β Scribed by Sandro Graffi; Vincenzo Grecchi
No coin nor oath required. For personal study only.
Existence of resonances in hydrogen Stark effect is proved. It is also proved that the divergent time-independent perturbation expansions are Borel summable to the resonances, and a simple application of the Borel-Pade" method for computing their position and width is indicated.
In 1955 Titchmarsh [1 ] gave a first proof of the continuity of the spectrum of the hydrogen-like Stark effect, described by the Hamiltonian operator:
(1) acting on L z (R3). Here 2F > 0 is the uniform electric field, and Z the atomic number. He also proved [2] that the spectrum of (t) on the negative real axis is asymptotically concentrated near the unperturbed eigenvalues [3]. The spectral concentration is a mathematical description of the 'weak quantization' experimentally observed in absence of eigenvalues, and must be originated by the occurrence of resonances.
In this letter we report on the existence of resonances, for a weak electric field, given by functions E m (F) which can be analytically continued to a complex region, reduce to the hydrogen eigenvalues for F = 0, and coincide with the Borel sum of the divergent time-independent perturbation expansion for Im(F) > 0.
Here we limit ourselves to a sketch of the proof. The complete details will appear elsewhere. The starting point is the well known separability of the Schroedinger equation H(F)t) = Eft in squared parabolic coordinates, which reduces the problem essentially to a system of two quartic anharmonic oscillators with a constraint on the total energy. This makes possible the application *Partially supported by G.N.F.M., C.N.R. **Partially supported by I.N.F.N., Sezione di Bologna.
π SIMILAR VOLUMES