GENERIC CONTINUOUS SPECTRUM FOR MULTI-DIMENSIONAL QUASIPERIODIC SCHRÖDINGER OPERATORS WITH ROUGH POTENTIALS

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1 GENERIC CONTINUOUS SPECTRUM FOR MULTI-DIMENSIONAL QUASIPERIODIC SCHRÖDINGER OPERATORS WITH ROUGH POTENTIALS YANG FAN AND RUI HAN Abstract. We stuy the mult-mensonal operator (H xu) n = m n = um + f(t n (x))u n where T s the shft of the torus T. When = 2 we show the spectrum of H x s almost surely purely contnuous for a.e. α an generc contnuous potentals. When 3 the same result hols for frequences uner an explct arthmetc crteron. We also show that general mult-mensonal operators wth measurable potentals o not have egenvalue for generc α.. Introucton In ths note we are ntereste n quasperoc Schrönger operators actng on l 2 (Z ) as follows (.) (H x u) n = u m + f(t n x)u n m n = where f : T R T n x = (x +n α... x +n α ) m = j= m j. We refer to f as the potental α (T\Q) as the frequency an x T as the phase. It s known that the spectral types of H x are almost surely nepenent of x. By the well known RAGE theorem fferent spectral types lea to fferent long-tme behavour of the solutons to tme-epenent Schrönger equatons. Thus one s naturally ntereste n fnng the way to entfy the spectral types of a gven operator. In ths note we wll show that for mult-mensonal operators purely contnuous spectrum s a generc phenomenon. In the one-mensonal case = a very useful crteron to exclue pont spectrum ue to Goron [3] states that f the potental can be approxmate n a reasonably fast sense by peroc ones then the operator oes not have pont spectrum. Followng ths ea Bosherntzan an Damank showe that gven any α for a generc contnuous potental H x has empty pont spectrum for a.e. x []. In the frst part of ths note we generalze ths result to mult-mensonal case. When = 2 we show ths result hols for an explct full measure set of α when 3 we show ths result s true uner an explct arthmetc crteron for α. It s nterestng whether ths result hols for a.e. α n the 3 case. In the secon part of ths note we explore the generc phenomenon n the frequency space. Accorng to Smon s Wonerlan Theorem [6] ense contnuous spectrum for a large class of metrc spaces of operators mples generc contnuous spectrum. Fxng a contnuous potental snce convergence n frequency mples operator convergence n the strong resolvent sense. Thus by Wonerlan Theorem generc contnuous spectrum follows from contnuous spectrum for ratonal frequences. However scontnuous potentals o not fall nto the crteron of Wonerlan Theorem. It was recently observe by Goron [4] that when = 2 one can prove contnuous spectrum for generc frequency even for measurable potentals. In ths note we generalze ths result to multmensonal operators. In ths note generc means ense Gδ. 2 The author actually ealt wth one-mensonal operator but wth mult-mensonal frequency.

2 2 YANG FAN AND RUI HAN A key ngreent that enables us to eal wth mult-mensonal operators s a crteron recently scovere by Goron an Nemrovsk [5]. Our results for generc contnuous potentals are as follows Theorem.. When = 2 f (α α 2 ) are not both of boune type then for generc contnuous potentals f H x has no pont spectrum for a.e. x T 2. The proof of Theorem. reles on the followng result about general mult-mensonal operators. Let x T = st (x Z). Theorem.2. Suppose there exsts an nfnte sequence Q = { = ( )} such that (.2) lm n α T = 0 for any =.... Then for generc contnuous potentals f H x has no pont spectrum for a.e. x T. Remark.. For = 2 (.2) hols f an only f (α α 2 ) are not both of boune type (see efnton 2.) see Lemma 4. n secton 4. However for 3 (.2) only hols for Lebesgue measure zero set of frequences ue to a smple argument by Borel-Cantell lemma see secton 6. Our result for measurable potentals s as follows. Theorem.3. Let the potental f be a measurable functon. For generc α T H x has empty pont spectrum for a.e. x. Remark.2. Ths theorem coul be easly generalze to the case of arbtrary (not necessarly equal to ) number of frequences. We organze ths note as follows: secton 2 serves as a preparaton for our proofs of Theorems.2.3 n sectons 3 an 5. The proof of Theorem. wll be scusse n secton 4. In secton 6 we show the smple argument by Borel-Cantell lemma that we mentone n Remark.. 2. Prelmnares For a Borel set U R we let U be ts Lebesgue measure. For x R let x T = st (x Z ). For a measurable functon f let f = {nf M 0 : f(x) M for a.e.x} be the L norm. 2.. Some facts from measure theory. Let f be a measurable functon on T. It s known that f( + y) converges to f( ) n measure as T y 0. We set (2.) F (y ɛ) = {x T : f(x + y) f(x) ɛ}. Then we have the followng fact Proposton 2.. For any ɛ > 0 an any η > 0 there s κ(ɛ η) > 0 such that f (2.2) y T < κ(ɛ η) then we have F (y ɛ) < η. We wll also set (2.3) E(M) = {x T : f(x) > M} Clearly E(M) 0 as M.

