Reflectionless PT-symmetric potentials in the one-dimensional Dirac equation

0.25 MB·English

by  Francesco Cannata

Reflectionless -symmetric potentials in the
PT
one-dimensional Dirac equation
0
1
0
Francesco Cannata(1) and Alberto Ventura(1,2)
2
(1) Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Italy
n
a (2) ENEA, Centro Ricerche Ezio Clementel, Bologna, Italy
J

2 January 22, 2010
2

]
h Abstract
p Westudytheone-dimensionalDiracequationwithlocalPT-symmetric
-
h potentialswhosediscreteeigenfunctionsandcontinuumasymptoticeigen-
t functionsareeigenfunctionsofthe PT operator,too: ontheseconditions
a
the bound-statespectra are real and the potentials are reflectionless and
m
conserveunitarityinthescatteringprocess. Absenceofreflectionmakesit
[ meaningful to consider also PT-symmetric potentials that do not vanish

asymptotically.
2
v
4
1 Introduction
6
9
2 Reflectionless potentials, i.e. potentials that are transparent to incident waves
. at all energies,have playeda special role in quantum mechanics since the basic
1
0 paperbyKayandMoses[1],whoformulatedtheproblemofconstructingaplane
0 stratified dielectric medium transparent to electromagnetic radiation in terms
1 of a one-dimensional Schr˝odinger equation with a potential with preassigned
:
v bound-state spectrum that transmits without reflection continuum wave func-
i tionsatallincidentenergies. Fromamathematicalpointofview,theKay-Moses
X
methodisequivalenttosolvinganon-linearSchr˝odingerequationwhosepoten-
r
a tial,V (x),isaquadraticfunctionofafixednumber,n,ofunknownbound-state
wave functions[2]; it can also be considered as a kind of Hartree-Fock potential
with n occupied states for a system of particles interacting through schematic
contact interactions in one space dimension[3].
More recent approaches to reflectionless potentials in non-relativistic quan-
tum mechanics make use, among others, of Darboux transformations[4], super-
symmetrichyerarchyderivationsfromthetriviallytransparentconstantpotential[5]
and Casimir invariants of non-compact Lie groups[6], the latter method giving
rise to large families of reflectionless potentials in implicit form, in addition to
explicit analytical forms derived in previous approaches.
In relativistic quantum mechanics,the Kay-Mosesmethod has been applied
totheone-dimensionalDiracequationwitheitherscalar[7][8][9]orpseudoscalar
1
potentials[10], since the presence of a vector component may break the trans-
parency of the potential at all energies (see Ref.[7] and Section 3 of the present
work), with notable exceptions, one of which will be discussed in detail in Sec-
tion 3. The relativistic extension of the Kay-Mosesmethod is equivalent to the
solution of an auxiliary non-linear Dirac equation.
Reflectionless potentials play an interesting role in non-Hermitian theo-
ries,too,suchasquasi-Hermitianquantummechanics[11], -symmetricquan-
PT
tum mechanics[12][13], or pseudo-Hermitian quantum mecanics[14],[15]. As is
known,ifanon-HermitianpotentialV (x)isinvariantundertheproductofpar-
ity andtimereversal ,sothat V (x)( )−1 V∗( x)=V (x),andthe
P T PT PT ≡ −
bound-stateeigenfunctionsoftheSchr˝odingerHamiltonianH = 1 d2 +V (x)
−2mdx2
areeigenfunctions of (exact symmetry),the correspondingeigenvalues
PT PT
are real. As for the continuum of scattering states, it was proved in Ref.[16]
for asymptotically vanishing potentials in the Schr˝odinger equation that if the
asymptotic wave functions are eigenstates of (exact asymptotic sym-
PT PT
metry),the -symmetric potentialis reflectionlessandunitarityis conserved.
PT
InSection2ofthe presentworkweextendthe prooftotheDiracequationwith
potentials that admit non-zero constant limits at x .
→±∞
Scattering fromreflectionlesspotentials withexactasymptotic symme-
PT
try can thus be treated by the methods of standard quantum mechanics, with-
out the need for an equivalent Hermitian formulation, which is neither exempt
from technical difficulties, nor from ambiguity of interpretation: it has been
shownthat the equivalent Hermitian description of scattering from strongly lo-
calizednon-Hermitianpotentials, aDiracdeltafunctionwithcomplexcoupling
strengthinRef.[17]anda -symmetriccombinationofdelta functions inRef.
PT
[18],impliesstronglynon-localmetricoperatorsand,consequently,anapparent
breaking of causality due to incoming waves in the exit channel. This seems
to be the price one has to pay in order to restore unitarity in the scattering
process, although a new formulation of the problem[19][20], based on the dis-
cretization of the Schr˝odinger equation on an infinite one-dimensional lattice,
has provided examples where the metric operator in the Hermitian equivalent
formulation can be chosen as a diagonal matrix, called a quasi-local operator,
which prevents the appearance of incoming waves in the exit channel, at the
cost of a change of scale of the probability density on the left and the right of
the scattering centre.
In the presentstate of formulationof quasi-Hermitiantheories,however,we
share the opinion expressed in Ref.[18], that it makes sense to treat a non-
Hermitian scattering potential as an effective one, accepting that it may well
involve the loss of unitarity when attention is restricted to the system itself
and not its environment, with which it can exchange probability flux (see also
Refs. [21][22][23]). On the other hand, reflectionless potentials are a special
class of -symmetric potentials that conserve unitarity even in the standard
PT
formulationofquantummechanics;therefore,we believe thatthey maydeserve
a study of their own, not only in the standard framework, i.e. with a trivial
metric operator, adopted here, since it does not give rise to unphysical aspects
2
for an isolated system, but possibly also as a test of alternative approaches.
The mainscope ofthe presentworkis to investigatethe behaviourofreflec-
tionlesspotentials inrelativisticquantummechanics,under differentconditions
of Lorentz covariance, i.e. when they appear as vector, scalar or pseudoscalar
components in the one-dimensional Dirac equation. Section 2 describes the
general formalism, examples of scalar-plus-vector potentials are worked out in
Section 3, pseudoscalar potentials in Section 4 and scalar potentials in Section
5. Finally, Section 6 is dedicated to conclusions and perspectives.
2 General formalism
The time-independent Dirac equation in (1+1) dimensions with vector , scalar
andpseudoscalarpotentials, V, S andP, respectively,reads,in units ~=c=1
[α p +β(m+S(x))+iα βP (x)+V(x)]Ψ(x)=EΨ(x) . (1)
x x x
Here, Ψ(x) is a two-dimensional spinor and p i d . α and β are two
x ≡ − dx x
anti-commuting Hermitian traceless matrices with the property α2 = β2 =
x
1 0
1 , which can be identified with two Pauli matrices.
0 1 ≡ 2
(cid:18) (cid:19)
We assume for generality’s sake that S, P and V can have non-zero limits
atx= : lim S(x)=S ,andanalogousnotationsforP andV. Inthe
x
±∞ →±∞ ±
0 1 1 0
Dirac representation[24], α =σ = and β =σ = , and
x x 1 0 z 0 1
(cid:18) (cid:19) (cid:18) − (cid:19)
the asymptotic Dirac equation reads
m+S +V E i d +P ψ (x)
i ±d +P±− m− Sdx+V± E ψ1(x) =0. (2)
(cid:18) −dx ± − −(cid:0) ± ±(cid:1)− (cid:19)(cid:18) 2 (cid:19)
Let us sear(cid:0)ch for a sol(cid:1)ution of the following form
ψ (x)=A eik±x+B e ik±x
ψ (x1)=A C±eik±x+B±D− e ik±x , (3)
2 −
± ± ± ±
where A , B C and D are complex numbers.
