Bloch function in an external electric field and Berry-Buslaev phase. (English) Zbl 0869.35093

Bony, J.-M. (ed.) et al., New trends in microlocal analysis. Tokyo: Springer. 143-156 (1997).
The perturbed Lamé equation of \[ \left(-{{d^2}\over{du^2}}+2\rho(u+\omega)+\varepsilon u-E_0\right) \psi(u, \varepsilon)=0 \] is considered as a model equation. The asymptotic solution of this equation is constructed (\(\varepsilon\downarrow 0\)) using V. S. Buslaev’s ideas [Theor. Math. Phys. 58, 153-159 (1984; Zbl 0557.34053)]. It has the form of the so-called Bloch solution \(e^{iku}b(u, z)\). The leading term of this solution contains a geometric phase factor of the form \[ U(z)=e^{i\int\theta(z)dz}. \] where the function \(\theta\) is calculated exactly. This factor may be interpreted as an adiabatic phase found by M. V. Berry [Proc. R. Soc. Lond. Ser. A 392, 45-57 (1984)] in his study of the quantum adiabatic theorem.
For the entire collection see [Zbl 0859.00023].


35Q60 PDEs in connection with optics and electromagnetic theory
74F15 Electromagnetic effects in solid mechanics


Zbl 0557.34053