Memoirs on Differential Equations and Mathematical Physics
Volume 35, 2005, 129–146
J. Shabani and A. Vyabandi
ON A NEW TWO PARAMETER MODEL
OF RELATIVISTIC POINT INTERACTIONS
IN ONE DIMENSION
Abstract. We introduce and study a new 2-parameter model of relativistic point interactions in one dimension formally given by
Dα,y = D + αδ(x − y); x ∈ R, y > 0,
where D is the free Dirac Hamiltonian and α is a 2×2 matrix. Dα,y provides
a generalization of two models of relativistic point interactions discussed in
[Lett. Math. Phys. 13 (1987), 345–358].
We define Dα,y using the theory of self-adjoint extensions of symmetric
closed operators in Hilbert spaces, derive its resolvent equation, analyze
its spectral properties and discuss scattering theory for the pair (Dα,y , D).
We also study the nonrelativistic limit of Dα,y which provides a special
2-parameter model of the one-dimensional generalized point interactions
introduced in [1].
2000 Mathematics Subject Classification. 81Q10, 47N50, 81Q05.
Key words and phrases. boundary conditions problem, one-dimensional Dirac operator, self-adjoint extensions, resolvent equation, spectral properties, nonrelativistic limit, scattering theory.
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On a New Two Parameter Model of Relativistic Point Interactions
131
1. Introduction
Relativistic point interactions in one dimension have been discussed for
a long time in various areas of physics, in particular in connection with the
Kronig–Penney type models and Saxon–Hutner conjecture (see, e.g., [2–10]
and references therein).
The first rigorous mathematical formulation of these interactions was
given in [10] using the theory of self-adjoint extensions of symmetric closed
operators in Hilbert spaces.
Indeed [10] defines two models Dα,y and Tβ,y of relativistic point interactions which provide natural generalisation of nonrelativistic one-dimensional
δ-interactions of the first and the second type [11].
This paper considers a 2-parameter model Dα,y of relativistic point interactions in one dimension formally given by
Dα,y = D + αδ(x − y),
x ∈ R,
y > 0,
where D is the free Dirac Hamiltonian and α is a 2 × 2 matrix of the form
α 0
α=
, α, α̃ ∈ R.
0 α̃
To the best of our knowledge, this model is new. It provides a straightforward generalisation of the models Dα,y and Tβ,y discussed in [10] which
correspond to the special cases α 6= 0, α̃ = 0 and α = 0, α̃ = −c2 β 6= 0,
respectively.
The paper is organized as follows. In Section 2, we define the quantum
Hamiltonian Dα,y following the strategy used in [12, 14] in the case of
relativistic δ-sphere interactions. We also derive the resolvent equation of
Dα,y , analyse its spectral properties and carry out a systematic study of the
scattering theory for the pair (Dα,y , D).
The nonrelativistic limit corresponding to Dα,y defines a 2-parameter
model ∆α,β,y of nonrelativistic point interactions in one dimension. It turns
out that this model is a special case of the one dimensional generalized point
interactions introduced in [1].
Section 3 is devoted to the study of ∆α,β,y . In a forthcoming paper [15]
we generalize the results of section 2 and 3 to finitely and infinitely many
relativistic point interactions.
2. The Relativistic Point Interaction
A. Definition of the Hamiltonian. The quantum Hamiltonian describing a relativistic point interaction is formally given by
H = D + αδ(x − y),
where α is a 2 × 2 matrix of the form
α 0
α=
,
0 α̃
x ∈ R,
y > 0,
α, α̃ ∈ R,
(1)
(2)
132
J. Shabani and A. Vyabandi
and the
N one-dimension free Dirac operator in the Hilbert space H =
L2 (R) C2 is defined by [10]
!
2O
d
c2
d O
c
−ic
2
dx
D = −ic
σ1 +
σ3 =
,
2
d
dx
2
−ic dx
− c2
(3)
O
2,1
2
D(D) = H (R)
C ,
where
(i) σ1 =
0 1
1 0
, σ3 =
1 0
0 −1
are Pauli matrices in C2 ;
(ii) c is the velocity of light;
(iii) H m,n (Ω) is the Sobolev space of indices(m, n).
We consider the symmetric closed operator Ḋy defined by
Ḋy = D,
O
g1
2,1
2
D(Ḋy ) = g =
∈ H (R)
C |g(y±) = 0 .
g2
The adjoint Ḋy∗ of Ḋy reads
Ḋy∗ = D,
O
g1
∗
2,1
2
∈ H (R)
C |g ∈ ACloc (R − {y}) .
D(Ḋy ) = g =
g2
ACloc (Ω) denotes the set of locally absolutely continuous functions on Ω.
