Electron in the field of flexural vibrations of a membrane: Quantum time, magnetic oscillations, and coherence breaking

We have studied the motion of an electron in a membrane under the influence of flexural vibrations with a correlator that decreases upon an increase in the distance in accordance with the law r-^2η. We have conducted a detailed consideration of the case with η < 1/2, in which the perturbation theory is inapplicable, even for an arbitrarily weak interaction. It is shown that, in this case, reciprocal quantum time 1/τ _q is proportional to g ^1/(1-η) T ^{(2-η)/(2-2η)}, where g is the electron-phonon interaction constant and T is the temperature. The method developed here is applied for calculating the electron density of states in a magnetic field perpendicular to the membrane. In particular, it is shown that the Landau levels in the regime with ω _c τ _q » 1 have a Gaussian shape with a width that depends on the magnetic field as B ^η. In addition, we calculate the time τ_φ of dephasing of the electron wave function that emerges due to the interaction with flexural phonons for η < 1/2. It has been shown that, in several temperature intervals, quantity 1/τ_φ can be expressed by various power functions of the electron-phonon interaction constant, temperature, and electron energy.