We prove the Pad?e (Stieltjes) summability of the perturbation series of any energy level En,1(?), n ? N, of the cubic anharmonic oscillator, H1(?) = p2 +x2 +i??x3, as suggested by the numerical studies of Bender and Weniger. At the same time, we give a simple proof of the positivity of every level of the PT -symmetric Hamiltonian H1(?) for positive ? (Bessis–Zinn Justin conjecture). The n zeros, of a state ?n,1(?), stable at ? = 0, are confined for ? on the cut complex plane, and are related to the level En,1(?) by the Bohr–Sommerfeld quantization rule (semiclassical phase-integral condition). We also prove the absence of non-perturbative eigenvalues and the simplicity of the spectrum of our Hamiltonians.
Grecchi V., Maioli M., Martinez A. (2009). Pade summability of the cubic oscillator. JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL, 42(42), 1-17 [10.1088/1751-8113/42/42/425208].
Pade summability of the cubic oscillator
GRECCHI, VINCENZO;MARTINEZ, ANDRE' GEORGES
2009
Abstract
We prove the Pad?e (Stieltjes) summability of the perturbation series of any energy level En,1(?), n ? N, of the cubic anharmonic oscillator, H1(?) = p2 +x2 +i??x3, as suggested by the numerical studies of Bender and Weniger. At the same time, we give a simple proof of the positivity of every level of the PT -symmetric Hamiltonian H1(?) for positive ? (Bessis–Zinn Justin conjecture). The n zeros, of a state ?n,1(?), stable at ? = 0, are confined for ? on the cut complex plane, and are related to the level En,1(?) by the Bohr–Sommerfeld quantization rule (semiclassical phase-integral condition). We also prove the absence of non-perturbative eigenvalues and the simplicity of the spectrum of our Hamiltonians.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