A straightforward computation shows that the equation
c2 i [ h c2
∗
∗
Ḋy g(z) = zg(z), g ∈ D(Ḋy ), z ∈ C −
− ∞, −
,∞
2
2
has the solutions
( 0
ζ ik (x−y)
,
1e
g
0
0 ,
( 0
i
0 ,
g (2) (z, x) =
ζ
ik0 (y−x)
2c
,
−1 e
(1)
where
i
(z, x) =
2c
r
x > y,
x < y,
x > y,
x < y,
c4
≡ k 0 (z),
4
1
c2
ζ = 0 [z + ], Im k 0 (z) ≥ 0,
ck (z)
2
1
k =
c
0
Im k 0 > 0,
z2 −
(4)
z ∈ C.
(5)
On a New Two Parameter Model of Relativistic Point Interactions
133
Thus Ḋy has deficiency indices (2,2) and hence it has a four-parameter
family of self-adjoint extensions. Let us now construct the self-adjoint extension corresponding to the free Dirac operator with the potential
α 0
V (x) = αδ(x − y), α =
, α, α̃ ∈ R.
0 α̃
Assume that g satisfies the equation
[D + αδ(x − y)]g = zg,
!
d
c2
−ic dx
2
D=
,
2
d
−ic dx
− c2
α=
α 0
,
0 α̃
α, α̃ ∈ R,
(6)
and the limits g(y±) exist. Integrating the equation (6) over (y − , y + )
and taking the limit as → 0, we get the following boundary conditions
(
α
g2 (y+) − g2 (y−) = − i2c
[g1 (y+) + g1 (y−)],
(7)
i
α̃
g1 (y+) − g1 (y−) = − 2c [g2 (y+) + g2 (y−)].
As indicated in [12], the boundary conditions in (7) defines a self-adjoint
extension of Ḋy iff α = α+ .
N
Consider in L2 (R) C2 the operator Dα,y defined by
Dα,y =
c2
2
d
− ic dx
2
d
−ic dx − c2
!
,
(8)
)
iα [g (y+) + g (y−)]
g
(y+)
−
g
(y−)
=
−
2
1
2c 1
D(Dα,y ) = g ∈ D(Ḋy∗ ) 2
.
g1 (y+) − g1 (y−) = − iα̃ [g2 (y+) + g2 (y−)]
2c
(
According to [12], the operator Dα,y provides the mathematical definition
of the formal expression (1).
The case α = 0 (i.e., α = α̃ = 0) in the equation (8) yields the free Dirac
Hamiltonian D0,y ≡ D.
The case α 6= 0, α̃ = 0 in the equation (8) yields the Hamiltonian Dα,y
which describes the relativistic δ-point interaction of the first type centered
at y ∈ R defined by [10]:
Dα,y = D,
O
D(Dα,y ) = g ∈ H 2,1 (R − {y})
C2 |g2 ∈ ACloc (R),
g1 ∈ ACloc (R − {y}); g2 (y+) − g2 (y−) = −(iα/c)g1 (y) ,
− ∞ < α ≤ ∞.
The case α = 0, α̃ = −c2 β 6= 0 in the equation (8) yields the Hamiltonian
Tβ,y which describes the relativistic δ-point interaction of the second type
134
J. Shabani and A. Vyabandi
centered at y ∈ R defined by [10]:
Tβ,y = D,
O
D(Tβ,y ) = g ∈ H 2,1 (R − {y})
C2 |g2 ∈ ACloc (R − {y}),
g1 ∈ ACloc (R); g1 (y+) − g1 (y−) = iβcg2 (y) ,
− ∞ < β ≤ ∞.
Following [12], we note that all the results corresponding to Dα,y could
be generalized to the model Dα̂,y formally given by
H = D + α̂δ(x − y),
x ∈ R,
y > 0,
where α̂ is a non-diagonal 2 × 2 matrix with α̂ = α̂+ .
