14th September (tue)

The theory of semi-classical approximation tells us that the energy levels of lagrangean tori satisfying the so-called the Maslov quantization condition provide approximate eigenvalues of the laplacian in an asymptotic sense. In the case of Liouville surface, since there are many such lagrangean tori, one might expect that almost all eigenvalues are approximated in such a way. However in the real situation, there is a singular lagrangean subset that is an intersected union of two tori, and when regular tori tends to the singular one, the rate of approximation is getting worse. Thus the usual semi-classical approximation become ineffective around there. In this talk, we shall show that there is another semi-classical approximation corresponding to this singular lagrangean subset, which gives effective approximate values for some eigenvalues of the laplacian.