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

Buslaev, Vladimir; Grigis, Alain

J. Anal. Math. 84, 67-143 (2001).

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.

Reviewer: Ion Onciulescu (Iaşi)

MSC:

35B10

Periodic solutions to PDEs

Keywords:

formal asymptotic solutions

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