B. The resolvent equation. From the Krein resolvent formula [16], after
a straightforward computation (see, e.g., [11]) we obtain
(Dα,y − z)−1 = (D − z)−1 −
1
α(f˜k0 (. − y), .)fk0 (. − y)+
−1
(2c + iαζ)(2c + iα̃ζ )
αα̃ ˜
(fk0 (. − y), .)fˆk0 (. − y)+
+ α̃(g̃k0 (. − y), .)gk0 (. − y) + i
2c
αα̃
+i
(g̃k0 (. − y), .)ĝk0 (. − y) , z ∈ ρ(Dα,y ), Im k 0 > 0,
(9)
2c
n
o
2 S
2
where Rk0 = (D − z)−1 , z ∈ C − (−∞, − c2 ] [ c2 , ∞) is the free Dirac
resolvent with integral kernel [10]
i
ζ
sgn(x − x0 ) ik0 |x−x0 |
0
e
Rk0 (x − x ) =
ζ −1
2c sgn(x − x0 )
−
and
fk0 (x − y) =
( ζ
f˜k0 (x − y) =
(
gk0 (x − y) =
(
g̃ (x − y) =
(
fˆk0 (x − y) =
(
k0
ik0 (x−y)
,
1 e
ζ
ik0 (y−x)
,
−1 e
−ζ ik0 (x−y)
,
1 e
−ζ ik0 (y−x)
,
−1 e
1
ik0 (x−y)
,
ζ −1 e
−1
ik0 (y−x)
,
ζ −1 e
1
ik0 (x−y)
,
−ζ −1 e
−1
ik0 (y−x)
e
,
−ζ −1
1
ik0 (x−y)
,
ζ −1 e
1
ik0 (y−x)
,
−ζ −1 e
x > y,
x < y,
x > y,
x < y,
x > y,
x < y,
x > y,
x < y,
x > y,
x < y,
On a New Two Parameter Model of Relativistic Point Interactions
ĝk0 (x − y) =
z ∈C−
( ζ
135
ik0 (x−y)
,
1 e
−ζ ik0 (y−x)
,
1 e
2i
− ∞, −
c
2
x > y,
x < y,
h c2
∪
,∞
, Im k 0 > 0.
2
Remark 1. From the equation (9), a straightforward computation shows
(i) As α̃ → 0, the Hamiltonian Dα,y converges in the norm resolvent
sense to Dα,y :
n · lim (Dα,y − z)−1 = (Dα,y − z)−1 , z ∈ ρ(Dα,y ) ∩ ρ(Dα,y ),
α̃→0
where [10]
α
(f˜k0 (. − y), .)fk0 (. − y),
2c(2c + iαζ)
z ∈ ρ(Dα,y ), Im k 0 > 0.
(Dα,y − z)−1 = (D − z)−1 −
(ii) Let α̃ = −βc2 , β ∈ R. Then as α → 0, the Hamiltonian Dα,y
converges in the norm resolvent sense to Tβ,y :
n. lim (Dα,y − z)−1 = (Tβ,y − z)−1 ,
α→0
z ∈ ρ(Dα,y ) ∩ ρ(Tβ,y ),
where [10]
β
(g̃k0 (. − y), .)gk0 (. − y),
2(2 − iβcζ −1 )
z ∈ ρ(Tβ,y ), Im k 0 > 0.
(Tβ,y − z)−1 = (D − z)−1 +
The following theorem gives the additional information on the domain of
Dα,y .
Theorem 2.1. The domain D(Dα,y ), −∞ < α, α̃ ≤ ∞, y ∈ R, consists
of all elements ψα of the type
2ic
ψα (x) = φk0 (x) −
×
(2c + iαζ) (2c + iα̃ζ −1 )
n
× αφk0 ,1 (y)fk0 (x − y) + α̃φk0 ,2 (y)gk0 (x − y)+
o
αα̃
αα̃
+i
φk0 ,1 (y)fˆk0 (x − y) + i
φk0 ,2 (y)ĝk0 (x − y) , x 6= y, (10)
2c
2c
N
φk0 ,1 2,1
where φk0 = φ 0 ∈ D(D) = H (R) C2 and Im k 0 > 0. The decompok ,2
sition (10) is unique and with ψα of this form we obtain
(Dα,y − z)ψα = (D − z)φk0 .
(11)
Let ψα ∈ D(Dα,y ) and assume that ψα = 0 in an open set ϑ ∈ R. Then
Dα,y ψα = 0 in ϑ, i.e., Dα,y describes a local interaction.
136
J. Shabani and A. Vyabandi
Proof. The following relation
D(Dα,y ) = (Dα,y − z)−1 (D − z)D(D) =
n
1
= Rk 0 −
[α(f˜k0 (. − y), .)fk0 (. − y)+
(2c + iαζ)(2c + iα̃ζ −1 )
αα̃ ˜
(fk0 (. − y), .)fˆk0 (. − y)+
+ α̃(g̃k0 (. − y), .)gk0 (. − y) + i
2c
o
αα̃
+i
(g̃k0 (. − y), .)ĝk0 (. − y)] (D − z)D(D),
2c
z ∈ ρ(Dα,y ), Im k 0 > 0,
proves (10).
