Spectrum of Schrödinger operator on a homogeneous tree (Q1063937)

Let \(B_{\nu}\) be the Bethe tree (infinite Cayley tree), i.e. connected non-oriented tree, and in every of its vertices g, excluding one vertex \(g_ 0\) (root), \(\nu\) sides meet. The root is the origin of unique side. Denote by \(| g|\) the minimal number of sides in the path connecting g and \(g_ 0\) and for a finite set \(S\subset B_{\nu}\), \(| s| =card S\), consider a space \(\Phi_ S\) of the functions f(g), \(g\in S\) with the scalar product \((f,h)=\sum_{g\in S}f(g)\overline{h(g)}\). Define the selfadjoint operator \(H_ s^{\omega}\) as follows: \[ (H_ s^{\omega}f)(g)=\sum_{| g- g'| =1}a(g,g')f(g')+\xi (g)f(g) \] where \(a(g,g')=0\) if \(g\in S\), g'\(\in S\), and \(=\nu^{-1}\) if g,g'\(\in S\), \(a(g,g')=a(g',g)\) and \(\xi\) (g), \(g\in B_{\nu}\), are i.i.d. random variables. Let \(N_ s^{\omega}(\lambda)\) be the number of eigenvalues of \(H_ s^{\omega}\) not exceeding \(\lambda\), \(V_ n=\{g:| g| \leq n\}.\) Theorem 1. With probability 1 the limit (1) \(\lim_{n\to \infty}| V_ n|^{-1}N^{\omega}_{V_ n}(\lambda)\equiv N(\lambda)\) exists and is nonrandom. Theorem 1'. There exists the limit (1) for \(\xi\) \(\equiv 0\) and is purely discontinuous function with dense set of jumps. The proof of Th. 1' follows from straightforward calculations. The proof of Th. 1 uses the scheme proposed by the reviewer [Teor. Mat. Fiz. 6, 415-424 (1971; Zbl 0208.129)] and is based on the representation of the Laplace transform of \(N^{\omega}_{V_ n}(\lambda)\) as the expectation of the functional \(\exp\{\int^{+}_{0}\xi (g\tau)d\tau \}\) with respect to the continuous time random value \(g_ t\) on the tree \(B_{\nu}\) (the analog of the Feynman-Kac formula).