4 A. V. SOBOLEV
such a reduction is a major issue. In [42] at Step 3 it was sufficient to have the
geometric estimate for δ = 1. In the present paper it is crucial to have such an
estimate for δ 1. Moreover, trace class estimates were derived in [42] under the
assumption that Λ was a half-space, which is clearly insufficient for our purposes.
As far as the 1-dim asymptotics are concerned (i.e. Step 2), our estimates are
perhaps somewhat more detailed, since apart from the parameter α, they allow one
to monitor the dependence on the scaling parameters as well.
The detailed structure of the paper is described at the end of Chapter 2.
Some notational conventions. We conclude the Introduction by fixing some
basic notations which will be used throughout the paper. For x Rd we denote
x = (1 +
|x|2)
1
2
. Very often we split x = (x1,x2,...,xd) in its components as
follows:
x = (ˆ,xd), x ˆ x = (x1,x2,...,xd−1),
and for some l = 1, 2,...,d, we write

x = (x1,x2,...,xl−1,xl+1,...,xd).
The notation B(x,R) is used for the open ball in
Rd
of radius R 0, centered at
x
Rd.
For some ρ 0 let
(1.11)
Cρn) (
= (−2ρ,
2ρ)n
be the n-dimensional open cube.
The characteristic function of the domain Λ Rd, d 1, is denoted by χΛ =
χΛ(x). To avoid cumbersome notation we write χz,(x) := χB(z,)(x).
For a function u = u(x), x
Rd
its Fourier transform is defined as follows:
ˆ(ξ) u =
1
(2π)
d
2
e−ix·ξu(x)dx.
The integrals without indication of the domain of integration are taken over the
entire Euclidean space Rd.
The notation Sp, 0 p is used for the standard Schatten-von Neumann
classes of compact operators in a separable Hilbert space, see e.g. [3], [34]. In par-
ticular, S1 is the trace class, and S2 is the Hilbert-Schmidt class. Unless otherwise
stated the underlying Hilbert space is assumed to be
L2(Rd).
By C, c (with or without indices) we denote various positive constants whose
precise value is of no importance.
Acknowledgments. Part of this paper was written during my stay at the
Erwin Schr¨ odinger Institute, Vienna in July 2009. I thank the organizers of the
Programme “Topics in Spectral Theory” for the invitation and to the staff of the
Institute for their hospitality.
I am grateful to H. Leschke, R. Helling and W. Spitzer for useful discussions and
their cordial hospitality at Erlangen in September 2009. Thanks go to A. Laptev,
Yu. Safarov and M. Shubin for stimulating discussions, and to A. ottcher, P.
Deift, B. Helffer, A. Its, N. Lerner and G. Rozenblum, for pointing out to me some
useful references. I am grateful to A. ottcher and J. Oldfield for reading parts
of the text and correcting some inconsistencies. I also thank the referee and I.
Krasovsky for their remarks.
This work was supported in part by EPSRC grants EP/F029721/1
and EP/D00022X/2.
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