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On Some Integral Operators Appearing in Scattering Theory, And On some integral operators appearing in scattering theory, and their resolutions S. Richard,∗ T. Umeda† Graduate school of mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan Department of Mathematical Sciences, University of Hyogo, Shosha, Himeji 671-2201, Japan E-mail: [email protected], [email protected] Abstract We discuss a few integral operators and provide expressions for them in terms of smooth functions of some natural self-adjoint operators. These operators appear in the context of scattering theory, but are independent of any perturbation theory. The Hilbert transform, the Hankel transform, and the finite interval Hilbert transform are among the operators considered. 2010 Mathematics Subject Classification: 47G10 Keywords: integral operators, Hilbert transform, Hankel transform, dilation operator, scattering theory 1 Introduction Investigations on the wave operators in the context of scattering theory have a long history, and several powerful technics have been developed for the proof of their existence and of their completeness. More recently, properties of these operators in various spaces have been studied, and the importance of these operators for non-linear problems has also been acknowledged. A quick search on MathSciNet shows that the terms wave operator(s) appear in the title of numerous papers, confirming their importance in various fields of mathematics. For the last decade, the wave operators have also played a key role for the arXiv:1909.01712v1 [math-ph] 4 Sep 2019 search of index theorems in scattering theory, as a tool linking the scattering part of a physical system to its bound states. For such investigations, a very detailed ∗Supported by the grantTopological invariants through scattering theory and noncommuta- tive geometry from Nagoya University, and by JSPS Grant-in-Aid for scientific research (C) no 18K03328, and on leave of absence from Univ. Lyon, Universit´eClaude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France. †Supported by JSPS Grant-in-Aid for scientific research (C) no 18K03340. 1 understanding of the wave operators has been necessary, and it is during such investigations that several integral operators or singular integral operators have appeared, and that their resolutions in terms of smooth functions of natural self- adjoint operators have been provided. The present review paper is an attempt to gather some of the formulas obtained during these investigations. Singular integral operators are quite familiar to analysts, and refined estimates have often been obtained directly from their kernels. However, for index theorems or for topological properties these kernels are usually not so friendly: They can hardly fit into any algebraic framework. For that reason, we have been looking for representations in which these singular integral operators have a smoother appearance. It turns out that for several integral operators this program has been successful, and suitable representations have been exhibited. On the other hand, let us stress that even though these formulas are necessary for a C∗-algebraic approach, it seems that for pure analysis they do not lead to any new refined estimates. Let us now be more precise for a few of these operators, and refer to the subsequent sections for more information. The Hilbert transform is certainly one of the most famous and ubiquitous singular integral operators. Its explicit form in L2(R) is recalled in (2.1). A nice representation of this singular operator does not take place directly in L2(R), but by decomposing this Hilbert space into odd and even functions, then it becomes possible to obtain an expression for the Hilbert transform in terms of the operators tanh(πA) and cosh(πA) where A denotes the 2 generator of dilations in L (R+). Such an expression is provided in Proposition 2 2.1. Similarly, the Hankel transform whose definition in L (R+) is recalled in (2.3) is an integral operator whose kernel involves a Bessel function of the first kind. This operator is not invariant under the dilation group, and so does the inversion 2 R 1 1 operator J defined on f L ( +) by [Jf](x) = x f( x ). However, the product of these two operators is invariant∈ under the dilation group and can be represented by a smooth function of the generator of this group, see Proposition 2.2 for the details. n For several integral operators on R+ or on R , the dilation group plays an important role, as emphasized in Section 2. On the other