± ± ± ±
Direct substitution of formulae (3) in the asymptotic Dirac equation (2)
yields
k +iP k +iP
C = ± ± , D = − ± ±
± m+S +E V ± m+S +E V
± − ± ± − ±
and the asymptotic momenta satisfy the relation
k2 =(E V )2 (m+S )2 P2 , (4)
± − ± − ± − ±
while A and B remain to be fixed on boundary conditions.
± ±
Like in our previous works[25],[26], which use the same representation of
Dirac matrices, the parity operator, , and the time reversaloperator, , are
P T
=P β =P σ ,
0 0 z
P
= β = σ ,
z
T K K
3
whereP changesxinto xand performscomplexconjugation. Theirproduct
0
− K
=P σ2 =P (5)
PT 0K z 0K
is thus the same as in non-relativistic quantum mechanics[16].
If the Dirac Hamiltonian on the left-hand side of Eq. (1) is -symmetric,
PT
S = S , V = V and P = P . symmetry of the potentials thus
+ ∗ + ∗ + ∗
implies −k2 = k2, w−hich means t−hat−eithPeTr k = k , or k = k . Using
∗ ∗ + ∗ +
formulae±(3), it i∓s easy to show that only with −the choice k −=k −the ratios of
∗ +
transmitted waves over incident waves remain finite even −if the amplitudes of
asymptotic wave functions may diverge at x = . This argument does not
±∞
hold for the reflection coefficient: therefore, only when reflection is identically
zeroitmakessensetotreat -symmetricpotentialsthatdonotvanishasymp-
PT
totically. In turn, k = k implies C = C and D = D . When all the
∗ ∗ ∗
potentials vanish asy±mptot∓ically, the w±ell-kno∓wn expre±ssions∓for free particles
are recovered: k2 =E2 m2, C = k , D = k .
− ± E+m ± −E+m
In general, A and B are linear combinations of the coefficients of asymp-
totic expansions±of two li±nearly independent solutions to Eq. (1), Ψ(1)(x) and
Ψ(2)(x)
1 1
xlim Ψ(i)(x)=ai±(k±) C eik±x+bi±(k±) D e−ik±x (i=1,2)
→±∞ (cid:18) ± (cid:19) (cid:18) ± (cid:19)
in the general asymptotic solution
lim Ψ(x)=α lim Ψ(1)(x)+β lim Ψ(2)(x) ,
x x x
→±∞ →±∞ →±∞
or
A =αa (k )+βa (k ) ,
1 2
± ± ± ± ±
B =αb (k )+βb (k ) .
1 2
± ± ± ± ±
α and β, in turn, can be fixed by boundary conditions. If Ψ(x) is a pro-
gressive wave, travelling from left to right, we must have, apart from a global
normalization constant, not relevant in this context,
1 1
lim Ψ(x)= eik−x+R e ik−x ,
x→−∞ C L→R D −
 (cid:18) − (cid:19) (cid:18) − (cid:19)
1
 limx→+∞Ψ(x)=TL→R C+ eik+x ,
(cid:18) (cid:19)
where the transmissionand reflection coefficients, T and R , have been
L R L R
→ →
introduced.
Therefore
A =αa +βa =1,
1 2
− − −
B =αb +βb =R ,
1 2 L R
 − − − →
B =αb +βb =0,
 A =+αa +1+βa 2=+ T ,
+ 1+ 2+ L R
→

4
whence
. ( RTLL→→RR== bbbb1111++++aaab2222−+−−−−−−aaba1111−+−−bbbb2222++++ ,. (6)
In the same way, if Ψ(x) is a regressive wave, travelling from right to left
1
lim Ψ(x)=T e ik−x ,
x→−∞ R→L D −
 (cid:18) − (cid:19)
1 1
 limx→+∞Ψ(x)= D+ e−ik+x+RR→L C+ eik+x .
