1 Introduction Let d 1begiven and let d1 = d,where = f0 1 2 :::g. Wedenote the points in d1 by(n x), n 2 d, x 0. Let V : d 7! R be given function and

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1 On the Spectrum of the Surface Maryland Model Vojkan Jaksic 1 and Stanislav Molchanov 2 1 Department of Mathematics and Statistics University of Ottawa, 585 King Edward Avenue Ottawa, ON, K1N 6N5, Canada 2 Department of Mathematics University of North Carolina Charlotte, NC 28223, USA March 30, 1998 Abstract We study spectral properties of the discrete Laplacian H on the half space d1 = d with a boundary condition (n ;1) = tan( n ) (n 0), where 2 [0 1] d. We denote by H 0 the Dirichlet Laplacian on d1. Whenever is independent over rationals (H) =R. Khoruzenko and Pastur [KP] have shown that for a set of 's of Lebesgue measure 1, the spectrum of H on R n (H 0 ) is pure point and that corresponding eigenfunctions decay exponentially. In this paper we show thatif is independent over rationals then the spectrum of H on the set (H 0 )ispurely absolutely continuous. Toappear in Letters in Mathematical Physics 1

2 1 Introduction Let d 1begiven and let d1 = d,where = f0 1 2 :::g. Wedenote the points in d1 by(n x), n 2 d, x 0. Let V : d 7! R be given function and let H be the discrete Laplacian on l 2 ( d1 )with the boundary condition (n ;1) = V (n) (n). When V 0 this operator reduces to the Dirichlet Laplacian which we denote by H 0. We recall that (H 0 )=[;2(d 1) 2(d 1)] and that the spectrum of H 0 is purely absolutely continuous. The operator H acts as (H )(n x) = ( P jn;n 0 j jx;x 0 j=1 (n 0 x 0 ) if x>0 (n 1) P jn;n 0 j =1 (n 0) V (n) (n 0) if x =0 where jnj = P d j=1 jn j j. Note that operator H can be viewed as the Schrodinger operator (1.1) H = H 0 V (1.2) where the potential V acts only along the d1 = d. More precisely, (V )(n x) =0 if x>0 and (V )(n 0) = V (n) (n 0). For many purposes, it is convenient to adopt this point of view and we will do so in the sequel. Since H 0 is bounded, the operator H is properly dened as aself-adjoint operator on l 2 ( d1 ). The spectral theory of operators H in the cases where V is a random or almost periodic function has been studied in [AM], [BS], [G], [G1], [JL], [JM], [JMP], [KP], [P], [M]. The principal physical motivation is to understand the formation and propagation properties of surface waves in regions with corrugated boundaries. For additional information on this program we refer the reader to a review article [JMP]. The Maryland model is the family of operators on l 2 ( d )oftheform h = h 0 V (n), where V (n) V = tan( n ) (1.3) h 0 is the usual free Laplacian on d, = ( 1 2 ::: d ) 2 [0 1] d and 2 [0 ]. To avoid singular cases, we will assume that for a given, is chosen so that 8n, n 6 0 mod =2: (1.4) The Maryland model has been extensively studied in [FP], [FGP], [FGP1],[GFP], [PRG], [S1], [S2]. We recall that = ( 1 ::: d ) is independent over rationals if for any choice of rational numbers r 1 :::: r d 2 Q, X k r k k 62 Q: It is not dicult to show that if is independent over rationals then for any 6= 0 and a.e. 2 [0 ], (h) =R (see e.g. [CFKS]). We say that has typical Diophantine properties if there 2

