Let H be the Schrödinger operator with potential V(n) = n^(−γ) cos(2πn^ρ), where ρ ∈ (1,2) and γ ∈ (0, 1/2 − ρ−1/2). I show that for almost every boundary condition H has pure-point spectrum

Krüger, Helge

English

Caltech Authors - Main

Cootempon,ry M<\ttwmaticfl: 
Volume 552. 2011 
Schrodinger operators with potential V(n) = n -1 cos(21rnP) 
Helge Kruger 
ABSTRACT. Let H be lhc Schrodinger opewtor with potential V (n) = n -"t cos(27rnP) , 
where p E (1, 2) and 1 E {0, ~- 9 ). I show that for almost every boundary 
condition H has pme-point spectrum. 
1. Introduction 
ln this short note, I wish Lo show that for l < p < 2 Lhe sequence nP (mod 1) 
generates sufficient randomness to obtain results for decaying potentials similar to 
the case of independent. identically distributed random variables as discussed by 
Simon in [16] and Kiselev, Last, and Simon in [8]: 
Theorem 1.1 ([8]). Let {wn. }~=l be ·independent, identically distributed, and non-
constant random variables. Then for 'Y E (0. i) the Schrodinger operator .6 + V on 
£2(Z+) with potential 
(1.1) V(n) = Wn 
n' 
has pure point spectrum for almost every { Wn };:::'=1 . 
Here .6.v.(n)- u(n+ 1) + u(n - 1) is the d iscrete Laplacian. 
1 will now introduce the operator studied in this paper and state the main 
result. Let p and 1' satisfy 
(1.2) 
and introduce the potential 
(1.3) 
For the houndru:y condition /3 E JR. the half-line SchrOdinger 
£2 (Z+)-+ e2 (Z+) is given by 
operator H ~-P 
(1.4) (H~ u)(n) = { u(n + 1) + u(n -1) + V1 .p(n)u(n), 
y,p tt.(2) + (V7 ,p(1) + ;?)u(l), 
n 2: 2; 
n = 1. 
2000 !l'fathematics Subject Classification. Primary 8lQlO; Secondary 47836. 
Key words and pht·ases. Schrooinger Operators, pure point spectrum, de.ca.ying potentials. 
This work was done while H. K. was an Erwin Schrooinger Junior Research Fellow. 
Parts of this work are suppv• ~"u l>y i:\ fellvw:;hip vf ~lie Si111uu:; Fvuuua~ivu. 
@20ll Am(trint.r'l Mathe:rnat.ic:aJ Society 
109 
110 
Since {; .,(n)--+ 0 as n --too. we haw that a,.,~ (H~7.p) = [-2.2]. 
Theorem 1.2. Let p and 1 obey (1.2). Then for almo$f every ;3 . H~.p has pure 
point spectrum in [- 2, 2]. 
Arg11ments similar to the one used in this proof W<'rc first u::;cd by Kirsch, 
Molchanov. and Pnstur [6], [7 ]. then Stoh [19], a.nd last Bourgain [1]. Bourgain's 
proof is closest to the one given in this paper. 
This result is moth·atcd by the work of Lukic [13], in which he showed that 
for V (n ) = n1 co:.;(21Tan) with 1 > 0 the operator If bas absolut<'ly continuous 
spectrum in - 2. 2]. Furthermore. he shows that the possible singular :-.pectrum 
[-2, 2] is contained in a finite set of points. That eigem·alues are possible is known 
since the work of\\'ign<' r and von Neumann on the continuous Schrodinger operator 
[14]. 
The condition on 13 in Theorem 1.2 is optimal. General results (see Section 12.4. 
in [17]) imply thaL for a dense Gli set of 13 the operator H~,fl has singular continuous 
spectrum. 
Remark 1.3. 