(cid:18) (cid:19) (cid:18) (cid:19)
Thus
A =αa +βa =0,
1 2
− − −
B =αb +βb =T ,
1 2 R L
 − − − →
A =αa +βa =R ,
 +B =1α+b +β2+b =R1→.L
+ 1+ 2+
whence ( TRRR→→LL==baa1b2+11−++abaa21−22−−−−−−−−aaaa1111−−−−babb2222−+++ ., (7)
Not surprisingly, the transmission and reflection coefficients (6) and (7) are
the same as in the non-relativistic case[16].
ψ(1)(x)
TheWronskianoftwosolutionsoftheDiracequation,Ψ(1)(x) 1
≡ ψ2(1)(x) !
(2)
ψ (x)
and Ψ(2)(x) 1 is defined as
≡ ψ2(2)(x) !
ψ(1)(x) ψ(2)(x)
W (x) 1 1 =ψ(1)(x)ψ(1)(x) ψ(2)(x)ψ(1)(x) . (8)
≡(cid:12) ψ(1)(x) ψ(1)(x) (cid:12) 1 2 − 1 2
(cid:12) 2 2 (cid:12)
(cid:12) (cid:12)
It is easy(cid:12)(cid:12)to check that dW(x)(cid:12)(cid:12)= 0, i.e. W (x) = const., by expressing the
dx
derivatives of the spinor components as linear combinations of the components
themselves,asdictatedbyEq. (1). Ifthetwosolutionsarelinearlyindependent,
W =0, of course.
6
Usingdefinition(8)andasymptoticwavefunctions,Ψ(i) lim Ψ(i)(x),
x
± ≡ →±∞
one easily obtains
lim W (x) W =(a b a b )(D C ) . (9)
1 2 2 1
x ≡ ± ± ±− ± ± ±− ±
→±∞
Remembering expressions (6-7) of the transmission coefficients, formula (9)
yields
W =(a b a b )T (D C ),
1 2+ 2 1+ R L
− − − − → −− −
W =(a b a b )T (D C ).
+ 1 2+ 2 1+ L R + +
− − − → −
and W =W is equivalent to
+
−
T (D C )=T (D C ),
R L L R + +
→ −− − → −
5
or
T D C D C
L→R = −− − = −− − =eiν , (10)
T D C D C
R L + + ∗ ∗
→ − −− −
Here,ν =2arg(D C )isarealphase. When,inparticular,allpotentials
−− −
vanish asymptotically, C = D are real numbers and the two transmission
− − −
coefficients are equal.
The phasedifference, ν, of the two transmissioncoefficients is differentfrom
zero when the imaginary components of the -symmetric potentials do not
PT
vanish asymptotically and is present in non-relativistic quantum mechanics,
too,as recentlyshownin Ref.[27] fora -symmetricversionofthe hyperbolic
PT
Rosen-Morse potential.
Itisworthwhiletopointoutthattheformalismjustdevelopedreferstolocal
potentials. For non-local potentials it has been shown that the ratio of the two
transmissioncoefficientsisnot1,butacomplexnumberofunitmodulus,evenif
the imaginarypotentials vanish asymptotically,both in non-relativistic[28] and
relativistic wave equations[26]. In this case the two reflection coefficients have
the same phase, but different modulus and unitarity is broken.
Let us now apply the operator(5) to the generalasymptotic wavefunc-
PT
tions
1 1
lim Ψ(x) Ψ (x)=A eik±x+B e ik±x ,
x ≡ ± ± C ± D −
→±∞ (cid:18) ± (cid:19) (cid:18) ± (cid:19)
or, more conveniently, to the following interpolating function, which coincides
with the asymptotic wave functions at large x
| |
1 1
Ψ (x)= (1+sgn(x))Ψ (x)+ (1 sgn(x))Ψ (x) .
int. +
2 2 − −
By definition (5) one gets
1 1
PTΨint.(x)=Ψ∗int.(−x)= 2(1−sgn(x))Ψ∗+(−x)+ 2(1+sgn(x))Ψ∗−(−x) .
Imposing Ψ (x)=eiϕΨ (x), with ϕ a real phase, yields
int. int.
PT
Ψ ( x)=eiϕΨ (x) .