3 Contnue fracton approxmants. Let { pn q n } be contnue fracton approxmants of α. The followng propertes of contnue fractons wll be use later. Frst { pn+ = a n p n + p (2.4) n q n+ = a n q n + q n Seconly for any k q n we have (2.5) Thrly we have (2.6) q n α T kα T. q n α T 2. q n+ q n+ Defnton 2.. α s sa to be of boune type f there exsts C > 0 such that a n C for any n N. Remark 2.. It s well known that boune type α form a Lebesgue measure zero set. (2.7) If α s of boune type by (2.4) (2.5) an (2.6) we have for some C > 0 kα T Ck for any k A key ngreent from [5]. We have combne Theorems 3. an 5. from [5] nto the followng form whch s more convenent for us to use n ths note. Theorem 2.2. Let V be a complex-value functon on Z. Suppose there exsts γ > δ > 0 an an nfnte set P N satsfyng lm P τ τ = = such that there s a (τ τ )-peroc functon V τ ( ) satsfyng the property that for some λ 0 > 0 where ρ τ = ρ τ < (2 + M τ + λ 0 ) (2+γ)τ τ max V V (n) ; M τ = max V (n). n (2+δ)τ τ n (2+δ)τ τ Then the equaton Hu = λu wth any λ λ 0 oes not have non-trval l 2 (Z ) solutons. 3. Proof of Theorem.2 For any > δ > 0 let Γ n = (2 + δ) j= j. Take γ >. For x y R let x y = (x y... x y ). Proof. For any k N take n k such that (3.) Γ nk τ (n k) τ (n k) α T < k 2 for =.... By Rokhln-Halmos Lemma (see Theorem on p.242 n [2]) there exsts a set O nk {O nk + j α} j kγ nk are sjont an such that O nk > 2 k (2kΓ nk + ). We further partton O nk nto sets S nk l l s nk such that (3.2) am (S nk l) < k.

4 4 YANG FAN AND RUI HAN Choose compact set K nk l S nk l such that (3.3) For 0 m τ (n k) s nk l= =... we efne U nk lm = Then by (3.) an (3.2) we have (3.4) Set F nk K nk l > 2 k (2kΓ nk + ). j kγ nk /τ (n k ) am (U nk lm) < 2 + k K nk l + m α + j τ (n k) α. for any l m above. = {f C(T ) : f s constant on each U nk lm} an let F nk be the k 2+γ 2+δ Γn k neghborhoo of Fnk n C(T ). Note that by (3.4) an the fact that contnuous functon on T s unformly contnuous we have for each t N s open an ense subset of C(T ). Thus k t F = t s a ense G δ subset of C(T ). If f F then f Fñk for a subsequence {ñ k } of {n k }. For k > f 0 let Tñk = (Kñk l + j α). l sñk j (k )Γñk Then by (3.3) (3.5) an for any x Tñk we have (3.6) max j Z j Γñk /τ (ñ k ) F nk k t F nk Tñk ( (2k 2)Γ ñ k + ) ( 2 k ) 2 k as k 2kΓñk + f(x + j τ (ñ k) α) f(x) < k 2+γ 2+δ Γñ k < (4 + 2 f 0 ) 2+γ 2+δ Γñ k. By Borel-Cantell lemma a.e. x T belongs to nfntely many Tñk thus Theorem 2.2 mples that for any λ 2 + f 0 the equaton H x u = λu has no l 2 (Z ) non-trval soluton. Absence of pont spectrum then follows from the fact the norm of H x s 2 + f Proof of Theorem. Theorem. follows from a quck combnaton of Theorem.2 an the followng Lemma. Lemma 4.. When = 2 (.2) hols f an only f (α α 2 ) are not both of boune type. Proof.

5 5 The f recton. Let { p() n q n () suffces to prove the followng lemma. Lemma 4.2. For any ɛ > 0 there exsts m n N such that (4.) max ( q(2) m q n () ) < ɛ. q () n+ q (2) m+ } be contnue fracton approxmants of α = 2. By (2.6) t We wll argue by contracton. Assume that for some ɛ 0 > 0 we have max ( q(2) m q () q () n ) ɛ n+ q (2) 0 for m+ any (m n) N 2. Fx any n N choose m such that q () n q (2) m+ Then by our assumpton we have q(2) m ɛ q () 0 thus n+ < ɛ 0 q() n ɛ 0 q () n+ q(2) m q (2) m. q() n. ɛ 0 Ths mmeately mples q () n+ q ɛ 2 n () for any n N whch means α s of boune type. Smlarly 0 we coul show α 2 s also of boune type whch s a contracton. The only f recton. We wll agan argue by contracton. Assume (α α 2 ) are both of boune type an that there exsts Q = { = ( 2 )} such that { 2 α T 0 (4.2) 2 α 2 T 0. However by (2.7) we have for some C > 0 2 α T 2 α 2 T Cτ (n) 2 Cτ (n) 2 whch obvously can not converge to 0 at the same tme contractng (4.2). 5. Proof of Theorem.3 We nee to ve nto two fferent cases: Case. f = or Case 2. f <. Here we prove a etale proof of Case an scuss brefly about Case 2 at the en of ths secton. Case. Let {e } = be the stanar bass for R. For a number M > 0 let E(M) be efne as n (2.3). For τ = (τ...τ ) N let M τ = nf{m : E(M) (τ τ ) }. Let κ be efne as n (2.2). The followng lemma yels Theorem.3 rectly. Lemma 5.. Suppose there exsts an nfnte sequence Q wth lm Q τ τ for =... Then f the frequences satsfy (5.) (τ α... τ α ) T < κ(m (2+γ)τ τ τ (τ τ ) 2 ) for some γ > 0 an any τ Q the operator H x has no pont spectrum for a.e. x T.