3 exists constants C k > 0such that jn ; mj >Cjnj ;k (1.5) for all n 2 d and m 2. The set of 's in [0 1] d for which (1.5) holds has Lebesgue measure 1. If has typical Diophantine properties then (h) = R for any which satises (1.4), the spectrum is pure point, the eigenvalues of h are simple and the corresponding eigenfunctions decay exponentially, see [FP] or [CFKS]. Thus, in any dimension and for typical, the deterministic potential (1.3) is strongly localizing. The surface Maryland model is a family of operators on l 2 ( d1 )denedby (1.1)-(1.3), H = H 0 V : Notation. In the sequel, whenever the meaning is clear within the context, we will drop subscripts. Thus, we write H for H, etc. We willalsousethe shorthand c d =2(d 1), so (H 0 )=[;c d c d ]. An easy Weyl's sequence argument yieldsthat for any 2 [0 1] d,andany which satises (1.4), (H 0 ) (H). If is independent over rationals, the standard argument (see e.g. [CFKS] or [JMP1]) yields that for a.e. 2 [0 ], (H) =R. Since the potential V models strongly corrugated boundary, it is natural to expect that the spectrum of H outside (H 0 ) is dense pure point for any which satises (1.5). Indeed, Khoruzenko and Pastur [KP] have proven the following result. Theorem 1.1 Assume that has typical Diophantine properties. Then for any 6= 0 and which satises (1.4), (H) =R and the spectrum of H on the set R n (;c d c d ) is pure point. On this set, the eigenvalues are simple and the corresponding eigenfunctions decay exponentially. In this paper, we study the spectrum of H on the set (H 0 ). One can show (see [JMP], [JMP1]) that for any 2 [0 1] d, the wave operators =s; lim t!1 eith e ;ith 0 exist for a.e. 2 [0 ], and consequently, that [;c d c d ] ac (H). This is not a surprising result: Due to the free propagation along the x-axis, the operator H should have some absolutely continuous spectrum. There where various speculations that H might have some point spectrum on (;c d c d ), and if is \extremely well approximated" byrationals even some singular continuous spectrum. Thus, the following result comes perhaps as a surprise. Theorem 1.2 If 2 [0 1] d is independent over rationals then for any 6= 0 and which satises (1.4), the spectrum of H on the set (;c d c d ) is purely absolutely continuous. 3

4 The relation between this result and the program of [JMP] will be discussed in [JMP1]. Acknoweldgments We are grateful to Y. Last, L. Pastur and B. Simon for useful discussions. The research of the rst author was supported by NSERC and of the second by NSF. Part of this work was done during the visit of the second author to University of Ottawa which was supported by NSERC. 2 Dimension reduction One of the basic ideas used in practically all works on the spectral theory of operators (1.1) is to \integrate" the x-variable and to reduce the d1-dimensional spectral problem to a d-dimensional problem, which will depend non-linearly on the spectral parameter E. In this section we carry out this dimension reduction for the surface Maryland model, and lay the ground for the proof of Theorem 1.3 The rst steps follow closely Section 2 in [JM]. We give details for readers convenience. We recall that the points in d1 are denoted by n =(n x), n 2 d, x 2. Let z 2 C, Im(z) 6= 0, be given and let These matrix elements satisfy the equation R(m (n x1) z)r(m (n x;1) z) if x>0, and if x = 0. R(m (n 1) z) X jn;n 0 j =1 R(m n z)=( m (H ; z) ;1 n ): X jn;n 0 j =1 R(m (n 0 x) z) = mn zr(m (n x) z) (2.6) R(m (n 0 0) z)(v (n) ; z)r(m (n 0) z) = mn (2.7) If m = (m 0) is the point on the boundary, Equation (2.6) can be \integrated". This is most conveniently done in the Fourier representation associated to the variable n. Let T = R=2 be the circle and T d the d-dimensional torus. We denote the points in T d by =( 1 ::: d ), and by d the usual Lebesgue measure. In the sequel we will use the shorthand () = P d k=1 2cos k. Let be aunitary map dened by the formula F : l 2 ( d1 ) 7! L 2 (T d ) l 2 ( ) (F )(n x) ^( x) = 1 (2) d=2 X n2 d (n x)exp(in ): In the new representation, Equations (2.6) and (2.7) are (recall that m =(m 0)) ^R(m ( x 1 z)) ^R(m ( x ; 1) z)(() ; z) ^R(m ( x) z) =0 (2.8) 4

