Quantum double well in magnetic field: tunnelling, libration, normal forms

We consider the spectral problem for 2-D magnetic Shrödinger operator with the potential having a form of symmetric double well and with constant magnetic field. This problem is very well studied when the magnetic field is absent. In particular, in semiclassical limit the distance between two lowest eigenvalues is exponentially small with respect to a corresponding small parameter $h$. The derivation of explicit formulas for the splitting of eigenvalues (V. Maslov, A. Poljakov, E. M. Harrel, B. Helfer, J. Sjostrand, B. Simon, etc.) is based on a passage from standard fast oscillating WKB-functions $A(x)e^{\frac{iS(x)}h}$ to fast decaying functions $A(x)e^{\frac{-S(x)}h}$. This passage changes the corresponding real-valued standard Hamiltonian $H=p^2/2+V (x)$ to (again!) the real-valued “tunnelling” Hamiltonian $H=-p^2/2+V(x)$, which allows one to use the theory of the classical Hamiltonian systems for description of tunnel effects. This idea does not work for the situation including the magnetic field: the “tunnelling” Hamiltonian becomes a complex-valued function and one cannot use the theory of the classical Hamiltonian systems. Our observation is that one can reduce the quantum double-well problem for the magnetic Shrödinger operator to the standard quantum double-well problem using the partial Fourier transform and mixed momentum-position coordinates. We show also that the splitting formula takes natural and simple form if it is based on so-called libration and normal forms coming from classical mechanics. We apply these results for description of tunnelling of wavepackets in quantum nanowires.
This work was done together with J. Brüning and R. V. Nekrasov and was supported by DFG-RAS project 436 RUS 113/990/0-1, Grant № 2.1.1/450 of Russian Federation Ministry of Sciences and Education and by RFBR grant № 11-01-00973.