Next let ψα = 0. Then
n
2ic
αφk0 ,1 (y)fk0 (x − y)+
(2c + iα)(2c + iα̃ζ −1 )
αα̃
+ α̃φk0 ,2 (y)gk0 (x − y) + i
φk0 ,1 (y)fˆk0 (x − y)+
2c
o
αα̃
+i
φk0 ,2 (y)ĝk0 (x − y)
2c
0
and φk0 ∈ C (R), implies φk0 = 0 which proves the uniqueness of (10). The
relation (11) follows from
φk0 (x) =
1
(Dα,y − z)−1 (D − z)φk0 = φk0 −
×
(2c + iαζ)(2c + iα̃ζ −1 )
n
× α(f˜k0 (. − y), (D − z)φk0 )fk0 (. − y)+
+ α̃(g̃k0 (. − y), (D − z)φk0 )gk0 (. − y)+
αα̃ ˜
+i
(fk0 (. − y), (D − z)φk0 )fˆk0 (. − y)+
2c
o
αα̃
+i
(g̃k0 (. − y), (D − z)φk0 )ĝk0 (. − y) =
2c
= ψα , z ∈ ρ(Dα,y ), Im k 0 > 0.
Let us now prove locality. We assume first y 6∈ ϑ. Then
((D − z)(αφk0 ,1 (y)fk0 (. − y) + α̃φk0 ,2 (y)gk0 (. − y)+
αα̃
αα̃
+i
φk0 ,1 (y)fˆk0 (. − y) + i
φk0 ,1 (y)ĝk0 (. − y)))(x) = 0
2c
2c
implies that
(Dα,y ψα )(x) = zψα (x) + ((D − z)φk0 )(x) =
=
2ic
((D − z)(αφk0 ,1 (y)fk0 (. − y)+
(2c + iαζ)(2c + iα̃ζ −1 )
αα̃
+ α̃φk0 ,2 (y)gk0 (. − y) + i
φk0 ,1 (y)fˆk0 (. − y)+
2c
On a New Two Parameter Model of Relativistic Point Interactions
+i
αα̃
φk0 ,1 (y)ĝk0 (. − y)))(x) = 0,
2c
137
x ∈ ϑ.
Second, if y ∈ ϑ, then ψα (y) = 0 and φk0 ∈ C 0 (R) implies φk0 = 0, and
hence
(Dα,y ψα )(x) = zψα (x) = 0, x ∈ ϑ.
C. Spectral properties. The spectral properties of Dα,y follow from (9).
For α, α̃ ∈ R the essential spectrum is purely absolutely continuous and co2 S
2
2
2
incides with (−∞, − c2 ] [ c2 , ∞). The point spectrum of Dα,y in [− c2 , c2 ]
contains the poles of the resolvent equation (9). Then Dα,y has two eigen2
2
values in [− c2 , c2 ] iff α, α̃ < 0:
( n 2 2 2
c (4c −α ) c2 (α̃2 −4c2 )
2(4c2 +α2 ) , 2(4c2 +α̃2 )
σp (Dα,y ) =
∅,
o
, α, α̃ < 0
α, α̃ ≥ 0, α = α̃ = ∞,
and two resonances iff α, α̃ > 0.
2
Following the strategy of [10], one proves that the operator (Dα,y − c2 )
converges in the norm resolvent sense to the Schrödinger operator ∆α,β,y
O 1 0 c2
−1
−1
n − lim (Dα,y −
− z) = (∆α,β,y − z)
, α, β ∈ R,
0 0
c→∞
2
where
d2
,
dx2
D(∆α,β,y ) =
0
(12)
g (y+) − g 0 (y−) = α [g(y+) + g(y−)]
2,2
2
= g ∈ H (R − {y}) ,
g(y+) − g(y−) = β2 [g 0 (y+) + g 0 (y−)]
− ∞ < α, β ≤ ∞.
∆α,β,y = −
The Hamiltonian ∆α,β,y defines a special exactly solvable model of nonrelativistic point interaction. In section 2.A we will discuss the properties of
the above Hamiltonian.
In particular, as c → ∞, the two eigenvalues of Dα,y (rest energy sub2
2
tracted) (Eα − c2 ), (Eα̃ − c2 ) give their respective nonrelativistic limits
2 2
c2
c (4c − α2 ) c2
lim Eα −
= lim
−
c→∞
c→∞
2
2(4c2 + α2 )
2
−1
α2
α2
=−
lim 1 + 2
4 c→∞
4c
2
α
=− ,
4 c2 (α̃2 − 4c2 ) c2
c2
lim Eα̃ −
= lim
−
c→∞
c→∞
2
2(4c2 + α̃2 )
2
138
J. Shabani and A. Vyabandi
−1
4
4
α̃
, β=− 2
= − 2 lim 1 + 2 2
β c→∞
β c
c
4
= − 2.