5 and ^R(m ( 1) z)() ^R(m ( 0) z) d VR(m ( 0) z) ; z ^R(m ( 0) z) =em () (2.9) where e m () =F( mn )=(2) ;d=2 exp(im ). It follows from Equation (2.8) that if x>0, ^R(m ( x) z) = ^R(m ( 0) z)r( z) x (2.10) where r( z) istheroot of the quadratic equation X 1 X () =z (2.11) such that jr( z)j < 1. The other root is given by ~r( z) = 1=r( z). Substituting (2.10) into (2.9) and using that r ; z = ;~r, we get the equation ; ^R(m ( 0) z)~r( z) VR(m d ( 0) z) =em (): (2.12) Let R(m n z) R((m 0) (n 0) z) =( (m 0) (H ; z) ;1 (n 0) ) R(m z) ^R((m 0) ( 0) z) and j(n z) = T d e;in ~r( z)d: Aplying F ;1 to (2.12) we get that matrix elements R(m n z) satisfy the equation ; X k j(n ; k z)r(m k z)v (n)r(m n z) = mn : (2.13) The above construction is of course applicable to any V. Toproceed we have touse the particular structure of the potential V. Note that 1 ; exp(;2i n ; 2i) V (n) = tan( n ) =;i 1exp(;2i n ; 2i) : Multiplying both sides of Equation (2.13) by 1exp(;2i n ; 2i) andapplying F again, we get after simple algebra where e ;2i R(m ; 2 z)[i ; ~r( ; 2 z)] ;R(m z)[~r( z)i] =h m () h m () =e m ()exp(;2i)e m ( ; 2): 5

6 In the sequel we will assume that > 0. Similar analysis applies if < 0. It follows from Equation (2.11) that if Imz > 0thenIm ~r( z) > 0 (write ~r in polar form). Thus, if Im(z) > 0 and we arrive at the equation where ~R(m z) =R(m z)[i ~r( z)] (2.14) e ;2i ~R(m ; 2 z)( ; 2 z) ; ~R(m z) =h m () (2.15) ( z) = i ; ~r( z) i ~r( z) : For latter applications, the following simple observation is critical. Recall that > 0. Let R =[0 1). Lemma If Im(z) > 0 then 8 2 T d, j( z)j < Let jej < 2(d 1) and let O be an open set inside f 2 T d : j() ; Ej < 2g. Then t E ( ") ( E i") is a continuous function on O R, and for any ( ") in this set, jt E ( ")j < 1. Proof: Since Im~r( z) > 0, an elementary computation yields Part 1. It follows from Equation (2.11) and basic complex analysis that ~r( E i") iscontinuous in variables and " on the set OR, and that for 2O,Im~r( E) > 0. This yields Part 2. 2 Finally, theproof of Theorem 1.2 will be based on Proposition 2.2 Assume that for any E 2 (;c d c d ), and any which satises (1.4), lim sup "!0 T d j ~R(0 E i")jd < 1: Then the spectrum of H on (;c d c d ) is purely absolutely continuous. Proof: Let H n be the cyclic subspace generated by the vector (n 0) and H. It is easy to show (see e.g. [JL]) that the linear span of H n 's is dense in l 2 ( d1 ). Let U n0 be the unitary operator of translation by the vector (n 0 0). If 0 = ; 2 n 0 then ( (n0 0) (H ; z) ;1 (n0 0)) = ( (0 0) U ;1 n 0 (H ; z) ;1 U n0 (0 0) ) = ( (0 0) (H 0 ; z) ;1 (0 0) ): This argument and Fatou's theorem (see e.g. [RS] or [S3]) yield that H has purely absolutely continuous spectrum on (;c d c d ) for all satisfying (1.4) if for such 's and all E 2 (;c d c d ), A(E) = lim sup j( (0 0) (H ; E ; i") ;1 (0 0) )j < 1: "!0 6

7 Note that ( (0 0) (H ; E ; i") ;1 (0 0) )=(2) T ;d=2 R(0 E i")d: d It follows from Relation (2.14) that and we have that The statement follows. 2 jr(0 E i")j 1 j ~R(0 E i")j A(E) lim sup(2) ;d=2 jr(0 E i")jd "!0 Td lim sup(2) ;d=2 ;1 "!0 T j ~R(0 E i")jd: d 3 Proof of Theorem 1.2 We are now ready to nish the proof of Theorem 1.2. We will use the shorthand ~R( z) ~R(0 z). Let E 2 (;c d c d )begiven and let ">0. It follows from Equation (2.15) that j ~R( E i")j 2(2) ;d=2 j( ; 2 E i")jj~r( ; 2 E i")j: (3.16) Integrating over T d we get that Let T d j ~R( E i")jd 2(2) d=2 T d j( E i")jj ~R( E i")jd: C (1) E = f 2 T d : j() ; Ej 2g C (2) E = f 2 T d : j() ; Ej 2g: We will use the shorthand C 0 = 2(2) d=2. Splitting the integrals we get that C (1) E j ~R( E i")j(1 ;j( E i")j)d C 0 Since j( E i")j < 1, C (1) E C (2) E j ~R( E i")j(j( E i")j;1)d: j ~ R( E i")j(1 ;j( E i")j)d < C0 : (3.17) Let O be an open set such that its closure is properly contained in C (1) E. Then it follows from Part 2ofLemma 2.1 that there exists constant C > 0 such that for any 2Oand 0 "<1, 1 ;j( E i")j C > 0 7