∗
± − ∓
Remembering the behaviourofk , C andD under complex conjugation,
± ± ±
we obtain the following constraints on A and B
± ±
A =eiϕA ,
B∗± =eiϕB∓ . (11)
∗
± ∓
For a progressive wave (A = 1, B = 0), this is equivalent to T =
+ L R
− →
A = e iϕ− and R = B = 0, while, for a regressive wave (A = 0,
+ − L R
B =1),oneobtainsT→ =B− =e iϕe andR =A =0. Inother−words,
+ R L − R L +
the potentials are reflect→ionless and conserve un→itarity, since the tranesmission
ceoefficients have unit modulus.e e
6
In the non-relativistic limit, C , D 1 and the lower components of
| ±| | ±| ≪
Dirac spinors are negligible with respect to the higher ones. Non-vanishing
potentials at x only affect asymptotic momenta k and the preceding
→ ±∞ ±
discussion and its conclusions remain valid, thus generalizing the case of short-
range potentials treated in Ref.[16].
It is worthwhileto pointoutthat potentials thatbehaveasymptoticallylike
-symmetric step functions (P = P and so on) may admit asymptotic
+ ∗
wPaTvefunctionsthatareeigenstatesof − ,−unlikethestepfunctionsthemselves,
PT
whicharenotreflectionless,becausetheasymptoticbehaviourofwavefunctions
is determined by the behaviour of the potentials in their whole domain.
In the following sections, we specialize the general interaction of Eq. (1)
to scalar-plus-vector, pseudoscalar and scalar potentials and, for each type of
potential, work out some examples in detail.
3 Scalar-plus-vector potentials
In the present section, we specialize Eq. (1) to a scalar-plus-vector potential
with the same x dependence: S(x)=c f(x), V (x)=c f(x), with c and c
S V S V
real coupling constants.
[α p +βm+(c β+c )f(x)]Ψ(x)=EΨ(x) , (12)
x x S V
WefinditconvenienttoadopttheDiracrepresentationα =σ andβ =σ .
x x z
ψ (x)
The Dirac equation (12) satisfied by the spinor Ψ(x) = 1 is thus
ψ (x)
2
(cid:18) (cid:19)
written explicitly in matrix form
m E+(c +c )f(x) i d ψ (x) 0
− V S − dx 1 = ,
i d m E+(c c )f(x) ψ (x) 0
(cid:18) − dx − − V − S (cid:19)(cid:18) 2 (cid:19) (cid:18) (cid:19)
(13)
which reduces to a system of two first-order equations for the unknown spinor
components ψ (x) and ψ (x) . In order to obtain analytic solutions, we limit
1 2
ourselves to the particular cases c = c and c = c , which correspond to
V S V S
−
spin symmetry and pseudo-spin symmetry in three space dimensions[29]. Let
us consider the case c =c =c first. It is easy to see that the two equations
V S ′
obtained from formula (13) reduce to the simple system
d2 ψ (x)+2c (E+m)f(x)ψ (x)= E2 m2 ψ (x) k2ψ (x)
−dx2 1 ′ 1 − 1 ≡ 1 .
ψ (x)= i d ψ (x)
( 2 −E+mdx(cid:0) 1 (cid:1)
(14)
Here,theequationsatisfiedbyψ (x)isSchr¨odinger-like,withthesame -
1
PT
symmetric form f(x) as the original Dirac equation and an energy-dependent
potentialstrengths (E)=2c (E+m)andψ (x)isobtainedbyderivingψ (x)
′ ′ 2 1
withrespecttox. Onther.h.s. ofthefirstequation,k2 >0forscatteringstates,
whileforboundstates,k2 <0impliesanimaginaryvalueofk,correspondingto
7
poles of the transmission coefficient. In the limiting case k =0, both normaliz-
ableboundstatesandnon-normalizablehalf-boundstates[30],correspondingto
transmission resonances, are possible, depending on the potentials under con-
sideration.