(1.5) 
It is possible to consider more grnrml potentials of the form 
V (n) = ~ cos(21TnP) + f (n), 
n"' n"'+E 
where 1 and p obey (1.2). f (n ) is a bounded sequence. and € > 0. The proof of the 
theorem is essentially lht same. 
it should also be possible to replace .T H cos(21TX) with a sufficiently nice !-
periodic function. Th<' main r-equirement is extending Theor·em 3.1 on the spectrum 
of the Almost Mathieu. operator. whose proof is not the topic of this paper. 
Remar k 1.4. The proof shows that for some ~ > 0. we have for all generalized 
eigenfunctions 1/• 
{1.6) 
for n large enough. 
Let me now discuss. if the range of 1 and p given in ( l.2) is optimal. For 
simplicity, I restrict myself to 1 > 0, when the potential is decaying 
By the result~ of Christ and Kiselev [2], Dcift and Killip [4], or Renlling [15], 
we have that H~,p has absolutely continuous spectrum for 1 > ~. l~1rthennore, 
the results of Stolz from [18] imply thai II~.p has purely absolutely continuous 
spectrum for p E (0. 1) and 'Y > 0. The already mentioned result by Lukic [13] 
imply absolutely continuous spectrum for p = 1 and 1 > 0. 
However, I would ex-peeL that in the complement of this range. that is p > 1 and 
1 E (0, !) rhe operator H~.p has pure point spectrwn for almost every {3. The main 
reason for this is that lhe Lyapunov expoucut L>. (E ) associated to the potential 
V (n ) = 2>.cos(2rrnP) is expected to behave like L>.(E) ~ ~t>.2 for some 1 > 0 as 
>. -+ 0. See the work of Bourgain [1], and my own in [10] and [11] for some positive 
results in this direction. 
Jf one compares Theorem 1.2 with the main result of [16], one notices that the 
result of this paper requires an almost sure choice of boundary condition, wherca.<; 
[16J holds for almost every random parameter. It b an interesting question. if one 
SCHRODI:>IGER OPBRA1 OR.S \\ !TH I'OTEI'>TL\1. \ '(n ) = n - · cosl21w1 ) 1! I 
could obtain a rebult ::-imilar to ,16; in our cont<>xt. In order lo add a random 
parameter. one ~hould modify the potential to 
1 (1.7) v,..p,1J(n) = n'i cos(2rr(n + 1J)P). 
Then the quest io11 is: Does Theorem 1.2 still hold with almo::;t every ,B replaced by 
almost every (}'? The rca.>on for the choice of pot.cnLial is that this implies some 
stability of the sets I 1 coru;lructed in Propos ition 3.2. 
The rest of t.he paper splits into two sections. [u the next one. I discuss some 
general properties of th<' resolvent equation. Then l usc these in Section 3 to prove 
T heorem 1. 2. 
2. The resolvent equa tion 
In this section, J will discuss a method to obtain bounds on the Green function. 
Let me begin by introducing the necessary notation. I will denote by II = D. + V 
a Schrodinger operator either on e2 (Z) or on e2 (Z+)· For A~ Z denote by HA lhc 
restriction of H to {l (A). l denote by {ex} .rEZ l he ~tandard basis of €2 (Z). t hat is 
(2.1) ( ) {
1, x = n; 
ex n = 
0, othcndsc. 
ForE¢. u(H·') and .r. y E :\. t he Green's function is defined hy 
(2.2) GA(E. x. y) = (ex . (HA E) - 1e11). 
For either x ¢. A or y ¢.A, we set G'"(E, ;r, y) = 0. For:=:~ A. 1 denote by x:: the 
restriction map f2 (A) -t f2(:=:). 
Let now H = 6. + V and fi = D. + V be Schrodinger operators and I ~ A n:=:. 
A computation shows 
(2.3) (H2 - E)- 1'1., - 'r (II_..- E)-1 = (ii=-- £) - 1(\, H A- f(=-':o)(H'\- £)- 1. 