hand, on a finite in- terval (a,b) this group does not play any role (and is even not defined on such a space). For singular operators on such an interval, the notion of rescaled energy representation can be introduced, and then tools from natural operators on R can be exploited. This approach is developed in Section 3. As a conclusion, this short expository paper does not pretend to be exhaustive or self-contained. Its (expected) interest lies on the collection of several integral operators which can be represented by smooth functions of some natural self- adjoint operators. It is not clear to the authors if a general theory will ever be built from these examples, but gathering the known examples in a single place seemed to be a useful preliminary step. 2 2 Resolutions involving the dilation group The importance of the dilation group in scattering theory is well-known, at least since the seminal work of Enss [6]. In this section we recall a few integral operators which appeared during our investigations on the wave operators. They share the common property of being diagonal in the spectral representation of the generator of the dilation group. Before introducing these operators, we recall the action of 2 n 2 this group in L (R ) and in L (R+). In L2(Rn) let us set A for the self-adjoint generator of the unitary group of dilations, namely for f L2(Rn) and x Rn : ∈ ∈ [eitAf](x) := ent/2f(etx). 2 There also exists a representation in L (R+) which is going to play an important itA t/2 t 2 role: its action is given by [e f](x) := e f(e x) for f L (R+) and x R+. Note that we use the same notation for these generators∈ independently∈ of the dimension, but this slight abuse of notation will not lead to any confusion. 2.1 The Hilbert transform Let us start by recalling that the Hilbert transform is defined for f in the Schwartz space (R) by the formula S 1 1 i [Hf](x) := P.v. f(y)dy = eikx kˆ [F f](k)dk. (2.1) π R x y −√2π R Z − Z Here P.v. denotes the principal value, kˆ := k for any k R∗ and F f stands for |k| ∈ the Fourier transform of f defined by 1 [F f](k) := e−ikxf(x)dx. (2.2) √2π R Z It is well known that this formula extends to a bounded operator in L2(R), still denoted by H. In this Hilbert space, if we use the notation X for the canonical self-adjoint operator of multiplication by the variable, and D for the self-adjoint d realization of the operator i dx , then the Hilbert transform also satisfies the equality H = i sgn(D), with− sgn the usual sign function. Based on− the first expression provided in (2.1) one easily observes that this operator is invariant under the dilation group in L2(R). Since this group leaves the odd and even functions invariant, one can further decompose the Hilbert space in order to get its irreducible representations, and a more explicit formula for H. Thus, let us introduce the even/odd representation of L2(R). Given any function ρ on R, we write ρe and ρo for the even part and the odd part of ρ. We then introduce the unitary map 2 fe 2 2 : L (R) f √2 L (R+; C ). U ∋ 7→ fo ∈ 3 h1 2 2 Its adjoint is given on L (R+; C ) and for x R by h2 ∈ ∈ ∗ h1 1 (x) := [h1( x ) + sgn(x)h2( x )] . U h2 √2 | | | | A new representation of the Hilbert transform can now be stated. We refer to [13, Lem. 3] for the initial proof, and to [19, Lem. 2.1] for a presentation corresponding to the one introduced here. Proposition 2.1. The Hilbert transform H in L2(R) satisfies the following equal- ity: 0 tanh(πA) i cosh(πA)−1 H ∗ = i − . U U − tanh(πA)+ i cosh(πA)−1 0 Note that this representation emphasizes several properties of the Hilbert trans- form. For example, it makes it clear that the Hilbert transform exchanges even and odd functions. By taking the equality tanh2 + cosh−2 = 1, one also easily deduces that the norm of H is equal to 1. This latter property can certainly not be directly deduced from the initial definition of the Hilbert transform based on the principle value. The Hilbert transform appears quite naturally in scattering theory, as empha- sized for example in the seminal papers [4, 26, 27]. The explicit formula presented in Lemma 2.1 is used for the wave operators in [19, Thm. 1.2]. A slightly different version also appears in [13, Eq. (1)]. 2.2 The Hankel transform Let us recall that the Hankel transform is a transformation involving a Bessel function of the first kind. More precisely, if Jm denotes the Bessel function of the first kind with m C and (m) > 1, the Hankel transform m is defined on f C∞(R ) by ∈ ℜ − H ∈ c + [ mf](x)= √xy Jm(xy)f(y)dy.
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