When c = c =c , ψ and ψ exchange their role, since ψ now satisfies
V S ′′ 1 2 2
−
a Schr¨odinger-like equation with the original f(x) and the energy-dependent
strength s (E)=2c (E m), while ψ is proportionalto the space derivative
′′ ′′ 1
−
of ψ .
2
d2 ψ (x)+2c (E m)f(x)ψ (x)= E2 m2 ψ (x) k2ψ (x)
−dx2 2 ′′ − 2 − 2 ≡ 2
ψ (x)= i d ψ (x)
( 1 −E mdx(cid:0)2 (cid:1)
−
(15)
Energy dependence of the coupling strengths in Eqs. (14-15) may affect the
reflection properties of a -symmetric potential. An example of this general
PT
behaviour is provided by the hyperbolic Scarf potential with integer coupling
constants l and n
l2+n(n+1) 1 il(2n+1) sinhx
f(x)= + , (16)
− 2m cosh2x 2m cosh2x
whichisknowntobereflectionlessintheSchr˝odingerequation[16](notethatthe
quoted reference uses units 2m = 1, as is common in non-relativistic quantum
mechanics). When inserted in the Dirac equation, it gives rise to an equivalent
Schr˝odinger-like equation (14) where the potential maintains the same shape,
but is no more reflectionless, because of the energy dependence of the coupling
strengths.
Onthe contrary,if f(x) exhibits anexact symmetry inthe Schr¨odinger
PT
problem,itmaintainsitintheDiracproblemwiththeappropriatesuperposition
of vector and scalar components, provided it is not connected with a particular
value of the coupling strength, s or s , which becomes a function of E.
′ ′′
A notable example is provided by the -symmetric potential
PT
1
f(x)= , (17)
(x+iǫ)2
where ǫ is an arbitrary real number, regularizing f at x = 0, which is a well-
known example of reflectionless potential in the Schr¨odinger case[16]. Let us
consider the case c = c = c first, so that the equation (14) satisfied by ψ
V S ′ 1
reads
d2 2c (E+m)
ψ (x)+ ′ ψ (x)= E2 m2 ψ (x) k2ψ (x) (18)
− dx2 1 (x+iǫ)2 1 − 1 ≡ 1
(cid:0) (cid:1)
for scattering states (E2 >m2). The above equation is Schr¨odinger-like and is
quicklysolvedbyintroducingthecomplexvariablez =k(x+iǫ)andfactorizing
ψ (z)=z1/2ϕ(z): in fact, the equation satisfied by ϕ
1
d2 d 1
z2 ϕ(z)+z ϕ(z)+ z2 2c(m+E) ϕ(z)=0 (19)
′
dz2 dz − − 4
(cid:20) (cid:21)
8
is a Bessel equation of index ν2 = 2c(m+E)+ 1. Note that ν is imaginary
′ 4
when 2c(m+E)+ 1 < 0, which can happen, for instance, for positive c and
′ 4 ′
large negative E, or viceversa.
Two linearly independent solutions of Eq. (19) with asymptotic behaviour
appropriate to scattering states are the Hankel functions of first and second
type, H(1)(z) and H(2)(z), respectively
ν ν
lim H(1)(z)= 2 1/2exp i z πν π ,
lim |z|→∞H(2ν)(z)= 2πz1/2exp i z− 2πν− 4π , (20)
|z|→∞ ν (cid:0)πz (cid:1) (cid:2)−(cid:0) − 2 − 4(cid:1)(cid:3)
(cid:0) (cid:1) (cid:2) (cid:0) (cid:1)(cid:3)
valid for ν > 1/2, argz < π, this latter condition being ensured by the
ℜ − | |
non-zero imaginary part of z, i.e. z = kǫ. According to formulae (14), the
ℑ
corresponding linearly independent solutions of the Dirac equation are
Ψ(k)(x) ψ1(k)(x) =z1/2 Hν(k)(z)
≡ −E+imddxψ1(k)(x) ! −iλ ddzHν(k)(z)+ 21zHν(k)(z) !