If V(n) = V(n) for n E I. this formula hecomes 
(2.4) (ff2- E)- 1 \J- \r(H'\ - E) - 1 = (fi:=: - E)-1 [,,. 6.2 ](H " - E)- 1, 
where [A, B] = AB - BA denotes the commutator. [n the case of I an interval 
[a , b] this commut(:l.tor ran be computed explicitly to be 
(2.5) [\la,b, D.Ju(n) = 
-u(a), 
1t(a- 1). 
tt(b ..L- 1), 
-u(b). 
n=c.L-l; 
n=a: 
n =b; 
n = b+l: 
0, otherwise. 
Combining these com;idcrations, we obtain 
Lemma 2.1. Let x E [a, bJ andy E A \ [a, b]. As.wme V(n) = V(n) for n E [a, b]. 
Then 
(2.6) 105 (£. k. x)l. 
kE { a- l ,a,b,b+ 1} 
112 H . KHUCErt 
If H. is just defined on €2 ([a, b]) , then the. equation becomes the more familiar 
IGA(E.x,y)l ~ II(HA - E)-111 L IGfa,bi(E.k. x)l. 
kE{".b} 
I should digress here a little hit, the computation <.lone to prove Lemma 2.1 should 
be fami liar from the context of continuum operators. ln the discrete case, these are 
usually unnecessary, since everything is as smooth as one wants. However, in this 
work (2.6) is essential, since it, allows us to avoid getting extra eigenvalues inside of 
gaps of opera.tors, when restricting to some space interval. Using Lemma 2.1, we 
will show the following theorem. 
Theorem 2.2. Let H be a Schrodinger ope·rator·, c E (0, l), N 2: 1 large enmtgh, 
& ~ IR. c < c < d < d such that 
[c, d] = /\._ U Ac U A+ 
with A_ = [c, cj, Ac = [c + 1, d- 1], and A+ = [d, d] disjoint intervals. Assttme 
for E E & that theTe exist [a±, b± J ~ J\± and Schrodinger operator's H ± such that 
b± - a.± 2: N . 
1 1 l 
(2.7) u(H±) n [E- JVl-o; , E + JVl-g] = 0, IIV- V± lle.x([a± ,b± ]) s 2JV1- f:' 
Also assume 
(2 .8) 
Then there exists !3 ~ & such that 
:. (i) 1!31 S e- N 4 • 
( ii) For x E { c, d}, y E Ar;, and E E £ \ !3, we have 
(2.9) IG[c.dl (E,x,y)i::; e-N ~ . 
In words this theorem says local gaps imply decay of the Green's function. (2. 7) 
makes precise what I mean by a local gap. 
For the proof, we will need the Combes- Thomas estimate, [3]: 
Lemma 2.3. Lett: > 0. Then theTe exists "' = ~~:(t:) > 0, Oo = o0 (e) > 0 snch that 
the following hold: 
ForEE IR, 8 E (0, 80 ), H : f 2 (A) ~ £2 (/\.) a SchrodingeT opemtoT with 
(2.10) u(H) n [E- t5, E + 6] = 0, 
and x, yE A with ix- Vi 2: (p\,., we have 
(2.11) fG(E,x, y) l S e-~~;<l j x-y l_ 
Let E E & and a±, b± as in ( 2. 7) . Define 
(2.12) H± = H± + X[a±,b±J(V - V±)· 
T hen by assumption IIH±- H±ll $ 2NL. and thus 
~ 1 1 
(2.13) u (H±) n [E- 2l'P-€, E + ZNl-&] = 0. 
By the Combes-T homas estimate, we can conclude for E E I , 
y = l a± ; b± J , x E {a±, b±} 
SCliRODJNGER OPERATORS WITH POTJ:~NTJAJ, V(n) = ,. -.,. cos(Z1rr> 1') 113 
that 
(2.14) IG--;Ja±.b±i(E . )I< - KN' ± ,x,y _ e 
as long as N is large enough. Here, a±, b± depend on E, but 1'<, dues not. 
Lemma 2.4. Define B by 
(2.15) !3 = 
Then !BI ~ e-N ~ . 
