A functional equation for semiclassical Fredholm determinant for strongly chaotic billiards

We derive a functional equation for the Fredholm determinant of the boundar y element method. By assuming that the functional equation holds for the se miclassical Fredholm determinant for strongly chaotic billiards, we obtain a real function whose zeros are the semiclassical eigenenergies. We also sh ow by the numerical experiment of concave triangle billiards that the semic lassical eigenenergies are very close to the exact eigenenergies.