Spectral properties of zeroth-order pseudodifferential operators

Kinks and rotations are studied in long Josephson junctions for small and large surface losses. Geometric singular perturbation theory is used to prove existence for small surface losses, while numerical continuation is necessary to handle large surface losses. A survey of the system behaviour in terms of dissipation parameters and bias current is given. Linear orbital stability for kinks is proved for small surface losses by calculating the spectrum of the linearized problem. The spectrum is split into essential spectrum and discrete spectrum. For the determination of the discrete spectrum, robustness of exponential dichotomies is used. Puiseux series together with perturbation theory for linear operators are an essential tool. In a final step, a smooth Evans function together with geometric singular perturbation theory is used to count eigenvalues. For kinks, non‐linear orbital stability is shown. For this purpose, the asymptotic behaviour of a semigroup is given and the theory of centre and stable manifolds is applied. Copyright © 2001 John Wiley & Sons, Ltd.

Pseudo-differential operators, Wigner transform and Weyl transform on the affine Poincaré group

This paper is the second part of a two-paper series whose aim is to give a detailed description of Connes’ pseudodifferential calculus on noncommutative [Formula: see text]-tori, [Formula: see text]. We make use of the tools introduced in the 1st part to deal with the main properties of pseudodifferential operators on noncommutative tori of any dimension [Formula: see text]. This includes the main results mentioned in [2, 5, 11]. We also obtain further results regarding action on Sobolev spaces, spectral theory of elliptic operators, and Schatten-class properties of pseudodifferential operators of negative order, including a trace-formula for pseudodifferential operators of order [Formula: see text].

We study the method of layer potentials for manifolds with boundary and cylindrical ends. The fact that the boundary is non-compact prevents us from using the standard characterization of Fredholm or compact pseudo-differential operators between Sobolev spaces, as, for example, in the works of Fabes-Jodeit-Lewis [10] and Kral-Wedland [18]. We first study the layer potentials depending on a parameter on compact manifolds. This then yields the invertibility of the relevant boundary integral operators in the global, non-compact setting. As an application, we prove a well-posedness result for the non-homogeneous Dirichlet problem on manifolds with boundary and cylindrical ends. We also prove the existence of the Dirichlet-to-Neumann map, which we show to be a pseudodifferential operator in the calculus of pseudodifferential operators that are “almost translation invariant at infinity.”

Can Boutet de Monvel’s algebra on a compact manifold with boundary be obtained as the algebra \Psi^0(G) of pseudodifferential operators on some Lie groupoid G ? If it could, the kernel {\mathcal G} of the principal symbol homomorphism would be isomorphic to the groupoid C*-algebra C^*(G) . While the answer to the above question remains open, we exhibit in this paper a groupoid G such that C^*(G) possesses an ideal \mathcal{I} isomorphic to {\mathcal G} . In fact, we prove first that {\mathcal G}\simeq\Psi\otimes{\mathcal K} with the C*-algebra \Psi generated by the zero order pseudodifferential operators on the boundary and the algebra \mathcal K of compact operators. As both \Psi\otimes \mathcal K and \mathcal{I} are extensions of C(S^*Y)\otimes \mathcal{K} by \mathcal{K} ( S^*Y is the co-sphere bundle over the boundary) we infer from a theorem by Voiculescu that both are isomorphic.

Cauchy problems related to integrable deformations of pseudo differential operators