PROOF. The numbex of po:.;sible choices for a, b is bounded by (d - c)2 and also 
#a(Hia.bJ) ::; (d- c) for all possible choices of a , b. Hence 
IBI ::; 2(d- c)3e-N~ . 
The claim follows. D 
PROOF OF THEOREM 2.2. Let E E 1 \ B. Then by applying (2.6) with A = 
[c, d] and [a, b] = [Y-, Y+], we can conclude that 
IGic,dJ(E,x,y) l ~ eN l . (IGic,dl (E,x,y_)l + IG{c,dJ(E,x,y+)i). 
Now using the result obtained by the Combes-Thomas estimate, and (2.6) once 
more, we obtain 
IG~c.d] (E, .1:, Y) i S 8e2N~ · e-~<N'. 
Choosing N large enough, the result follows. D 
3. P roof of Theore m 1.2 
Introduce for a irrational. w E [0 , 1]. and 1\ > 0 the Almost-:Mathieu operator 
fh,.a.w : €2 (Z) , €2 (2), (3.1) 
H>..,c. ,wu(n) = u(n + 1) + u(n - 1) + 2A cos(27r(w + na))u(n). 
We will need the following fact about the spectrum of this operator. 
Theorem 3.1. Let 6 > 0. There exists a constant /'i, = f'<,(b') > 0 and >.a = Ao(6) > 0 
such that for A E (0, Ao) , w E [0, 1], and a satisf'l}ing 
(3.2) 2 cos(1ra) E [-2 + 6, - oj U (o, 2- o], 
we have 
(3.3) dist(± 2cos(7ra), a(H>.,a,w)) ~AI'<,. 
PROOF. See [1], {5], [10], [12). 
For k ~ 2, we introduce the disjoint sets 
(3.4) A% = [2k I 2k+l], 
(3.5) AJ; = [2k-t, 2k - 1], 
(3.6) 
and Ak = AJ; U A'k U At. We also define 
1 (3.7) c = 6(2- p- 2')'). 
0 
114 H.KR0GE;R 
If (1.2) holds, t.hen f > 0. Also note ; + 3.: = 1- ~ E (0. I). 
Proposition 3 .2. Let 8 > 0, k 2: 1.:1 (6) and o: E [0. l ] with 
(3.8) 2 cos( no:) E [-2 + 8, -8] U [6, 2- 6]. 
Then there exist intervals J:.r. ~ !'it satisfying 
( i) #(I±) ;::: 2 . 2(-r+Z£ )k + l. 
(ii) IIV...,..p - V>.,cc.w lle""(J:::) = 0( 2k,Lp1 ) forsomew and A E [ 2-,L 11 , 2~<!+21 ] . 
PROOF. The arguments for f _ and It are similar, so I restrict myself to L . 
Define c = l 2~·-• t2k - l J and o:m = pmP- l . It is easy to check that 
d p-1- 0( 1 ) 
-d px - -2-. X :J: -p 
Using this. one concludes that for k large enough there exists rh such that 
lrh - cl <_ 2£~ ., lpmP- 1 a:l < 4 
- - z<k- 1 H2-p) · 
Let e = f2h+le-)kl and define L = [rh- £. 1n + P.]. By some computations the claim 
follows. 0 
B:v Theorem 2.2 with N = 2(")'_._Z<lk, we now obtain 
Corollary 3. 3. Let o > 0 and k 2: k2 ( o). There exists a set Eo.k such that 
(i) IEa,k I ::; k\ . 
