Physics Letters A 376 (2012) 2217–2221
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Physics Letters A
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An upper bound for asymmetrical spinless Salpeter equations
Claude Semay
1
Service de Physique Nucléaire et Subnucléaire, Université de Mons, Académie universitaire Wallonie-Bruxelles, Place du Parc 20, 7000 Mons, Belgium
article info abstract
Article history:
Received 29 March 2012
Received in revised form 22 May 2012
Accepted 22 May 2012
Available online 24 May 2012
Communicated by P.R. Holland
A generic upper bound is obtained for the spinless Salpeter equation with two different masses. Analytical
results are presented for systems relevant for hadronic physics: Coulomb and linear potentials when a
mass is vanishing.
©
2012 Elsevier B.V. All rights reserved.
1. Introduction
The spinless Salpeter equation (SSE), whose general form is
given by (
¯
h
=
c
=
1)
p
2
+
m
2
1
+
p
2
+
m
2
2
+
V
(
r
)
|φ=
M
|φ,
(1)
where M is the mass of the system, is the simplest relativistic
eigenvalue equation. It is sometimes denoted semirelativistic since
it is not a covariant formulation. This equation can be considered
as a Schrödinger equation with its nonrelativistic kinetic part re-
placed by a relativistic counterpart. More rigorously, it is obtained
from the covariant Bethe–Salpeter equation [1] with the follow-
ing approximations: Elimination of any dependences on timelike
variables and neglect of particle spin degrees of freedom as well
as negative energy solutions [2]. It has also been shown that a
Nambu–Goto string meson with a short-range Coulomb interaction
is described by a SSE with a time component vector funnel poten-
tial, for fixed angular momentum and large radial excitation [3].
Numerous techniques have been developed to solve numerically
this equation [4–8]. Though a quite high precision can be reached,
it is always interesting to obtain bounds on eigenvalues. First, they
can be used as verifications of the computation. Second, they are
generally a simple and fast way to obtain information about the
spectra. Sometimes, bounds yield analytical results which can give
precious details about the solutions of the SSE. With some rare
exceptions [9,10], most of the analytical results have been obtained
for the symmetrical version of the SSE [11–20]
σ
p
2
+
m
2
+
V
(
r
)
|φ=
M
|φ.
(2)
E-mail address: claude.semay@umons.ac.be.
1
FRS-FNRS Senior Research Associate.
The one-body (two-body) case is treated with
σ
=
1 (2). Note that
an arbitrary positive value of
σ
can be considered to study duality
relations between different many-body systems [21].
The main purpose of this work is to obtain a generic upper
bound on the eigenvalues of the general SSE. The computation are
presented in Section 2 after a brief description of the method used.
In Section 3, some analytical results obtained with this bound are
given and compared with numerical solutions. Concluding remarks
are given in Section 4.
2. Upper bound
The method used in this Letter is known as the envelope the-
ory. It was introduced a long time ago for nonrelativistic kinemat-
ics and then generalized for semirelativistic systems [12–14,22].
It was rediscovered more recently under the name of the auxil-
iary field method [10,21,23,24]. Both techniques are equivalent [25]
but they were introduced from completely different starting points.
In particular, the auxiliary field method uses the notion of auxil-
iary fields (also called einbein fields), first introduced to get rid of
the square root kinetic operator in calculations for semirelativistic
eigenvalue equations [26,27].
We will consider here a potential V
(
r
)
depending only on the
radial distance. The idea is to replace the Hamiltonian in (1) by the
following one
˜
H
=
ν
2
1
+
m
2
1
2
ν
1
+
ν
2
2
+
m
2
2
2
ν
2
+
p
2
2
μ
+
ρ
P
(
r
)+
V
I
(
ρ
)
−
ρ
P
I
(
ρ
)
,
(3)
with three real parameters
ν
1
,
ν
2
and
ρ
, and
μ
=
ν
1
ν
2
ν
1
+
ν
2
,
I
(
x
) =
K
−
1
(
x
),
K
(
x
) =
V
(
x
)
P
(
x
)
.
(4)
The kinematics is nonrelativistic, P
(
r
)
is the auxiliary potential and
the prime denotes the derivative. An eigenvalue of
˜
H is given by
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©
2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.physleta.2012.05.046