Multidimensional tunneling between potential wells at non degenerate minima

Abstract

6 pages, 1 figureInternational audienceWe consider tunneling between symmetric wells for a 2-D semi-classical Schrödinger operator for energies close to the quadratic minimum of the potential V in two cases: (1) excitations of the lowest frequency in the harmonic oscillator approximation of V; (2) more general excited states from Diophantine tori with comparable quantum numbers

N wells at a circle. Splitting of lower eigenvalues

“…The case d > 1 is much more complicated. There are many results obtained in this area (see [9][10][11][12][13][14][15][16][17] and the list is far from exhaustive). The semiclassical asymptotics of the discrete spectrum and strict estimates of the splittings are described in [9] and other works of these authors (using the theory of pseudo differential operators).…”

“…The semiclassical expansion for the eigenfunctions and the rigorous asymptotics for the splitting widths in the lowest levels were obtained in [10] (with the use of Maslov's canonical operator). The possibility to solve this problem in that case was discussed during the Diffraction Day Conference 2014 in the talk of A. Anikin and M. Rouleux [12].…”

Нижняя Часть Спектра Двумерного Оператора Шредингера С Периодическим По Одной Переменной Потенциалом И Приложения К Квантовым Димерам

“…Теперь можно применить известные оценки [7] и получить утверждение "г" и асимптотики (см. [10], [11], [20]) для доказательства утверждения "д". А именно, случай 0 при m = 0 был рассмотрен в [10], случай 1 -позднее в [11].…”

“…А именно, случай 0 при m = 0 был рассмотрен в [10], случай 1 -позднее в [11]. Обобщение для произвольного m получено в [20]. Теперь выведем дисперсионные соотношения.…”

On semi-classical spectral series for an atom in a periodic polarized electric field

“…So we have to compute w near minimal geodesics γ E ′ (h) between ∂U ± E ′ (h) . Such (finitely many) minimal geodesics are also called librations [2], [6], [1]. Within the required accuracy on tunneling rates, we could again replace E ′ (h) by E ′ , which amounts to replace the librations by the instanton between U ± E ′ = {y ± 0 }.…”