(ii) Let y E Ak, x E {2k-l,2k+l +2k-l), and 
(3.9) E E ([-2 + 8, - oJ U [8, 2- 8]) \ Ea,k· 
We have 
(3.10) 
\Ve now proceed to derive Theorem 1.2. The stralegy of proof is often called 
spectral averaging. See for example Section 12.3. in the book [17] by Simon for 
another implementation of this strategy. Fix some/. p satisfying (1.2). For 8 > 0, 
introduce 
(3.11) Es = n ( LJ Es k) . t~k2(J> k~e ' 
The Borel- Cantelli argument shows that Eo I = 0. In particular E = Ui22 £:} also 
has zero measure. Jt, is well-known tha1 there exists a unique probability measure 
p,f:J chat satisfies 
(3.12) 
for Im(z) > 0. This measure is known a the spectral measure. 
Lemma 3.4. There exists a set B such that IR \ Bl = 0 and for f3 E B , we have 
J.LP( f ) = 0. 
SCHROf)[:-.IGEH OPERATORS WITH POTENTIAL t (t~ ) = ,, -~ cos(:.!1rnP) 115 
PROOF. By Tlworem 11.8. in [17], we haw that J J1 8 dJ is th<' Lebesgue mea-
sure. Thus j 11 "~(£)dJ = O. 
Since Jti\£) ~ 0. Lhe claim follows. 0 
P ROOF OF THEORE~I 1.2. Let :3 E B. For p!7 almost every E E ( -2, 2) \ {0} 
there exisrs a generalized eigenfunction (see Lemma 3.1. in [20J), that is a nonzero 
solut.ion v of H~,p11 = Eu satisfying l7t(n) l :S n for n ~ 1 and u(O) = 0. If we show 
that u is in f2(Z+). we obtain Lhat 11 8 almost every E is an eigenwuue. thus that 
f.L8 is pure point. 
By construction of £. we o1n choose 8 > 0 and e ;::: 1 sucll that 
E E [-2 + 6. - <5] U fo, 2- 6J. 
and fork ~ f 
E E Eo.k· 
For x E.'\~= [21.. 2k+ 1!, we h~we 
u(x) = -G~~p(E, x. 2k)u(2" 1 -1)- G~.~(E, x, 2k+l + 2" 1)u(2k+l + 2k-l + 1) . 
Corollary 3.3. we obtain that lu(:r) l $ ~ for k large enough. This implies that 
u E f 2 (Z-t) finishing the proof. 0 
Acknowledgements 
I am thanking to Daniel t'eltschi and Robert Sims for their kind invitation to 
the second Arizona School of Analysis wit h Applications, where the idea for this 
project originated. Furthermore, I thank ~vlilivoje Lukic for useful discussions. I 
also like ro thank Gunter Stolz for useful discussions. which lead to the choice of 
(1. 7). 
R e ferences 
[1] J. Bourgain, Posttwe Lyapovrwv expon<:nls for mo.~t energtes, Geometric a-spects of func-
t ional analysis, 37- 66, Lecture Notes in 1\·lath. 1745, Springer, Berlin, 2000. 
[2] M. Christ, A. Kiselev, Absolutely continuous spectrum for one-dimetMHmal Schrodmger 
operators u.-ith slowly decaymg potenttals: some optimal results. Journal of the .'\MS. 11 
(1998), 771-797. 
[3] J.M. Combes. 1.. Thomas. Asymptotic behavicur of eigenfunctions for multiparticle 
SchrOdmger opemtors. Comm. }.lath. Phys. 34 (l9i3) . 251 270. 
(<!] P. Deift. R. Killip , On the ab:;olutely continuous ~;pectrom of one-dtmenstonal Schrodmger 
operators with .5quare summable potentials. Comm . .\lath. Phys. 203, 341 347 (1999). 
(5] B. Hclffer, P. Kerdelhue. J . Sj&trand. f,e papillon de Hofsladter revts1.t.e. Mem. Soc. Math. 
France (1'\.S.). No. -l3 (1990). 7 pp. 
[6] \V. Kirsch. S.A. !\lolcbanov. L.A. Pa.stur. The one-dimensional Schrdmger operator unth 
unbounded potenttal: the pttN! 1JOint spectrum. F\mktsional. Anal. i Prilozhcn. 24:3 (1990). 