β
2
In Section 2.C we will show that − α4 and − β42 are the two eigenvalues of
∆α,β,y [see the equation (27)].
D. Scattering theory of the pair (Dα,y , D). From Theorem 2.1, the
scattering wave functions of Dα,y are defined by
ψα,y (k, σ, x) =
0
eik σx
−
σζ −1 eik0 σx
( ( 0
)
0
ζ ik (x−y)
2iceik σy
e
,
x>y
1
−
α
+
ζ
ik0 (y−x)
, x<y
(2c + iαζ)(2c + iα̃ζ −1 )
−1 e
(
)
ik0 (x−y)
1
, x>y
−1
ζ −1 e
+ α̃σζ
+
−1
ik0 (y−x)
, x<y
ζ −1 e
(
)
ik0 (x−y)
1
,
x>y
αα̃
ζ −1 e
+
+i
1
ik0 (y−x)
2c
, x<y
−ζ −1 e
ζ ik0 (x−y)
σζ −1 αα̃
e
,
x>y
1
+i
,
−ζ ik0 (y−x)
2c
, x<y
1 e
x, y ∈ R, k 0 ≥ 0, σ = ±1, −∞ < α, α̃ ≤ ∞.
A straightforward computation shows that ψα,y (k, σ) are eigenfunctions associated with Dα,y corresponding to left (σ = +1) and right (σ = −1)
incidence [11].
The asymptotic forms of ψα,y are defined by [11, 17]
ψα,y (z, +1, x) =
(
l
Tα,y
(z)ψ(z, +1, x)
as x → ∞,
ψ(z, +1, x) + Rlα,y (z)ψ(z, −1, x) as x → −∞,
ψα,y (z, −1, x) =
(
ψ(z, −1, x) + Rrα,y (z)ψ(z, +1, x) as x → ∞,
r
Tα,y
(z)ψ(z, −1, x)
as x → −∞,
(13)
where ψ(z, σ, x) is the solution of Dψ = zψ given by
iσk0 x
e
ψ(z, σ, x) =
, σ = ±1,
0
σζ −1 eiσk x
with k 0 and ζ defined by (4) and (5), respectively. Then the reflection and
transmission coefficients from the left (σ = +1) and the right (σ = −1) are
139
On a New Two Parameter Model of Relativistic Point Interactions
defined by
"
ik0 x #
0
0
1
e
eik x , −ζeik x ψα,y (z, +1, x) − −1 ik0 x
,
Rlα,y (z) = lim
x→−∞ 2
ζ e
"
#
0
1 −ik0 x −ik0 x e−ik x
r
Rα,y (z) = lim
,
e
, ζe
ψα,y (z, −1, x)−
x→+∞ 2
−ζ −1 e−ik0 x
1 −ik0 x −ik0 x l
e
, ζe
ψα,y (z, +1, x),
Tα,y
(z) = lim
x→+∞ 2
0
1 ik0 x
r
Tα,y
(z) = lim
e , −ζeik x ψα,y (z, −1, x),
x→−∞ 2
k 0 ≥ 0, −∞ < α, α̃ ≤ ∞, y ∈ R.
After a straightforward computation, one obtains
Theorem 2.2. Let α, α̃ ∈ R − {0}, y ∈ R. Then the unitary on-shell
scattering matrix Sα,y (z) in C2 associated with the pair (Dα,y , D) reads
"
#
l
Tα,y
(z) Rrα,y (z)
Sα,y (z) =
, k 0 ≥ 0, −∞ < α, α̃ ≤ ∞, y ∈ R,
r
Rlα,y (z) Tα,y
(z)
with
2ic
l
×
Tα,y
(z) = 1 − 2c + iαζ 2c + iα̃ζ −1
αα̃ r
× αζ + α̃ζ −1 + i
= Tα,y
(z),
c
0
2ic
αζ − α̃ζ −1 e2ik y ,
Rlα,y (z) = − 2c + iαζ 2c + iα̃ζ −1
0
2ic
αζ − α̃ζ −1 e−2ik y .