H(k)(z) (cid:16) (cid:17)
= z1/2 ν , (k =1,2)
−iλ Hν(k−)1(z)+ 1−2z2νHν(k)(z) !
(cid:16) (cid:17)
where λ k . In order to obtain the final form of the r.h.s., use has been
≡ E+m
made of the relation d H(k)(z) = H(k) (z) νH(k)(z). The asymptotic be-
haviour of the Dirac spdzinoνrs is ν−1 − z ν
H(k)(z)
lim Ψ(k)(x)= lim z1/2 ν ,
x→±∞ |z|→∞ −iλHν(k−)1(z) !
or, more explicitly, using formulae (20)
2 1/2 1 π π
lim Ψ(1)(x)= exp ikx kǫ i ν i (21)
x π λ − − 2 − 4
→±∞ (cid:18) (cid:19) (cid:18) (cid:19) (cid:16) (cid:17)
and
1/2
2 1 π π
lim Ψ(2)(x)= exp ikx+kǫ+i ν+i . (22)
x π λ − 2 4
→±∞ (cid:18) (cid:19) (cid:18) − (cid:19) (cid:16) (cid:17)
Formulae (21-22) are particular cases of the asymptotic formulae of Section
2, whose constants now are
1/2
2 π π
a =a = exp kǫ i ν i ,
1 1+
− π − − 2 − 4
(cid:18) (cid:19) (cid:16) (cid:17)
b =b =0,
1 1+
−
a =a =0,
2 2+
−
9
2 1/2 π π
b =b = exp kǫ+i ν+i ,
2 2+
− π 2 4
(cid:18) (cid:19) (cid:16) (cid:17)
so that formulae (6-7) immediately yield
T =T =1, R =R =0.
L R R L L R R L
→ → → →
Ofcourse,potential(17)doesnotsustainboundstateswith k=0, because
6
the transmission coefficients are independent of k and cannot have poles in k,
or E. When k =0, a non-trivialsolution can exist at E =m: in this case, Eq.
(18) reduces to
d2 4cm
′
ψ (x)+ ψ (x)=0. (23)
− dx2 1 (x+iǫ)2 1
ThesolutiontoEq. (23)canbesearchedforintheformofapower,(x+iǫ)γ,
thus leading to an algebraic equation for γ
γ2 γ 4cm=0,
′
− −
whose solutions are
1 (1+16cm)1/2
′
γ = ± .
1,2
2
Dependingonwhether1+16cm≷0,thetworootsareeitherrealorcomplex
′
conjugate: inbothcases,thegeneralsolutiontoEq. (23)canbeputintheform
ψ (x)=α (x+iǫ)γ1 +α (x+iǫ)γ2 ,
1 1 2
where α (i=1,2) are to be fixed on boundary conditions. It is easy to un-
i
derstand that a normalizable solution, i.e. a bound state, can exist only when
1+16cm > 1, or c > 0, by choosing α = 0. In this case, the solution for
′ ′ 1
ψ (x) reads
1
1−β
ψ1(x)=α2(x+iǫ) 2 . (24)
Here, β = √1+16cm > 1 and α can be determined by normalization of
′ 2
the complete Dirac spinor
+
∞ ψ (x)
dx ψ (x) i d ψ (x) 1 =1. (25)
1∗ 2mdx 1∗ i d ψ (x)
Z (cid:18) −2mdx 1 (cid:19)
−∞ (cid:0) (cid:1)
Both integrals in formula (25) can be computed analytically in terms of
asymptoticexpansionsofthehypergeometricfunction,F (A,B,C;z),sincethey
can be reduced to the integral representation
1+y2 −ady =yF 1,a,3; y2 +const. a> 1
2 2 − ℜ 2
Z (cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
yielding
+ Γ a 1
∞ 1+y2 −ady √π − 2 ,
≃ Γ(a)
Z−∞ (cid:0) (cid:1) (cid:0) (cid:1)
10