Spectral properties of the scaled quartic anharmonic oscillator

Unfolding the Quartic Oscillator

“…18) does not look like Shanley's [Sh1] should not be a surprise, because our conventions for choosing the cuts are different Fig. 18.…”

Spectral inversion of an indefinite Sturm–Liouville problem due to Richardson

“…It appears that N d goes slowly to infinity withẼ ij . The last entry with N d = 10 involves maxima with coordinates (i, j) = (6, 64), (13,63), (28, 58), (30, 37), (43,48), and the number doubles by exchanging factors. The quantum numbers involved are n = 69, 75, 85, 86 and 90.…”

“…In Schrödinger this requires ∆ = ∆ E = 0 while in Richardson we need ∆ = ∆ λ = 0. There are numerical ways [13] of searching for such points while having no prior estimate of their location. The above method has problems near points where the logarithmic derivatives are singular.…”

“…An example of a problem that is both right-and left-definite in its Schrödinger and Sturmian versions is a quartic anharmonic oscillator [13] with V (x) = λx 2 + x 4 . In this problem on x ∈ [−∞, ∞], the additional function w(x) = x 4 serves to stabilize the spectrum at λ = 0.…”

“…If there were additional cases off the real axis, these would invert to become branch points of λ n (E) for complex E which could lead to coupling crossing. Hundreds of searches using the methods described in [13] have been carried out which yield only critical points for real λ. This implies that Richardson branch points occur only for real values of E. Level crossing branch points for λ and E n (λ) complex will invert to critical points in the complex domain of E. Since they play no important role in the Richardson problem, we have not studied them.…”