The Salpeter Equation: Nonlocality and Pseudodifferential Operators
105
In fact, once the parameter A in Eq. (11) gains the status of operator
1
ˆ
A 2 + 1
f (x) =
∞
0
e
−t ˆ
A J 0 (t)f (x)dt,
(12)
where f (x) is a given x-dependent differentiable function and J 0 is the Bessel
function of the first kind of first order [30], it is possible to apply the theory of the
evolution operator shown in [31, 32] and define the operator ˆ
A to obtain a recursive
series solution highlighting the nonlocal nature embedded in the Hamiltonian of the
spinless Salpeter equation.
The formal structure of Eq. (12) is extremely attractive since it can be applied to
the initial value problem (9) after rearranged it as:
i
∂
∂τ
ψ(ξ, τ ) =
1 −
∂ 2
∂ξ 2
1
1 −
∂ 2
∂ξ 2
ψ(ξ, τ ).
(13)
Moreover, by definition
¯
ψ(ξ, τ ) =
1
1 −
∂ 2
∂ξ 2
ψ(ξ, τ ),
(14)
the initial value problem (13) can be written as
i
∂
∂τ
ψ(ξ, τ ) =
1 −
∂ 2
∂ξ 2
¯
ψ(ξ, τ ).
(15)
Assuming ˆ
A = i
∂
∂ξ , ¯
ψ is immediately defined
¯
ψ(ξ, τ ) =
∞
0
J 0 (y)ψ(ξ − iy, τ )dy ψ(ξ, 0) = ψ 0 (ξ ),
(16)
where J 0 is a modified Bessel function of the first kind [30]. Then the solution
of (16) reads
ψ(ξ, τ ) =
∞
n=0
(iτ ) n
n!
ψ n (ξ ).
(17)
The initial condition ψ(ξ, 0) is the zero-order term, whereas the nth-term ψ n (ξ ) can
be defined in recursive way
ψ n (ξ ) =
1 −
∂ 2
∂ξ 2
+∞
0
J 0 (y)ψ n−1 (ξ − iy)dy.
(18)