In this note we study pseudo-multipliers associated to the harmonic oscillator (also called Hermite multipliers) belonging to Schatten classes on @math . We also investigate the spectral trace of these operators.
Now, we include some references on the subject. Sufficient conditions for the @math -nuclearity of spectral multipliers associated to the harmonic oscillator, but, in modulation spaces and Wiener amalgam spaces have been considered by J. Delgado, M. Ruzhansky and B. Wang in @cite_1 @cite_14 . The Properties of these multipliers in @math -spaces have been investigated in the references S. Bagchi, S. Thangavelu @cite_7 , J. Epperson @cite_2 , K. Stempak and J.L. Torrea @cite_3 @cite_16 @cite_11 , S. Thangavelu @cite_18 @cite_5 and references therein. Hermite expansions for distributions can be found in B. Simon @cite_21 . The @math -nuclearity and Grothendieck-Lidskii formulae for multipliers and other types of integral operators can be found in @cite_6 @cite_14 . On Hilbert spaces the class of @math -nuclear operators agrees with the Schatten-von Neumann class @math in this context operators with integral kernel on Lebesgue spaces and, in particular, operators with kernel acting of a special way with anharmonic oscillators of the form @math @math has been considered on Schatten classes on @math in J. Delgado and M. Ruzhansky @cite_23 . The proof of our results will be presented in the next section.
@cite_18 @cite_14 @cite_7 @cite_21 @cite_1 @cite_3 @cite_6 @cite_23 @cite_2 @cite_5 @cite_16 @cite_11
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In this note we present the metric approximation property for weighted mixed-norm L-w((p1,...pn)) and variable exponent Lebesgue type spaces. As a consequence, this also implies the same property for modulation and Wiener-Amalgam spaces. We then characterise nuclear operators on such spaces and state the corresponding Grothendieck-Lidskii trace formulae. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on R-n and functions of the harmonic oscillator in terms of global symbols. 1. IntroductionIn this work we address to the problem of the membership to the ideal of r-nuclearoperators for integral operators of the form(1.1) Tf(x) =Z G(x)F(xy)H(y)f(y)dy;where Gand H are bounded functions and F is a smooth function admitting asuitable Taylor expansion. A such kind of operator arises in the theory of laser,particularly in the form of the Fox-Li operator which has attracted the interest ofresearchers in recent years(cf. [BBIN10], [BGI12], [BGHI12]). In those works, theauthors were focused in the investigation of the asymptotic distribution of the sin-gular values and eigenvalues. More recently, the r-nuclearity and its connection withGrothendieck-Lidskii’s formula on compact Lie groups and applications to some dif-ferential operators have been studied in [DR13]. Applications of nuclear operators tothe study of periodic pseudo-di erential operators have been established in [DW13].The nuclearity of operators on spaces of Bochner integrable functions(cf. [DE09])and the corresponding relation with the martingale vector-maximal function for thestudy of traces have been considered in [Del]. Recent progress on the nuclearity andsingular values can also be found in [FMO09] and [MO12].The Fox-Li operator was introduced in [FL61], A. G. Fox and T. Li considered therepeated reection of an electromagnetic wave between two plane-parallel rectangularmirrors. The current density at the surface of each reector satis es the eigenvalueintegral equationTf(x) = f(x):For a good account on the physics of such kind of operators we recommend the chapterXI, section 60 in the book [NE05] by Trefethen and Embree. The Fox-Li operator is In this work we establish sharp kernel conditions ensuring that the corresponding integral operators belong to Schatten-von Neumann classes. The conditions are given in terms of the spectral properties of operators acting on the kernel. As applications we establish several criteria in terms of different types of differential operators and their spectral asymptotics in different settings: compact manifolds, operators on lattices, domains in @math of finite measure, and conditions for operators on @math given in terms of anharmonic oscillators. We also give examples in the settings of compact sub-Riemannian manifolds, contact manifolds, strictly pseudo-convex CR manifolds, and (sub-)Laplacians on compact Lie groups. We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting "scattering matrix" mapping radiation fields on the past horizon and past null infinity to radiation fields on the future horizon and future null infinity is a bounded operator. The latter allows us to give a time-domain theory of superradiant reflection. The boundedness of the scattering matrix shows in particular that the maximal amplification of solutions associated to ingoing finite-energy wave packets on past null infinity is bounded. On the frequency side, this corresponds to the novel statement that the suitably normalised reflection and transmission coefficients are uniformly bounded independently of the frequency parameters. We further complement this with a demonstration that superradiant reflection indeed amplifies the energy radiated to future null infinity of suitable wave-packets as above. The results make essential use of a refinement of our recent proof [M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case |a|<M, arXiv:1402.6034] of boundedness and decay for solutions of the Cauchy problem so as to apply in the class of solutions where only a degenerate energy is assumed finite. We show in contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerate energy solutions. This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shift instability when solving the wave equation backwards.