14- 25 (Russian). Translation in Funct. Anal. Appl. 24:3 (1990), 176186. 
(7) W. Kirsch. S.A. ~lolchanov, L.A. P~lur. One-dmtetMional Schrdinger operators wtth htgh 
potentia( barners. Operator calculus and spectral theory (Lambrecht, 1991), 163 170, 
Oper. Theory Adv. Appl. , 57. Birkhuser, Flasel, 1992. 
[8] A. Kiselcv, Y. Last. B. Simon, Modified Prufe7' and BFGP transforms and the spectral 
analysiS of one-dtmenstonal Schrodmg('r operators. Comm. :Math. Pbys. 191 (1998), 1-45. 
[9j H. Kriiger, A famtly of Schrodmger Oper-ators whose spectrum ts an interval, Comm. Math. 
Phys. 290:3, 935-939 (2009). 
116 H. I<Ri.JGEB 
[10) H. Kriiger, Probabilisl:ic averages of Jacobt operators. Comm. );lath. P hys. 295:3, 853- 875 
(2010) . 
!11] H. KrUger. f\.1ultiscale .4nalystS fo1· Ergodic Schrod~nger opemtor·s and poszftmty of Lyapttnov 
e:vponents, J. d'Analyse !\·lath. (to appear). 
(12] II. Kriiger, Semtcla.ssu;al analys~s of the largest gap of qua.s?.-per-todic Schrodtnger operators. 
Math. tvlodel. Nat. Phcnom. 5:4 , 256-26S (2010). 
[13] M. Lukic, Orthogonal polynomials with mcursion coefficients of gene1·alized bounded varia-
tzon. 
[14] J. von Neumann, E. Wigner, Uber merkwiir-dige diskrete Etgenwerte. Phy:;. z. 30 (1929), 
467 -470. 
[15] C. Remliug. The a/n;uluLely cuntin·uuus spectrum of one-dimensional Sch.ri:idinger openztor·s 
with deca.ying potentials. Comm. :--1ath. Phys. 193 (1998), 151- 170. 
[16] B. Simon, Some Jacobt matrr.ces with decaying potential and dense point spectrum, Comm. 
i'VIath . Phys. 8 7 (1982) . 253 258. 
[17] B. Simon, Trace Ideals and Thei1· Applications. Second edition, Mathematical Surveys and 
Monographs, 120. American Mathematical Society, 2005 
[18] G. Stolz, Spectral theory for slowly oscillating potenti.al.s. f. Jacobi matrice.~- Manuscript a 
Math. 84, no. 3-4, 245-260 (1994). 
[19] G. Stolz, Localization for Schrodinge1· operators with effectwe barrieTs, J. Funct. Anal. 
146:2 (1997), 416-129. 
[20] G. Tesch!. Jacobi Operators and Comple1.ely Integrable Nonlinea'r Lattices. Math. Surv. and 
Mon. 72 , Amer. Math. Soc., Rhode Island , 2000. 
MATHEMATICS 253- :37, CALTRCH, PASAOE\I A. CA 91125 
E-mail address: helge@calt ech . e du 
URL: http://-www. i ts .ca l tec h. edu;-helge / 


Schrödinger operators with potential V(n)=N^(-y) cos (2πn^p)

Cootempon,ry M<\ttwmaticfl: 
Volume 552. 2011 
Schrodinger operators with potential V(n) = n -1 cos(21rnP) 
Helge Kruger 
ABSTRACT. Let H be lhc Schrodinger opewtor with potential V (n) = n -"t cos(27rnP) , 
where p E (1, 2) and 1 E {0, ~- 9 ). I show that for almost every boundary 
condition H has pme-point spectrum. 
