The integrated density of states for the difference Laplacian on the modified Koch graph

Abstract

We consider the integrated density of statesN(λ) of the difference Laplacian −Δ on the modified Koch graph. We show thatN(λ) increases only with jumps and a set of jump points ofN(λ) is the set of eigenvalues of −Δ with the infinite multiplicity. We establish also that

$$0< C_1 \leqslant \mathop {\lim }\limits_{\lambda \to 0} \frac{{N(\lambda )}}{{\lambda ^{d_s /2} }}< \overline {\mathop {\lim }\limits_{\lambda \to 0} } \frac{{N(\lambda )}}{{\lambda ^{d_s /2} }} \leqslant C_2< \infty$$

whered _s =2log5/log(40/3) is the spectral dimension of MKG.