8 and from Relation (3.17) that O j ~R( E i")jd C 0 =C: Let T : T d 7! T d be the translation map, T () = 2. For any positive integer m let O m T m (O). It follows from Equation (3.16) that j ~R( E i")j C 0 C 0 =C O 1 for all 0 <"<1, and arguing inductively, that O m j ~R( E i")jd mc 0 C 0 =C: Since is independent over rationals, T is ergodic, and the open sets O m cover T d. Picking a nite subcover, we derive that Theorem 1.2 now follows from Proposition 2.2. lim sup j ~R( E i")jd < 1: "!0 T d References [AM] [BS] Aizenman M., Molchanov S., Localization at Large Disorder and at Extreme Energies: An Elementary Derivation. Commun. Math. Phys. 157, 245 (1993) Boutet de Monvel A., Surkova A., Localisation des etats de surface pour une classe d'operateurs de Schrodinger discrets a potentiels de surface quasi-periodiques, preprint. [CFKS] Cycon H., Froese R., Kirsch W., Simon B., Schrodinger Operators. Springer-Verlag, Berlin-Heidelberg [FP] [FGP] Figotin A., Pastur L., An Exactly Solvable Model of a Multidimensional Incommensurate Structure. Commun. Math. Phys. 95, (1984) Fishman S., Grempel D., Prange R., Chaos, quantum recurrences, and Anderson localization. Phys. Rev. Lett. 49, 509 (1082). [FGP1] Fishman S., Grempel D., Prange R., Localization in a d-dimensional incommensurate structure, Phys. Rev. B 29, (1984) [FP1] Figotin A., Pastur L.,Spectra of random and almost periodic operators. Springer-Verlag, Berlin

9 [GFP] Grempel D., Fishman S., Prange R., Localization in an incommensurate potential: An exactly solvable model. Phys. Rev. Lett. 49, 833 (1982). [G] Grinshpun V., Dopovidi Akademii Nauk Ukrainy (Proc. Acad. Sc. Ukraine), 8, 18, (1992) 9, 26 (1993) [G1] [JL] [JM] [JMP] Grinshpun V., Localization for random potentials supported on a subspace. Lett. Math. Phys. 34, (1995). Jaksic V., Last Y., in preparation. Jaksic V., Molchanov S., On the Surface Spectrum in Dimension Two. Preprint. Jaksic V., Molchanov S., Pastur L., On the Propagation Properties of Surface Waves. IMA Preprint 1316, (1995) [JMP1] Jaksic V., Molchanov S., Pastur L., in preparation. [KP] [M] Khoruzhenko, B.A., Pastur L., The localization of surface states: an exactly solvable model. Physics Reports 288, (1997). Molchanov S., Lectures on Random Media. In Lectures on Probability, ed. P. Bernard, Lecture Notes in Mathematics, 1581, Springer-Verlag, Heidelberg (1994). [P] Pastur L., Surface Waves: Propagation and Localization. Journees \Equations aix Derivees Partielles" (Saint-Jean-de Monts, 1995), Exp VII, Ecole Polytech., Pallaiseau, (1995). [PR] [RS] [S1] [S2] [S3] Prange R., Grempel D., Fishman S., A solvable model of quantum motion in an incommensurate potential. Phys Rev. B 29, (1984). Reed M., Simon B., Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic Press Inc, Boston Simon B., The equality ofthe density ofstates in a wide class of tight bindinglorentzian models. Phys. Rev. B 27, (1983). Simon B., Almost periodic Schrodinger operators, IV. The Maryland model. Ann. Phys. 159, (1985). Simon B., Spectral Analysis of Rank One Perturbations and Applications. CRM Proc. Lecture Notes, 8, AMS, Providence, RI,

University of Minnesota, 514 Vincent Hall Minneapolis. Kharkov, Ukraine

On the Propagation Properties of Surface Waves V. Jaksic 1, S. Molchanov 2,L.Pastur 3 1 Institute for Mathematics and its Applications University of Minnesota, 514 Vincent Hall 55455-0436 Minneapolis Minnesota,