethe soft–corepotential[82,83] V = andsech–squaredpotential[84–87] V = . If
a −(rq +aq)1/q b −(ebr+e−br)2
α=0.8, a=1.6, q =3, β =0.5, and b=0.31 the potentials are ordered V V . Then, by Theorem 3, we conclude
a b
≤
E E , whichis verifiedby accuratenumericaleigenvaluesE =0.77260 E =0.81648form=1, τ = 1, d=5,
a b a b
≤ ≤ −
and j =1/2.
We note that expression (17) is exactly the same as (11) from the recent work [88]. Therefore Theorem 3 can be
refined in the same manner and corresponding corollaries can be derived mutatis mutandis.
We also note that one can derive similar theorem in one dimension. That is to say, we can obtain the expression
∞ ∞
(E E ) (ϕ ϕ +ϕ ϕ )dx = (V V )(ϕ ϕ +ϕ ϕ )dx from (4) and conclude that if S = S and
b − a 0 1a 1b 2a 2b 0 b − a 1a 1b 2a 2b a b
Va Vb, thRen Ea Eb. R
≤ ≤
B. Classes of potentials
Here we characterize the relationship between the eigenvalue E and mass of the particle m depending on the type
of the potential V. As in one dimension, equations (12a)–(12b) can be written in a Schr¨odinger–like form
k (k +s)
ψ′′+ d d +2(E+sm)V ψ = (m2 E2)ψ, (18)
− (cid:18) r2 (cid:19) − −
where ψ =ψ if s=1 and ψ =ψ if s= 1. We shall consider the following three classes of potential:
1 2
−
(i) V is finite near infinity and
(1) sV 0 and lim V =0. This implies m<E <m;
≤ r→∞ −
(ii) V is unbounded near infinity and
(2) sV 0 and lim V = s . This implies sE > m
≥ r→∞ ∞
or
(3) sV 0 and lim V = s . This implies sE < m.
≤ r→∞ − ∞ −
Following a similar path as in one dimension, one can verify that the above classes of potentials and relations
between energy E and mass m are valid for the system of the Dirac coupled equations (12a)–(12b) under spin and
pseudo–spin symmetry.
C. Refined comparison theorems in d dimensions
In that section we refine relativistic comparison theorems in a way that the graphs of the potentials can crossover
in a controlledmanner with the preservation of spectral ordering. Our establishment of refined comparisontheorems
requiresmonotonebehaviourofthe wavefunctionandconsequentlyaconstantsignofitsderivative. Butboundstate
wave functions are zero at the origin and vanish at infinity. Thus even if the wave function has constant sign, its
derivative changes sign. The following lemma helps us to allow for this.
Refined comparison theorems for the Dirac equation with spin and pseudo–spin symmetry in d dimensions. 9
Lemma 2: At the bottom of an angular–momentum subspace labelled by j, the functions ψ rkd and ψ r−kd are
1 2
monotone in the spin and pseudo–spin symmetric cases respectively.
Proof: In the case s=1, using (12a)–(12b), we find
ψ rkd ′ =(m+E)ψ rkd.
1 2
(cid:0) (cid:1)
Clearly ψ rkd ′ has constant sign since m+E is constant and ψ is either nonpositive or nonnegative. The case
1 2
s= 1 c(cid:0)an be(cid:1)proven similarly.
− (cid:3)
As in thee one–dimensional case, we need to know some characteristics of the nodeless state of the Dirac coupled
equations(12a)–(12b). Forexample,considerthecase(1)withs=1: accordingtotheprevioussection,thepotential
V 0, lim V =0, and m<E <m. The system (12a)–(12b) then takes the following form
≤ r→∞ −
k
ψ′ =(m+E)ψ dψ , (19a)
 1 2− r 1
 ψ′ =(m E+2V)ψ + kdψ . (19b)
2 − 1 r 2

Asymptotically near infinity the above equations become,
ψ′ =(m+E)ψ , (20a)
1 2
(cid:26) ψ′ =(m E)ψ . (20b)
2 − 1
The components ψ and ψ of the Dirac spinor vanish at infinity. Suppose that ψ 0 before vanishing, then ψ′ 0
and it follows from1the sy2stem above that ψ 0 and ψ′ 0. The assumption1ψ≥ 0 before vanishing, lea1ds≤to
2 ≤ 2 ≥ 1 ≤
ψ 0 and ψ′ 0. Consequently ψ and ψ must vanish with different signs.
2N≥ear the or2ig≤in if k > 0, we set ψ1 0 s2o ψ′ 0, then equation (19a) leads to ψ 0. The assumption ψ 0
d 1 ≥ 1 ≥ 2 ≥ 1 ≤
would give ψ 0. The quantity m E +2V has to change sign to guarantee the necessary behaviour of ψ , i.e.