1. Introduction 
ln this short note, I wish Lo show that for l < p < 2 Lhe sequence nP (mod 1) 
generates sufficient randomness to obtain results for decaying potentials similar to 
the case of independent. identically distributed random variables as discussed by 
Simon in [16] and Kiselev, Last, and Simon in [8]: 
Theorem 1.1 ([8]). Let {wn. }~=l be ·independent, identically distributed, and non-
constant random variables. Then for 'Y E (0. i) the Schrodinger operator .6 + V on 
£2(Z+) with potential 
(1.1) V(n) = Wn 
n' 
has pure point spectrum for almost every { Wn };:::'=1 . 
Here .6.v.(n)- u(n+ 1) + u(n - 1) is the d iscrete Laplacian. 
1 will now introduce the operator studied in this paper and state the main 
result. Let p and 1' satisfy 
(1.2) 
and introduce the potential 
(1.3) 
For the houndru:y condition /3 E JR. the half-line SchrOdinger 
£2 (Z+)-+ e2 (Z+) is given by 
operator H ~-P 
(1.4) (H~ u)(n) = { u(n + 1) + u(n -1) + V1 .p(n)u(n), 
y,p tt.(2) + (V7 ,p(1) + ;?)u(l), 
n 2: 2; 
n = 1. 
2000 !l'fathematics Subject Classification. Primary 8lQlO; Secondary 47836. 
Key words and pht·ases. Schrooinger Operators, pure point spectrum, de.ca.ying potentials. 
This work was done while H. K. was an Erwin Schrooinger Junior Research Fellow. 
Parts of this work are suppv• ~"u l>y i:\ fellvw:;hip vf ~lie Si111uu:; Fvuuua~ivu. 
@20ll Am(trint.r'l Mathe:rnat.ic:aJ Society 
109 
110 
Since {; .,(n)--+ 0 as n --too. we haw that a,.,~ (H~7.p) = [-2.2]. 
Theorem 1.2. Let p and 1 obey (1.2). Then for almo$f every ;3 . H~.p has pure 
point spectrum in [- 2, 2]. 
Arg11ments similar to the one used in this proof W<'rc first u::;cd by Kirsch, 
Molchanov. and Pnstur [6], [7 ]. then Stoh [19], a.nd last Bourgain [1]. Bourgain's 
proof is closest to the one given in this paper. 
This result is moth·atcd by the work of Lukic [13], in which he showed that 
for V (n ) = n1 co:.;(21Tan) with 1 > 0 the operator If bas absolut<'ly continuous 
spectrum in - 2. 2]. Furthermore. he shows that the possible singular :-.pectrum 
[-2, 2] is contained in a finite set of points. That eigem·alues are possible is known 
since the work of\\'ign<' r and von Neumann on the continuous Schrodinger operator 
[14]. 
The condition on 13 in Theorem 1.2 is optimal. General results (see Section 12.4. 
in [17]) imply thaL for a dense Gli set of 13 the operator H~,fl has singular continuous 
spectrum. 
Remark 1.3. 
(1.5) 
It is possible to consider more grnrml potentials of the form 
V (n) = ~ cos(21TnP) + f (n), 
n"' n"'+E 
where 1 and p obey (1.2). f (n ) is a bounded sequence. and € > 0. The proof of the 
theorem is essentially lht same. 
it should also be possible to replace .T H cos(21TX) with a sufficiently nice !-
periodic function. Th<' main r-equirement is extending Theor·em 3.1 on the spectrum 
of the Almost Mathieu. operator. whose proof is not the topic of this paper. 
Remar k 1.4. The proof shows that for some ~ > 0. we have for all generalized 
eigenfunctions 1/• 
{1.6) 
for n large enough. 
Let me now discuss. if the range of 1 and p given in ( l.2) is optimal. For 
simplicity, I restrict myself to 1 > 0, when the potential is decaying 
By the result~ of Christ and Kiselev [2], Dcift and Killip [4], or Renlling [15], 
we have that H~,p has absolutely continuous spectrum for 1 > ~. l~1rthennore, 
the results of Stolz from [18] imply thai II~.p has purely absolutely continuous 
spectrum for p E (0. 1) and 'Y > 0. The already mentioned result by Lukic [13] 
imply absolutely continuous spectrum for p = 1 and 1 > 0. 
