## Turning points for adiabatically perturbed periodic equations.(English)Zbl 0987.35013

This work consists of two parts. Part I is devoted to the formal solutions of the general equation $i\varepsilon {dy\over dt}=A(t)y,\;t\in\Delta =[\alpha,\beta], \tag{1}$ where $$A(t)$$ is a differential operator defined on a space $$X$$ of $$a$$-periodic $$(a>0)$$ functions of $$x$$; $$\varepsilon >0$$, $$\varepsilon \to 0$$. There are introduced some suitable general assumptions on the structure of the spectrum of $$A(t)$$; it is defined the turning point as a point with a certain spectral property. Under these assumptions some properties of formal asymptotic solutions are studied.
In Part 2 the authors discuss the special features of the formal solutions of the equation (1) corresponding to the equation $-\psi_{xx}+ p(x)\psi+ v(\varepsilon x)=0, \tag{2}$ where $$p$$ and $$v$$ are smooth real functions; $$p$$ is also $$a$$-periodic. The existence of exact solutions of (2) whose asymptotic behavior is given by the constructed formal solutions, is proved.

### MSC:

 35B10 Periodic solutions to PDEs

### Keywords:

formal asymptotic solutions
Full Text: