Estimates for the singular values of the operator T fg := f (H)g(x) are investigated for suitable functions f (λ), λ ∈ R, g(x), x ∈ R d , and a selfadjoint operator H in L 2 (R d ). It is assumed that the kernel of the semigroup e −tH satisfies special conditions. Power-like estimates for the singular values of the operator T fg are obtained, in particular, in the case where T fg ∈ S 2 . Conditions for the operator T fg to belong to the trace class are established. Neither any smoothness conditions for the kernel of the operator T fg , nor any knowledge of the (partial) diagonalization of the operator H are required. The results admit further refinement under additional conditions imposed on the generalized eigenfunctions of the operator H. 835 Licensed to Stockholm University. Prepared on Sat Aug 1 15:53:46 EDT 2015 for download from IP 130.237.165.40. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 836 V. A. SLOUSHCHfor integral operators to belong to the classes S p,q , p < 2, which are contained in the Hilbert-Schmidt class, have been studied to a lesser extent. To my best knowledge, without additional assumptions about the smoothness of the kernel, such results have been available only for the operator f (i∇)g(x) (see [5, 6] and [7]) and also for the Dirac operator (see [8]). An overview of some of the results mentioned above can be found in [7] and [4]; the paper [7] covers some applications.In the present paper we deal with estimates for the singular values of operators of two similar types. First, consider an operator of the formIt is assumed that the semigroup e −tH satisfies the so-called Nash-Aronson estimate also known as the upper Gaussian bound (see Subsection 2 of the Introduction); in particular, the semigroup generated by a selfadjoint uniformly elliptic second order differential operator with uniformly bounded coefficients fulfills this requirement. We require neither any smoothness conditions for the coefficients of the operator H, nor any knowledge of its (partial) diagonalization. In Theorem 1.3 we obtain conditions for the operator T fg to be in the classes S p,q , p ∈ (0, 2), q ∈ (0, +∞]. We also give conditions (see Theorem 1.2) for the operator T fg to belong to the classes S p,q , p ∈ (2, +∞), q ∈ (0, +∞].Moreover, in the present paper we consider certain integral operators bordered by suitable scalar weights. Namely, let (K, dμ) be a separable measurable space with σ-finite measure, let t( · , · ) ∈ L ∞ (K × R d , dμ dx) be the kernel of a bounded integral operator T : L 2 (R d ) → L 2 (K, dμ) and let f : K → C, g : R d → C be suitable measurable functions. It is assumed that the operator T (partially) diagonalizes the selfadjoint operator H acting in L 2 (R d ) and satisfying the conditions described above. In Theorem 1.5 we give estimates for the singular values of the integral operator T fg = fT g with the kernel f (k)t(k, y)g(y). Conditions for the operator T fg to belong to the classes S p,q , p ∈ (0, 2), q ∈ (0, +∞], inclu...

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Abstract. We study discrete spectrum in spectral gaps of an elliptic periodic second order differential operator in L 2 (R d ) perturbed by a decaying potential. It is assumed that a perturbation is nonnegative and has a power-like behavior at infinity. We find asymptotics in the large coupling constant limit for the number of eigenvalues of the perturbed operator that have crossed a given point inside the gap or the edge of the gap. The corresponding asymptotics is power-like and depends on the observation point.

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