Spectral asymptotics for infinite order pseudo-differential operators

Abstract

We study spectral properties of a class of global infinite order pseudodifferential operators and obtain the asymptotic behaviour of the spectral counting functions of such operators. Unlike their finite order counterparts, their spectral asymptotics are not of power-log-type but of log-type. The ultradistributional setting of such operators of infinite order makes the theory more complex so that the standard finite order global Weyl calculus cannot be used in this context.

Infinite order $$\Psi $$DOs: composition with entire functions, new Shubin-Sobolev spaces, and index theorem

“…As we have recently shown in [29] and we explain in Subsection 2.5, Γ * ,∞ Ap,ρ -hypoellipticity, plus suitable elliptic type bounds on the symbols, plays an important role for the validity of Weyl asymptotic formulae in the context of infinity order ΨDOs. In order to motivate our results from Section 3, let H = |x| 2 − ∆ be the harmonic oscillator and consider the infinite order operator…”

“…We also mention that in [15] the composition of a function and a pseudo-differential operator is treated in a completely different way in comparison with the approach we shall employ in this article. In fact, the symbolic calculus developed in [5,6,26,27,29,31] (see also [3,4]) provides a tool for studying the power series of Shubin operators under consideration.…”

Complex powers for a class of infinite order hypoelliptic operators

“…To motivate our investigations, we note that in [24,25] we studied the asymptotic behaviour of the eigenvalue counting function of a class of infinite order ΨDOs. One of the main building blocks for the development of the theory are the results presented in this article; most notably the precise estimates we obtain here on the symbol of the heat parametrix.…”

“…As it turns out, the heat parametrix plays an essential role in our analysis. With our technique, we derive precise estimates on the heat kernel which are of independent interest because, as we mentioned before, we applied them in [24,25] to derive asymptotic formulae for the eigenvalue counting function and the analysis of spectral properties of hypoelliptic operators of infinite order. In this context, the result on the complex powers of pseudodifferential operators of infinite order of our class also looks promising for applications in deriving such asymptotic formulae.…”

“…This can be particularly applied when A is the closure of n c n H n , the operator we mentioned above (see Theorem 7.12 and [25,Corollary 3.5]); the operator is non-negative since it is self-adjoint (cf. [24,Proposition 4.6]) and (Au, u) ≥ 0, u ∈ D(A) (see [6,Proposition 1.3.6,p. 21]).…”