Rrα,y (z) = − 2c + iαζ 2c + iα̃ζ −1
In particular, as c → ∞, the unitary on-shell scattering matrix Sα,y (k 2 +
2
c
2
) gives its nonrelativistic limit Sα,β,y (k) [see the equation (28)]. Indeed,
l
lim Tα,y
(z) =
c→∞
= lim
c→∞
(
)
2ic
αα̃ −1
αζ + α̃ζ + i
1−
.
c
2c + iαζ 2c + iα̃ζ −1
2
(14)
Let z = k 2 + c2 , k > 0 and α̃ = −βc2 , β ∈ R, then after a straightforward
computation (2) reads
αβ
c2
l
2
4 −1
lim T
k +
=i
=
β
α
c→∞ α,y
2
4k 2 4k
− 2i 2k12 − i 4k
140
J. Shabani and A. Vyabandi
r
= lim Tα,y
(k 2 +
c→∞
and
c2
),
2
(15)
c2
lim Rlα,y (k 2 + ) =
c→∞
2
2ic
−1 2ik0 y
αζ
−
α̃ζ
e
=
= lim −
c→∞
(2c + iαζ)(2c + iα̃ζ −1 )
α
k + βk
=−
e2iky
β
α
8k 2 4k
− 2i 2k12 − i 4k
(16)
c2
=
lim Rrα,y k 2 +
c→∞
2
2ic
−1 −2ik0 y
= lim −
αζ
−
α̃ζ
e
=
c→∞
(2c + iαζ)(2c + iα̃ζ −1 )
α
k + βk
=−
(17)
e−2iky .
β
α
i
1
2
8k 4k − 2 2k2 − i 4k
We note that in the low-energy limit k → 0 (k 0 → 0 and z =
Sα,y
y ∈ R,
c2
k +
2
2
→
k→0
−∞ < α, β ≤ ∞,
c2
2 )
0 −1
,
−1 0
0 0
α 6=
.
0 0
In the high energy we obtain
Sα,y
 4c 1 − αβ
1
4
→

k→∞
(2c + iα)(2 − iβc) −2i(α + βc2 )
−∞ < α, β ≤ ∞,
y ∈ R,

−2i(α + βc2 )
,
4c 1 − αβ
4
and
Sα,y
c→∞
→
k→∞
0 1
.
1 0
Remark 2. It turns out that the equation (19) can be obtained as a special
case of the one-dimensional generalized point interactions introduced in [1].
We note that the pole of Sα,y (z) coincides with the bound state (α, α̃ < 0)
or resonance (α, α̃ > 0) of Dα,y .
141
On a New Two Parameter Model of Relativistic Point Interactions
3. The Nonrelativistic Point Interaction
A. Basic properties. Consider in the Hilbert space L2 (R) the closed and
nonnegative operator H̃y defined by
d2
,
dx2
D(H̃y ) = {g ∈ H 2,2 (R)|g(y) = g 0 (y) = 0}.
H̃y = −
The adjoint H̃y∗ of H̃y is defined by
d2
,
dx2
D(H̃y∗ ) = H 2,2 (R − {y}),
H̃y∗ = −
y ∈ R.
Hence the equation
H̃y∗ f (k) = k 2 f (k),
f (k) ∈ D(H̃y∗ ),
has two linearly independent
f1 (k, x) =
f2 (k, x) =
k 2 ∈ C − R,
Im k > 0,
solutions
eik(x−y) , x > y,
0,
x < y,
0,
x > y,
eik(y−x) , x < y,
(18)
Im k > 0.
Therefore H̃y has deficiency indices (2,2) and hence it has a four-parameter
family of self-adjoint extensions. We consider in L2 (R) the operator ∆α,β,y
defined by the equation (12)
d2
,
dx2
D(∆α,β,y ) =
0
g (y+)−g 0 (y−) = α [g(y+)+g(y−)]
2,2
2
= g ∈ H (R − {y}) ,
g(y+)−g(y−) = β2 [g 0 (y+)+g 0(y−)]
− ∞ < α, β ≤ ∞.
∆α,β,y = −
(19)
Let αβ−4 = 0, α, β ∈ R. Then the integration by parts shows that ∆α,β,y
is symmetric and since H̃y has deficiency indices (2,2) and the 2-boundary
conditions in (19) are symmetric and linearly independent, it follows that
∆α,β,y is self-adjoint ([18], Theorem XII.4.30). We will accept those α, β
which satisfy the condition αβ − 4 = 0, α, β ∈ R.
The case α = 0, β = 0 in the equation (19) yields the kinetic energy
Hamiltonian ∆0 in L2 (R)
d2
, D(∆0 ) = H 2,2 (R).
dx2
The case α 6= 0, β = 0 in the equation (19) gives the δ-point interaction
of the first type, whereas α = 0, β 6= 0 leads to a δ-point interaction of the
second type [11].
∆0 = −
142
J. Shabani and A. Vyabandi
B. Resolvent equation. The resolvent of ∆α,β,y is given by the following
theorem.
