Viscosity limits for 0th order pseudodifferential operators

Abstract

Motivated by the work of Colin de Verdière and Saint-Raymond [CS-L20] on spectral theory for 0th order pseudodifferential operators on tori we consider viscosity limits in which 0th order operators, P , are replaced by P + iν∆, ν > 0. By adapting the Helffer-Sjöstrand theory of scattering resonances [HeSj86], we show that, in a complex neighbourhood of the continuous spectrum, eigenvalues of P +iν∆ have limits as the viscosity, ν, goes to 0. In the simplified setting of tori, this justifies claims made in the physics literature -see for instance [RGV01].P ν := P + iν∆, where P is a 0th order pseudodifferential operator on the torus (1.1) satisfying (1.2) and the dynamical assumption (1.3). The operator ∆ is the usual Laplacian on the torus.The assumption (1.3) guarantees continuity of the spectrum at 0 [CS-L20], [DyZw19b]. We then show that as ν → 0+ the eigenvalues of P ν in a complex neighbourhood of 0,tend to a discrete set associated to P alone -see Figure 1 for a numerical illustration.

Resonances as viscosity limits for black box perturbations

“…For applications of fixed complex absorbing potentials in generalized geometric settings see for instance Nonnenmacher-Zworski [NoZw09], [NoZw15] and Vasy [Va13]. The analogous results to Theorem 1 were proved for Pollicott-Ruelle resonances in [DyZw15], for kinetic Brownian motion by Drouot [Dr17], for gradient flows by Dang-Rivière [DaRi17] (following earlier work of Frenkel-Losev-Nekrasov [FLN11]), and for 0th order pseudodifferential operators, motivated by problems in fluid mechanics, by Galkowski-Zworski [GaZw19].…”

Resonances and viscosity limit for the Wigner-von Neumann type Hamiltonian

“…As mentioned above, Theorem 1.2 was proved by Zworski [27] in the case of the dilation analytic case. Analogous results were proved for Pollicott-Ruelle resonances by Dyatlov-Zworski [8] (see also [5], [7]), and for 0th order pseudodifferential operators by Galkowski-Zworski [11]. For the numerical results and original approach in physical chemistry, see the references in [25], [27].…”

Dynamics of resonances for 0th order pseudodifferential operators

“…since I −δ is the dual space of I δ relative L 2 pairing on T n and I δ ⊂ H Λ ⊂ I −δ (see for instance [GaZw19,§4]). This implies R ± are well-defined and bounded operators.…”

“…As in [GaZw19, (7.13)], we put P q,t := P + it∆ T n − iQ, Q := S Λ Π Λ qΠ Λ T Λ (5.1) with q ∈ C ∞ c (Λ; [0, ∞)) satisfies conditions in [GaZw19, Lemma 7.6]. For the definition of S Λ , Π Λ , see [GaZw19,§4,§5]. By [GaZw19, Lemma 7.6], for any > 0 sufficiently small, there exists q = q( ) such that R q,t := (P q,t − ζ) We now consider the Grushin problem…”

“…We are also interested in embedded eigenvalues λ as limits of eigenvalues λ(t) of P (t) = P + it∆. Galkowski and Zworski [GaZw19] defined the set of resonances of P and showed the resonances can be approximated by the eigenvalues of P (t). In the case when λ is simple, we compute the first derivative of λ(t) at 0.…”