2 2
≤ −
increasing then decreasing if it is nonnegative or decreasing then increasing if it is nonpositive. Since lim V = 0,
r→∞
then lim(m E+2V)= m E > 0, so lim (m E+2V)<0, thus m E+2V changes sign exactly once from
r→∞ − − r→0+ − −
negativeto positive(more details canbe foundin[89]). Thenfor k <0 equation(19b)leads to ψ 0if ψ 0and
d 1 2
≤ ≥
vice versa. Hence, if k > 0 both wave function components start at the origin with the same sign, but at infinity
d
they must have different signs: thus one of the wave function components will have at least one node in the lowest
state (Figure 2, left graph). When k < 0, ψ and ψ start with different signs and then vanish with different signs:
d 1 2
thus neither of them has a node in the groundstate (Figure 2, rightgraph). Similarly analysing the case s= 1 and
−
other types of potential, we get that ψ and ψ have no nodes if k > 0. Finally, we infer: the Dirac radial wave
1 2 d
functions ψ and ψ , which satisfy (12a)–(12b), are node free in the case S = sV if sk < 0. We note that Alberto
1 2 d
et. al. in the recent work [93] derived general result: n =n if sk <0, where n and n are the numbers of nodes
1 2 d 1 2
of ψ and ψ respectively. Now, using this result, we state and prove the refined comparison theorem.
1 2
Refined comparison theorems for the Dirac equation with spin and pseudo–spin symmetry in d dimensions. 10
3e+10 1500
2.5e+10 1000 Psi_1
3*Psi_2
2e+10 500
1.5e+10 0 5 10 15 20 25
1e+10 –500
Psi_1
2*Psi_2
5e+09 –1000
0 2 4 6 8 10 12 14 –1500
FIG. 2: Ground state of theDirac coupled equations (12a)–(12b) in thespin–symmetric case, V =S,for thecut–off Coulomb
v
potential [90–92] V = . Left graph: τ = 1, m = 1, d = 4, j = 1/2, v = 1.5, a =0.01, and E = 0.47399. Right graph:
−r+a
τ = 1, m=1, d=7, j =5/2, v=2.5, a=1.2, and E =0.69329.
−
Theorem 4: The potential V belongs to one of the classes (1)–(3) and has r−2skd–weighted area, S =sV, sk <0,
d
and
r
ρ(r)= (V (t) V (t))t−2skddt, r [0, ). (21)
b a
Z − ∈ ∞
0
Then if ρ 0, the eigenvalues are ordered, i.e. E E .
a b
≥ ≤
Proof: We prove the theorem for the spin symmetric case, i.e. s=1; for the other case the proof is similar. Let us
integrate by parts the right side of (16) in the following way
∞ ∞
(V V )ψ ψ dr = ψ ψ ρr2kd ∞ ρ ψ ψ r2kd ′dr,
Z b− a 1a 1b 1a 1b 0 −Z 1a 1b
0 (cid:12) 0 (cid:0) (cid:1)
(cid:12)
where ρ is defined by (21). Since ρ(0)=0 and lim ψ =0, relation (16) becomes
1
r→∞
∞ ∞
(E E ) (ψ ψ +ψ ψ )dr = ρ ψ ψ r2kd ′dr. (22)
b a 1a 1b 2a 2b 1a 1b
− Z −Z
0 0 (cid:0) (cid:1)
Sinceψ vanishesatinfinity,thefunctionψ rkd vanishesaswell. Thus,accordingtoLemma2,thefunctions ψ rkd ′
1 1 1
and ψ rkd have different signs, which leads to ψ ψ r2kd ′ 0. Then it follows from expression (22) (cid:0)that th(cid:1)e
1 1a 1b
≤
nonnegativity of ρ and the nodeless form of the w(cid:0)ave functio(cid:1)ns result in Ea Eb.
≤ (cid:3)
As in the one–dimensionalcase,ifwe know moredetails concerningthe behaviourofthe comparisonpotentials, we
can state simpler sufficient conditions:
Corollary 4: Let the comparison potentials belong to one of the classes (1)–(3). If the potentials cross over once,
say at r , V V for r [0, r ], and
1 a b 1
≤ ∈
∞
ρ( )= (V V )r−2skddr,
b a
∞ Z −
0
or if the potentials cross over twice, say at r and r , r <r , V V for r [0, r ], and
1 2 1 2 a b 1
≤ ∈
r2
ρ(r )= (V V )−2skddr.
2 b a
Z −
0