Theorem 3.1. The resolvent of ∆α,β,y is given by
= Gk +
2
α
4k
(∆α,β,y − k 2 )−1 =
n α
1
i 2 (Gk (. − y), .)Gk (. − y)+
β
2k
− i 21 2k12 − i 4k
β ˜
αβ
˜
+ i (G̃
(Gk (. − y), .)Gk (. − y)−
k (. − y), .)G̃k (. − y) +
2
4k
αβ ˜
˜ (. − y) ,
(G̃k (. − y), .)G̃
−
k
4k
k 2 ∈ ρ(∆α,β ),
(20)
Im k > 0, −∞ < α, β ≤ ∞, y ∈ R,
where
i
Gk (x − y) =
2k
˜ (x − y) = i
G̃
k
2k
eik(x−y) , x > y,
eik(y−x) , x < y, Im k > 0,
(21)
eik(x−y) ,
x > y,
−eik(y−x) , x < y, Im k > 0.
(22)
Proof. We use the resolvent formula
(∆α,β,y − k 2 )−1 = Gk −
2
1 X
λij (k)(fj (−k̄), .)fi (k),
4k 2 i,j=1
(23)
where fj , j = 1, 2, are defined by (18).
Next consider h ∈ L2 (R) and define the function g ∈ D(∆α,β,y ) by
g(k, x) = ((∆α,β,y − k 2 )−1 h)(x).
After imposing the boundary conditions in (19), one obtains
α
1
i 2k2 + i β2
i 2kα2 + αβ
− i β2
2k
α
λ(k) = α
. (24)
β
i 2kα2 + i β2
i 2kα2 + αβ
− i 12
2 4k − i 12 4k
2k − i 2
Inserting (24) in (23), one obtains (20).
Remark 3. From (20) one obtains the following results.
(i) As β → 0, the Hamiltonian ∆α,β,y converges in the norm resolvent
sense to −∆α,y :
n · lim (∆α,β,y − z)−1 = (−∆α,y − z) z ∈ ρ(∆α,β,y ) ∩ ρ(−∆α,y ),
β→0
where [11]
2αk
(Gk (. − y), .)Gk (. − y),
iα + 2k
Im k > 0, −∞ < α ≤ ∞, y ∈ R,
(−∆α,y − k 2 )−1 = Gk −
k 2 ∈ ρ(−∆α,y ),
with Gk defined by (21).
143
On a New Two Parameter Model of Relativistic Point Interactions
(ii) As α → 0, the Hamiltonian ∆α,β,y converges in the norm resolvent
sense to −∆β,y :
n · lim (∆α,β,y − z)−1 = (−∆β,y − z) z ∈ ρ(∆α,β,y ) ∩ ρ(−∆β,y ),
α→0
where [11]
2βk 2 ˜
˜ (. − y),
(G̃k (. − y), .)G̃
k
2 − iβk
Im k > 0, −∞ < β ≤ ∞, y ∈ R,
(−∆β,y ) − k 2 )−1 = Gk −
k 2 ∈ ρ(−∆α,y ),
˜ defined by (22). The additional information on the domain of
with G̃
k
∆α, β, y is given by the following theorem.
Theorem 3.2. The domain D(∆α,β,y ), −∞ < α, β ≤ ∞,
consists of all elements ψα,β of the type
+
2
α
4k
y ∈ R3 ,
ψα,β (x) = ϕk (x)+
α
β 0
1
˜ (x − y)+
ϕ (y)G̃
1
i 2 ϕk (y)Gk (x − y) −
k
β
1
2k
2k
− i 2 2k2 − i 4k
αβ
αβ 0
˜
+ ϕk (y)Gk (x − y) − i 2 ϕk (y)G̃k (x − y) , x 6= y,
(25)
4k
4k
where ϕk ∈ D(∆0 ) = H 2,2 (R) and Im k > 0. The decomposition (3.2) is
unique and with ψα,β of this form we obtain
(∆α,β,y − z)ψα,β = (∆0 − z)ϕk .
(26)
Let ψα,β ∈ D(∆α,β,y ) and assume that ψα,β = 0 in an open set ϑ̃ ∈ R3 .
Then ∆α,β,y ψα,β = 0 in ϑ̃, i.e., ∆α,β,y describes a local interaction.
Proof. Similar to the proof of Theorem 2.1.
C. Spectral properties. For α, β ∈ R, the essential spectrum of ∆α,β,y is
purely absolutely continuous and coincides with [0, ∞), while the singular
spectrum is empty. The point spectrum of ∆α,β,y is given as the poles of
the resolvent equation (20). One obtains
( n 2
o
− α4 , − β42 , α, β < 0,
σp =
(27)
0,
α, β ≥ 0.
iα
For α, β > 0, ∆α,β,y has two resonances at k1 = − 2i
β and k2 = − 2 with
resonance functions respectively given by
α (x−y)
e2
, x > y,
ψk1 (x) =
α > 0,
α
e 2 (y−x) , x < y,
( 2
e β (x−y) ,
x > y,
ψk2 (x) =
β > 0.
