Spectrum of a uniform grid of strings (Q1589209)

The spectrum \(\Lambda \) of a homogeneous membrane, spanned in the unit square, is investigated by approximating it with a network of strings \(\Gamma_h\) of meshsize \(h\). It is shown that for each \(M\) the spectrum \(\Lambda \) in \(|\lambda |\leq M\) differs from the spectrum of \(\Gamma _h\) by \(O(h^2)\). The spectrum of \(\Gamma _h\) is shown to be the set of solutions of explicitly given equations involving trigonometric functions.