6 6 YANG FAN AND RUI HAN Proof. For any τ Q. Let Xj τ = F (j τ α ( j + j )M (2+γ)τ τ τ ) an j () = e j. One coul check rectly by trgonometrc nequalty that f(x + j τ α) f(x) f(x + (j () + + j () ) τ α) f(x + (j () + + j ( ) ) τ α) = where we set j () + + j ( ) = 0 when =. Thus (5.2) X τ j F (j () τ α j M (2+γ)τ τ τ ) + (j () + + j ( ) ) τ α. Snce e τ α T τ α T κ(m (2+γ)τ τ τ (τ τ ) 2 ) agan by trgonometrc nequalty we have (5.3) F (j () τ α j M (2+γ)τ τ τ ) j (τ τ ) 2. Hence puttng (5.2) an (5.3) together we have (5.4) Xj τ j + + j τ 3 τ 3. Let Y τ j (5.5) = {x T : f(x + j τ α) > M τ } = E(M τ ) j τ α obvously Y τ j = E(M τ ). For any γ 2 > δ > 0 let I τ = {j Z : j (2 + δ)τ τ /τ for = 2 }. We enote Z τ = ( Iτ Xj τ ) ( I τ Yj τ ). Combnng (5.4) (5.5) we get (5.6) Z τ j + + j I τ (τ τ ) as Q τ. (τ τ ) j I τ By Borel-Cantell lemma for a.e. x T there s an nfnte sequence {τ} τ Px such that x / Z τ for any τ P x. Defne a τ-peroc potental by settng f τ (x + n α) = f(x + m α) where n j m j (mo τ j ) wth 0 m j τ j. Then snce x / Z τ we have (5.7) max f τ (x + n α) f(x + n α) < M (2+γ)τ τ τ n (2+δ)τ τ where M τ max n (2+δ)τ τ f(x+n α). Note that snce f = we have lm Px τ M τ =. Together wth (5.7) ths mples that for any λ 0 > 0 for τ P x large we have (5.8) max f τ (x + n α) f(x + n α) < (2 + M τ + λ 0 ) (2+ γ n (2+δ)τ τ 2 )τ τ. By Theorem 2.2 H x has no pont spectrum. Case 2. Note that when f < one coul choose M = f so that E(M) = 0. Then one can prove Lemma 5. wth (5.) replace by (5.9) (τ α... τ α ) T < κ((4 + 2M) (2+γ)τ τ (τ τ ) 2 ).

7 7 Lemma 6.. For any ɛ > 0 let (6.) Then A ɛ = lm sup A ɛ m...m 6. Arthmetc conton for > 2 A ɛ m...m = {α T : max (m m m α T ) < ɛ}. =... has Lebesgue measure zero. Ths Lemma clearly mples that when 3 α s such that (.2) hols form a Lebesgue measure zero set. Proof. Clearly A ɛ (2ɛ) m...m = (m m. Thus ) (6.2) m N By Borel-Cantell lemma A ɛ = 0. A ɛ m...m = (2ɛ) ( m N m acknowlegement m ) < for 3. Ths research was partally supporte by the NSF DMS We are grateful to Svetlana Jtomrskaya for her guance an useful scussons. We woul lke to thank Alexaner Goron for hs comments on the earler verson of ths manuscrpt. Y. F was supporte by CSC of Chna (No ) an the NSFC (grant no ) an NSF of Shanong Provnce (grant no. ZR203AM026). References. Bosherntzan M. an Damank D Generc contnuous spectrum for ergoc Schrnger operators. Communcatons n Mathematcal Physcs 283(3) pp Cornfel I. Fomn S. an Ya G Ya. Sna Ergoc Theory. Grunlehren er mathematschen Wssenschaften Goron A.Y The pont spectrum of the one-mensonal Schronger operator. Uspekh Matematcheskkh Nauk 3(4) pp Goron A.Y Dscrete Schrnger operators wth potentals efne by measurable samplng functons over Louvlle torus rotatons. Journal of fractal geometry to appear. 5. Goron A.Y. an Nemrovsk A Absence of Egenvalues for Quas-Peroc Lattce Operators wth Louvlle Frequences. Internatonal Mathematcs Research Notces p.rnw Smon B Operators wth sngular contnuous spectrum: I. General operators. Annals of Mathematcs 4() pp.3-45.