2
−e β (y−x) , x < y,
144
J. Shabani and A. Vyabandi
D. Scattering theory of the pair (∆α,β,y , ∆0 ). From (3.2) one can define
the generalized function associated with ∆α,β,y by
ψα,β,y =
ik(x−y)
1
α ikσy
e
,
ikσx
− 2e
=e
+ ik(y−x)
β
e
,
α
1
1
2k
4k 4k − i 2 2k2 − i 4k
ik(x−y)
ik(x−y)
σβ ikσy
αβ ikσy
e
,
x>y
e
,
+
e
+i e
−eik(y−x) , x < y
eik(y−x) ,
2
4k
ik(x−y)
σαβ ikσy
e
,
x>y
+i
e
,
−eik(y−x) , x < y
4k
x, y ∈ R, k > 0, σ = ±1, −∞ < α, β ≤ ∞.
x>y
x<y
x>y
x<y
+
+
The corresponding reflection and transmission coefficients from the left (σ =
+1) and the right (σ = −1) are defined by [11]
Rlα,β,y (k) = lim eikx ψα,β,y (z, +1, x) − eikx ,
x→−∞
r
Rα,β,y (k) = lim e−ikx ψα,β,y (k, −1, x) − e−ikx ,
x→+∞
l
Tα,β,y
(k)
= lim e−ikx ψα,β,y (k, +1, x),
r
Tα,β,y
(k)
= lim eikx ψα,β,y (k, −1, x),
x→+∞
x→−∞
k ≥ 0, −∞ < α, β ≤ ∞, y ∈ R.
After a straightforward computation, one obtains
Theorem 3.3. Let α, β ∈ R − {0}, y ∈ R. Then the unitary on-shell
scattering matrix Sα,β,y (k) in C2 associated with the pair (∆α,β,y , ∆0 ) reads
l
Tα,β,y (k) Rrα,β,y (k)
Sα,β,y (k) =
,
r
Rlα,β,y (k) Tα,β,y
(k)
(28)
k ≥ 0, −∞ < α, β ≤ ∞, y ∈ R,
with
−1
1
i
αβ
4
r
= Tα,β,y
(k),
β
−
i
2k2
4k
α
+
βk
k
e2iky ,
Rlα,β,y (k) = −
β
α
i
1
2
8k 4k − 2 2k2 − i 4k
α
+ βk
k
e−2iky .
Rrα,β,y (k) = −
β
α
i
1
2
8k 4k − 2 2k2 − i 4k
l
Tα,β,y
(k) = i
4k 2
α
4k
−2
On a New Two Parameter Model of Relativistic Point Interactions
145
We note that the limit β → 0 (respectively α → 0 ) in the equation (28)
gives the unitary on-shell scattering matrix Sα,y (k) (Sβ,y (k)) associated with
the pair (−∆α,y , −∆) and (−∆β,y , −∆), respectively [11]. One obtains
2k
−iαe−2iky
−1
→
Sα,β,y (k) β→0 (2k + iα)
=
−iαe2iky 2k
= Sα,y (k),
Sα,β,y (k)
k ≥ 0, −∞ < α ≤ ∞, y ∈ R,
2
−iβke−2iky
−1
→
(2 − iβk)
=
α→0
−iβke2iky 2
= Sβ,y (k),
k ≥ 0, −∞ < β ≤ ∞, y ∈ R.
In the low-energy limit k → 0 we get
Sα,β,y (k)
→
k→0
and in the high energy limit we obtain
Sα,β,y (k)
→
k→∞
0 −1
−1 0
0 1
.
1 0
We note that the pole of Sα,β,y (k) coincides with the bound state (α, β < 0)
or resonance (α, β > 0) of ∆α,β,y .
Acknowledgement
The second author would like to thank Prof. Blanchard, Universität
Bielefeld, for his invitation to the Research Institute BIBOS and for interesting discussions. We are both thankful to Prof. Exner for useful comments
on a preliminary draft of this paper and for sharing with us a copy of an
unpublished preprint.
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(Received 7.03.2005)
Authors’ addresses:
J. Shabani∗
UNESCO Harare Cluster office
Harare, Zimbabwe
E-mail: j.shabani@unesco.org
A. Vyabandi
Université du Burundi
Institut de Pédagogie Appliquée(IPA)
BP 5223 MutangaI Bujumbura, Burundi
E-mail: vyabandi@hotmail.com
∗
On leave of absence from University of Burundi, Faculty of Science, BP 2700 Bujumbura, Burundi