The generalized spectral theory and its application to the Kuramoto conjecture (Recent developments in mathematics of integrable systems)

Abstract

A spectral theory of linear operators based on a Gelfand triplet (rigged Hilbert space) is developed under the assumptions that a linear operator 𝑇 on a Hilbert space 𝐻 is a perturbation of a self-adjoint operator, and the spectral measure of the self-adjoint operator has an analytic continuation near the real axis. It is shown that for a suitable dense subspace 𝑋 of 𝐻 and its dual space 𝑋¹, for any 𝜙 ∈ 𝑋, the resolvent (λ − 𝑇)⁻¹𝜙 of the operator 𝑇 has an analytic continuation from the lower half plane to the upper half plane as an 𝑋¹-valued holomorphic function even when 𝑇 has a continuous spectrum on 𝐑. The Gelfand triplet consists of three topological vector spaces 𝑋 ⊂ 𝐻 ⊂ 𝑋¹. Basic tools of the usual spectral theory, such as spectra, resolvents and Riesz projections are extended to those defined on a Gelfand triplet. They prove to have the same properties as those of the usual spectral theory. The results are applied to estimate exponential decays of the semigroups of linear operators and bifurcations of nonlinear dynamical systems. In particular, a conjecture on a bifurcation of the Kuramoto model (Kuramoto conjecture) will be solved.