International audience

Pseudodifferential Operators in Hyperfunction Theory

For a long time, numerous problems in various applied fields [1-3] have been investigated with the use of singular integral equations. For example, some problems of aerodynamics [1], diffraction of electromagnetic and acoustic waves [2], elasticity [3], etc. can be reduced to the solution of such equations. The theory of singular integral equations was developed long ago [3], but related applied problems required the solution of these equations in the class of absolutely integrable functions. It was mentioned in [4] that, under specific conditions imposed on the coefficients of a singular integral equation, one can find solutions in the class of fimctions with nonintegrable singularity of the form 1/x as x -* 0. But the absence of applied problems leading to such solutions, numerical methods for their construction, and a theory of distributions developed for singular integral equations hindered the investigation of such singular solutions. However, in the framework of aerodynamics [5, 6], the mechanics of wings has recently been analyzed with the use of ejection of an external flow. It was shown in [6] that, in the case of a thin profile, this problem can be reduced to solving a singular integral equation of the first kind on a closed interval in the class of fimctions that have a nonintegrable singularity of the above-mentioned type at the ejection point. By [7], the investigation of the diffraction of an electromagnetic wave on an ideal conductor with a current thread on the surface leads to a similar mathematical problem. Such problems were numerically solved in [1, 7, 8] in sorne special cases with the use of the discrete vortex and discrete current methods. However, the integral operators constructed in these papers are continuously invertible in related pairs of spaces but have the property that their squares map the entire subspace of singular solutions to zero. In the present paper, in view of [9, 10], we give another mathematical treatment of the same applied problems, which permits one to consider them in the framework of the theory of pseudodifferential operators on a closed curve. Then we describe the construction of Sobolev type weighted spaces and develop the theory of linear operators (similar to pseudodifferential ones) in these spaces for open curves. We present examples illustrating the difference between the notions of integral in the sense of the Cauchy principal value and pseudodifferential operators on distributions that are singular solutions. Similar linear operators are constructed for singular integral equations of the second kind with Hilbert kernel and constant coefficients on a closed interval. We indicate the possibility of the generalization of the suggested construction of linear operators to some spatial problems.

Property [formula omitted] and perturbations

In [Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99] Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd-dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin–Salamon [The spectral flow and the Maslov index, Bull. London Math. Soc. 27(1) (1995) 1–33, MR 1331677]. In [F. Gesztesy, Y. Latushkin, K. Makarov, F. Sukochev and Y. Tomilov, The index formula and the spectral shift function for relatively trace class perturbations, Adv. Math. 227(1) (2011) 319–420, MR 2782197], a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper [A. Pushnitski, The spectral flow, the Fredholm index, and the spectral shift function, in Spectral Theory of Differential Operators, American Mathematical Society Translations: Series 2, Vol. 225 (American Mathematical Society, Providence, RI, 2008), pp. 141–155, MR 2509781]. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and its perturbation by a relatively trace-class operator. In this paper, we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher pth Schatten class condition for [Formula: see text], thus allowing differential operators on manifolds of any dimension [Formula: see text]. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by [M.-T. Benameur, A. Carey, J. Phillips, A. Rennie, F. Sukochev and K. Wojciechowski, An analytic approach to spectral flow in von Neumann algebras, in Analysis, Geometry and Topology of Elliptic Operators (World Scientific Publisher, Hackensack, NJ, 2006), pp. 297–352, MR 2246773; A. Carey, H. Grosse and J. Kaad, On a spectral flow formula for the homological index, Adv. Math. 289 (2016) 1106–1156, MR 3439708]. We illustrate our results using Dirac type operators on [Formula: see text] for arbitrary [Formula: see text] (see Sec. 8). In this setting Theorem 6.4 substantially extends Theorem 3.5 of [A. Carey, F. Gesztesy, H. Grosse, G. Levitina, D. Potapov, F. Sukochev and D. Zanin, Trace formulas for a class of non-Fredholm operators: a review, Rev. Math. Phys. 28(10) (2016) 1630002, MR 3572626], where the case d = 1 was treated.

We define and study an algebra Ψ ∞ 1,0,V (M 0 ) of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold M 0 whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M 0 to a compact manifold with corners.We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞ 1,0,V (M 0 ).We also consider the algebra Diff * V (M 0 ) of differential operators on M 0 generated by V and C ∞ (M ), and show that Ψ ∞ 1,0,V (M 0 ) is a microlocalization of Diff * V (M 0 ).Our construction solves a problem posed by Melrose in 1990.Finally, we introduce and study semi-classical and "suspended" versions of the algebra Ψ ∞ 1,0,V (M 0 ).

Spectral Theory of Pseudo-Differential Operators

We investigate global microlocal properties of localization operators and Shubin pseudodifferential operators. The microlocal regularity is measured in terms of a scale of Shubin-type Sobolev spaces. In particular, we prove microlocality and microellipticity of these operators.

Mapping properties of operator-valued pseudo-differential operators