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I. Oguni. The secondary Novikov-Shubin invariants of groups and quasiisometry. J. Math. Soc. Japan, 59(1):223–237, 2007. html. [90] P. Pansu. Croissance des boules et des g´eod´esiques ferm´ees dans les nilvari´et´es. Ergodic Theory Dynam. Systems, 3(3):415–445, 1983. A. Pastur. Selfaverageability of the number of states of the Schr¨odinger equation with a random potential. Mat. Fiz. i Funkcional. , (Vyp. 2):111–116, 238, 1971. A. Pastur. Behaviour of some Wiener integrals as t → ∞ and the density of states of the Schr¨ odinger equation with a random potential.

J. Phys. A, 19(18):L1173–L1177, 1986. [19] F. Chung, A. -T. Yau. Higher eigenvalues and isoperimetric inequalities on Riemannian manifolds and graphs. Comm. Anal. , 8(5):969– 1026, 2000. K. Chung. Spectral graph theory, volume 92 of CBMS Regional Conference Series in Mathematics. Conference Board of the Mathematical Sciences, Washington, DC, 1997. [21] Y. Colin de Verdi`ere. Spectres de graphes, volume 4 of Cours Sp´ ecialis´es. Soci´et´e Math´ematique de France, Paris, 1998. [22] T. Coulhon and L.

A function y satisfying the differential equations may now be expressed as y = (C(λ, ·), S(λ, ·))ξ for an appropriate ξ ∈ C2r . In particular, the function y = (C(λ, ·), S(λ, ·))ξ satisfies the interface conditions (and hence is a solution of Ly = λy) precisely if ξ is in the kernel of the (2r − n0 ) × 2r-matrix J(λ) = I(C(λ, ·), S(λ, ·)). 4. Initial value problems Initial value problems do, in general, not have unique solutions on trees. This causes the main differences in the treatment of inverse problems on trees when compared to intervals.

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