Localized eigenfunctions in the asymptotic model of the spectral problem

Localized eigenfunctions in the two-dimensional spectral problem containing a small parameter with higher derivatives are constructed on the expected solution form. Localization in this context means that the solution exponentially decays in both direc-tions starting from the "weakest" point or line. The constructions are based on the algo-rithm introduced by V.P. Maslov. A modification of this algorithm for the thin shell theory problems is given as an application. The paper shows implementation of the algorithm to obtain formulas giving eigenvalues and corresponding eigenfunctions. An example of solving a specific problem is given, illustrating stages of the applied asymptotic model.