Painlevé IV Transcendents Generated from the Complex Oscillator
51
g(x) = −x − {ln[ψ E 3 (x)]}
,
thus Painlevé IV transcendents can be found by simply supplying the extremal states
of Hamiltonians ruled by second-degree PHA, having third-order differential ladder
operators.
4 Complex Oscillator
The complex oscillator potential is given by [17]
V (x) =
1
2
ω
2 x
2 , ω = e
iθ , −
π
2
≤ θ <
3π
2
.
The general solution to the associated Schrödinger equation reads
u(x, ε) = e
−
ωx 2
2
1 F 1
1
4
−
ε
2ω
,
1
2
; ωx
2
+ λ x 1 F 1
3
4
−
ε
2ω
,
3
2
; ωx
2
= e
ωx 2
2
1 F 1
1
4
+
ε
2ω
,
1
2
; −ωx
2
+ λ x 1 F 1
3
4
+
ε
2ω
,
3
2
; −ωx
2
,
where λ = 2 ν Γ
3
4 −
ε
2ω
/Γ
1
4 −
ε
2ω
. The square-integrable solutions of such
non-Hermitian Hamiltonian are given by
φ n (x) = C n H n (
√
ωx)e
−
1
2 ωx 2 , E n (θ ) = (n +
1
2 )e iθ ,
for
√
ω = e
i
θ
2 , −
π
2 < θ <
π
2 and
φ n (x) = D n H n (
√ −ωx)e
1
2 ωx 2 , E n (θ ) = (n +
1
2 )e i(θ−π) ,
for
√
−ω = e
i(
θ−π
2 ) ,
π
2 < θ <
3π
2 , where C n , D n are normalization factors and
H n (z) are the Hermite polynomials of complex argument z. Note that for θ = ±
π
2
there are no square-integrable solutions for the stationary Schrödinger equation
since the complex oscillator potential reduces then to the repulsive oscillator. As the
eigenvalues lie in the first or in the fourth quadrant in the complex E-plane we can
take −
π
2 < θ <
π
2 without loss of generality. Moreover, since E n (−θ ) = [E n (θ )]
∗
we can further restrict to 0 ≤ θ <
π
2 .
Similarly, as for the standard oscillator, the analogues of the annihilation and
creation operators
a ±
ω =
1
√
2
∓
d
dx + ωx
,