We prove the Pade (Stieltjes) summability of the perturbation series of anyenergy level En,1(β), n \in N, of the cubic anharmonic oscillator, H1(β) =p^2 +x^2 +i√βx^3, as suggested by the numerical studies of Bender andWeniger.At the same time, we give a simple proof of the positivity of every levelof the PT -symmetric Hamiltonian H1(β) for positive β (Bessis–Zinn Justinconjecture). The n zeros, of a state ψ_n,1(β), stable at β = 0, are confinedfor β on the cut complex plane, and are related to the level En,1(β) by theBohr–Sommerfeld quantization rule (semiclassical phase-integral condition).We also prove the absence of non-perturbative eigenvalues and the simplicityof the spectrum of our Hamiltonians.

Pade' summability of the cubic oscillator / Vincenzo, Grecchi; Maioli, Marco; Andre', Martinez. - In: JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL. - ISSN 1751-8113. - STAMPA. - 42:(2009), pp. 425208-425224. [10.1088/1751-8113/42/42/425208]

Pade' summability of the cubic oscillator

MAIOLI, Marco;
2009

Abstract

We prove the Pade (Stieltjes) summability of the perturbation series of anyenergy level En,1(β), n \in N, of the cubic anharmonic oscillator, H1(β) =p^2 +x^2 +i√βx^3, as suggested by the numerical studies of Bender andWeniger.At the same time, we give a simple proof of the positivity of every levelof the PT -symmetric Hamiltonian H1(β) for positive β (Bessis–Zinn Justinconjecture). The n zeros, of a state ψ_n,1(β), stable at β = 0, are confinedfor β on the cut complex plane, and are related to the level En,1(β) by theBohr–Sommerfeld quantization rule (semiclassical phase-integral condition).We also prove the absence of non-perturbative eigenvalues and the simplicityof the spectrum of our Hamiltonians.
2009
42
425208
425224
Pade' summability of the cubic oscillator / Vincenzo, Grecchi; Maioli, Marco; Andre', Martinez. - In: JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL. - ISSN 1751-8113. - STAMPA. - 42:(2009), pp. 425208-425224. [10.1088/1751-8113/42/42/425208]
Vincenzo, Grecchi; Maioli, Marco; Andre', Martinez
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/629058
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