Stability Operators on Self-Similar Hierarchical Hilbert Spaces

AbstractWe construct and analyze a stability operator on a weighted hierarchical Hilbertspace whose scaling structure is governed by the golden ratio ϕ = 1+√52 . The space His a weighted ℓ2-direct sum of copies of a base Hilbert space H0, with weights ϕ−2k atlevel k. Via the isometric change of variables uk = ϕ−kψk, the inter-level coherenceoperator S becomes the tridiagonal operator (Suu)k = ϕ uk+1 − ϕ−1uk in standardℓ2, and S†uSu is the symmetric tridiagonal operator 3I − Tdisc with spectrum [1, 5].The stability operator L = −∆ + αS†S is self-adjoint by the Kato–Rellich theoremfor all α > 0. Its spectrum is the union of bands σ(L) = Sn≥1[μn + α, μn + 5α],where {μn} are the eigenvalues of −∆ on H0. Band n and band n + 1 are separatedby a gap (μn+1 − μn) − 4α, positive iff α < (μn+1 − μn)/4.