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. B¨ 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. B¨ 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.