Spectral analysis of pseudodifferential operators

The subject of this paper is the spectral analysis of pseudodifferential operators in the framework of perturbation theory. We build up a closed extension (the closure, or the Friedrichs extension) of the perturbed operator. We also prove Weyl-type theorems on the invariance of the essential spectrum of the unperturbed operator. In the case when the perturbed operator is symmetric we obtain a self-adjoint extension. Finally, we consider the case of the relativistic, spin-zero Hamiltonian, with a large class of interactions containing both local potentials, like the Coulomb and Yukawa, and nonlocal ones.     

Spectral asymptotics of differential and pseudodifferential operators. II

In the article we obtain asymptotic formulae with remainder estimates for the distribution function of the eigenvalues of degenerate elliptic differential operators and Schrödinger operators with singular potential.Translated from Trudy Seminara imeni I. G. Petrovskogo, Vol. 10, pp. 78–106, 1984.     

Spectral invariance for pseudodifferential operators on weighted function spaces

The spectrum of each symmetric ψ DO of the symbol class S⁰ _{1, γ}, 0≤γ<1, acting on B³ _p,q(w(x)) and F³ _p,q(w(x)), is independent of the choice ofs ∈ℝ, 0<p≤∞ (p<∞ in the F-case), 0<q≤∞ and the weight w(x)∈W.     

Spectral invariance for algebras of pseudodifferential operators on besov-triebel-lizorkin spaces

The algebra of pseudodifferential operators with symbols inS _1,δ ⁰ , δ<1, is shown to be a spectrally invariant subalgebra of ℒ(b _p,q ^s ) and ℒ(F _p,q ^s ). The spectrum of each of these pseudodifferential operators acting onB _p,q ^s orF _p,q ^s is independent of the choice ofs, p, andq.     

Microlocal properties for pseudodifferential operators of type 1,1

The author studies the action of a new selfadjoint algebra of pseudodifferential operators of type (1,1) on Sobolev wave front sets for distributions.     

Rational matrix pseudodifferential operators

The skewfield $\mathcal{K }(\partial )$ of rational pseudodifferential operators over a differential field $\mathcal{K }$ is the skewfield of fractions of the algebra of differential operators $\mathcal{K }[\partial ]$ . In our previous paper, we showed that any $H\in \mathcal{K }(\partial )$ has a minimal fractional decomposition $H=AB^{-1}$ , where $A,B\in \mathcal{K }[\partial ],\,B\ne 0$ , and any common right divisor of $A$ and $B$ is a non-zero element of $\mathcal{K }$ . Moreover, any right fractional decomposition of $H$ is obtained by multiplying $A$ and $B$ on the right by the same non-zero element of $\mathcal{K }[\partial ]$ . In the present paper, we study the ring $M_n(\mathcal{K }(\partial ))$ of $n\times n$ matrices over the skewfield $\mathcal{K }(\partial )$ . We show that similarly, any $H\in M_n(\mathcal{K }(\partial ))$ has a minimal fractional decomposition $H=AB^{-1}$ , where $A,B\in M_n(\mathcal{K }[\partial ]),\,B$ is non-degenerate, and any common right divisor of $A$ and $B$ is an invertible element of the ring $M_n(\mathcal{K }[\partial ])$ . Moreover, any right fractional decomposition of $H$ is obtained by multiplying $A$ and $B$ on the right by the same non-degenerate element of $M_n(\mathcal{K } [\partial ])$ . We give several equivalent definitions of the minimal fractional decomposition. These results are applied to the study of maximal isotropicity property, used in the theory of Dirac structures.     

Hilbert modular pseudodifferential operators

We introduce Jacobi-like forms of several variables, and study their connections with Hilbert modular forms and pseudodifferential operators of several variables. We also construct Rankin-Cohen brackets for Hilbert modular forms using such Jacobi-like forms.

     

On some properties of a class of degenerate pseudodifferential operators

A new class of pseudodifferential operators with degeneration is considered. The operators are constructed using a special integral transform mapping a weighted differentiation operator to a multiplication operator. The composition and boundedness properties of such operators in special weighted spaces are examined. Theorems on commutation of such operators with differentiation operators are obtained. The behavior of these operators as t → 0and t → +∞ is investigated. The properties of adjoint operators are studied, and an analogue of Gårding’s inequality is proved.     

Beals-Cordes-type characterizations of pseudodifferential operators

We show that, if is the representation of on given by (2.11), and is a bounded operator on , then belongs to if and only if

is a function on with values in the Banach space .

     

Rankin-Cohen brackets on pseudodifferential operators

Pseudodifferential operators that are invariant under the action of a discrete subgroup Γ of SL(2,R) correspond to certain sequences of modular forms for Γ. Rankin-Cohen brackets are noncommutative products of modular forms expressed in terms of derivatives of modular forms. We introduce an analog of the heat operator on the space of pseudodifferential operators and use this to construct bilinear operators on that space which may be considered as Rankin-Cohen brackets. We also discuss generalized Rankin-Cohen brackets on modular forms and use these to construct certain types of modular forms.     

Spectral properties of continuous refinement operators

This paper studies the spectrum of continuous refinement operators and relates their spectral properties with the solutions of the corresponding continuous refinement equations.