## 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].

### MSC:

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

### Keywords:

perturbed Lamé equation; Bloch solution; adiabatic phase

Zbl 0557.34053