The Fox–Li Operator as a Test and a Spur for Wiener–Hopf Theory

The paper is a concise survey of some rigorous results on the Fox–Li operator. This operator may be interpreted as a large truncation of a Wiener–Hopf operator with an oscillating symbol. Employing theorems from Wiener–Hopf theory one can therefore derive remarkable properties of the Fox–Li operator in a fairly comfortable way, but it turns out that Wiener–Hopf theory is unequal to the task of answering the crucial questions on the Fox–Li operator. 1 Masers, Lasers and the Fox–Li Operator The story begun 50 years ago. Fox and Li [13] considered the repeated reflection of an electromagnetic wave of wave length between two plane-parallel rectangular mirrors. By a tensor product phenomenon, it suffices to suppose that the mirrors are infinite strips of height 2a with distance b between them. A distribution u.x/, x 2 . a; a/, of the field on one mirror goes over into the distribution given by .Au/.x/ D e i =4 2 p Z a a .x y/u.y/dy; x 2 . a; a/; (1) A. Böttcher Faculty of Mathematics, TU Chemnitz, 09107 Chemnitz, Germany S. Grudsky Department of Mathematics, CINVESTAV , Mexico-City, Mexico A. Iserles ( ) Department of Applied Mathematics and Theoretical Physics Centre for Mathematical Sciences University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA, United Kingdom, e-mail: [email protected] DOI 10.1007/978-3-642-28821-0 3, © Springer-Verlag Berlin Heidelberg 2012 37 P.M. Pardalos and T.M. Rassias (eds.), Essays in Mathematics and its Applications, 38 A. Böttcher et al. on the other mirror. Here is the function .t/ D e ikpt 2Cb2 .t2 C b2/1=4 1C b p t2 C b2 (2) where k D 1= denotes the wave number. What Fox and Li were interested in were the eigenvalues and eigenfunctions of the operator A: if Au D u, then the distribution u.x/ will after n reflections be transformed into u.x/. The number 1 j j2 is the energy loss of the mode u at one step. This setup is called a maser in the case of microwaves ( 1 cm) and a laser when working with light waves, in the range 5 10 5 cm. Let us consider the integral operator A given by (1) on L. a; a/. Being compact, it has at most countably many eigenvalues with the origin as the only possible cluster point. Cochran [11] and Hochstadt [16] provided a rigorous argument which proves that A has at least one eigenvalue. However, there is no theorem that would imply more or anything else of interest about the operator A. Well, A has a difference kernel and hence one would expect that for large a the eigenvalues of A somehow mimic the values of the Fourier transform of , O . / WD Z 1 1 .t/e dt; 2 R: The function O . / is even, exponentially decaying as j j ! 1, and in L.R/. Had it been in C.R/, we would have had a theorem implying that the eigenvalues of A cluster along the range O .R/ as a ! 1. However, O . / behaves like r 2bj kj Œ1C i sign. k/ as ! k and hence it is not even in L1.R/. In addition we should mention that the case a b is not the really interesting case in physics. One is therefore left with tackling the eigenvalue problem for A numerically, the big problem in this connection being that the kernel is highly oscillating: note that k 20;000 cm 1 for light waves. Fox and Li found an ingenious way out. The physically relevant case is the one where a b. They wrote exp. ik p t2 C b2/ D exp ikb 1C t 2 2b2 CO t b4 ; and since jt j < a, one may ignore the O term if kba=b 1, that is, if a b. As b, this assumption automatically implies that a b, and therefore .t C b/ and b= p t2 C b2 may be replaced by pb and 1, respectively. In summary, the operator A may be approximated by the operator The Fox–Li Operator as a Test and a Spur for Wiener–Hopf Theory 39 .A1u/.x/ D e i e ikb p b Z a a e i.k=2b/.x y/2u.y/dy; x 2 . a; a/: The change of variables x ! ax, y ! ay yields the operator .A2u/.x/ D ae i e ikb p b Z 1 1 e i.ka2=2b/.x y/2u.y/dy; x 2 . 1; 1/; (3) and abbreviating ! WD ka=.2b/ D a=.2 b/ and pi WD ei =4 we arrive at the equality A2 D p 2 e ikbF ! with F ! and F! defined on L. 1; 1/ by .F !u/.x/ D r !i Z 1 1 e i!.x y/2u.y/dy; .F!u/.x/ D r ! i Z 1 1 e y/2u.y/dy: Note that F ! is really the adjoint of F! . The operator F! is now called the Fox–Li operator, and the eigenvalues and eigenfunctions of this operator are what one wants to know. After the change of variables x ! x=p! 1, y ! y=p! 1 the operator F! becomes the operator given by .F!u/.x/ D 1 p i Z 2p! 0 e y/2u.y/dy; x 2 .0; 2p! /; (4) on L.0; 2 p ! /, and since ! D a=.2 b/ may also be assumed to be very large, F! is a very large truncation of a Wiener–Hopf operator. In summary, the Fox–Li operator is a reasonable approximation to the original physical problem and at the same time a large truncated Wiener–Hopf operator whenever b a b. Using the dimensionless parameters O aWD ka and O bW D kb, these inequalities read O b O a O b, and ! becomes O a=.2 O b/. Fox and Li themselves showed that already the moderate choice O a D 25, O b D 100 leads to acceptable numerical results. 2 Wiener–Hopf Operators An integral operator on L2.0;1/ of the form .W u/.x/ D Z 1 0 %.x y/u.y/dy; x 2 .0;1/; is called a Wiener–Hopf operator. Such an operator is bounded on L2.0;1/ if and only if the Fourier transform a WD O %, taken in the distributional sense, is a function in L1.R/. The function % is uniquely determined by its Fourier transform 40 A. Böttcher et al. a, henceforth we denote the operator W by W.a/. The function a is usually referred to as the symbol of W.a/. Note that W.a/ is the compression to L2.0;1/ of the operator which acts on L.R/ by the following rule: take the Fourier transform, multiply the result by a, and then take the inverse Fourier transform. For 2 .0;1/, the truncated Wiener–Hopf operator W .a/ is defined on L.0; / by .W u/.x/ D Z 0 %.x y/u.y/dy; x 2 .0; /: (5) The Fourier transform of %.t/ D eit 2 is O %. / D p i e i 2=4. Thus, letting . / D e i 2=4, we see that the Fox–Li operator F! given by (4) is nothing but W2p!. /, and the problem is to find the eigenvalues and eigenfunctions of W . / as D 2 p ! ! 1. The spectral theory of Wiener–Hopf operators is well developed, one could say that Wiener–Hopf operators and their discrete analogues, Toeplitz operators, are the best understood nontrivial classes of non-selfadjoint operators. We refer to [4] for a presentation of the matter. However, as already said, no result of this theory is immediately applicable to provide any deeper insight into the spectrum spW . / of W . /. The best that is available to date is the following result. Theorem 1. We have spW. / D D and spW . / D for every > 0, where D is the closed unit disc in the complex plane. This was established in [7]. The nontrivial part of the theorem is that spW. / is all of D. In [7] it is actually shown that spW . / is contained in the open unit disc D and that each point 2 D belongs to the essential spectrum of W. /, which means that W. / I is not even invertible modulo compact operators.