However, I would ex-peeL that in the complement of this range. that is p > 1 and 
1 E (0, !) rhe operator H~.p has pure point spectrwn for almost every {3. The main 
reason for this is that lhe Lyapunov expoucut L>. (E ) associated to the potential 
V (n ) = 2>.cos(2rrnP) is expected to behave like L>.(E) ~ ~t>.2 for some 1 > 0 as 
>. -+ 0. See the work of Bourgain [1], and my own in [10] and [11] for some positive 
results in this direction. 
Jf one compares Theorem 1.2 with the main result of [16], one notices that the 
result of this paper requires an almost sure choice of boundary condition, wherca.<; 
[16J holds for almost every random parameter. It b an interesting question. if one 
SCHRODI:>IGER OPBRA1 OR.S \\ !TH I'OTEI'>TL\1. \ '(n ) = n - · cosl21w1 ) 1! I 
could obtain a rebult ::-imilar to ,16; in our cont<>xt. In order lo add a random 
parameter. one ~hould modify the potential to 
1 
(1.7) v,..p,1J(n) = n'i cos(2rr(n + 1J)P). 
Then the quest io11 is: Does Theorem 1.2 still hold with almo::;t every ,B replaced by 
almost every (}'? The rca.>on for the choice of pot.cnLial is that this implies some 
stability of the sets I 1 coru;lructed in Propos ition 3.2. 
The rest of t.he paper splits into two sections. [u the next one. I discuss some 
general properties of th<' resolvent equation. Then l usc these in Section 3 to prove 
T heorem 1. 2. 
2. The resolvent equa tion 
In this section, J will discuss a method to obtain bounds on the Green function. 
Let me begin by introducing the necessary notation. I will denote by II = D. + V 
a Schrodinger operator either on e2 (Z) or on e2 (Z+)· For A~ Z denote by HA lhc 
restriction of H to {l (A). l denote by {ex} .rEZ l he ~tandard basis of €2 (Z). t hat is 
(2.1) ( ) {
1, x = n; 
ex n = 
0, othcndsc. 
ForE¢. u(H·') and .r. y E :\. t he Green's function is defined hy 
(2.2) GA(E. x. y) = (ex . (HA E) - 1e11). 
For either x ¢. A or y ¢.A, we set G'"(E, ;r, y) = 0. For:=:~ A. 1 denote by x:: the 
restriction map f2 (A) -t f2(:=:). 
Let now H = 6. + V and fi = D. + V be Schrodinger operators and I ~ A n:=:. 
A computation shows 
(2.3) (H2 - E)- 1'1., - 'r (II_..- E)-1 = (ii=-- £) - 1(\, H A- f(=-':o)(H'\- £)- 1. 
If V(n) = V(n) for n E I. this formula hecomes 
(2.4) (ff2- E)- 1 \J- \r(H'\ - E) - 1 = (fi:=: - E)-1 [,,. 6.2 ](H " - E)- 1, 
where [A, B] = AB - BA denotes the commutator. [n the case of I an interval 
[a , b] this commut(:l.tor ran be computed explicitly to be 
(2.5) [\la,b, D.Ju(n) = 
-u(a), 
1t(a- 1). 
tt(b ..L- 1), 
-u(b). 
n=c.L-l; 
n=a: 
n =b; 
n = b+l: 
0, otherwise. 
Combining these com;idcrations, we obtain 
Lemma 2.1. Let x E [a, bJ andy E A \ [a, b]. As.wme V(n) = V(n) for n E [a, b]. 
Then 
(2.6) 105 (£. k. x)l. 
kE { a- l ,a,b,b+ 1} 
112 H . KHUCErt 
If H. is just defined on €2 ([a, b]) , then the. equation becomes the more familiar 
IGA(E.x,y)l ~ II(HA - E)-111 L IGfa,bi(E.k. x)l. 
kE{".b} 
I should digress here a little hit, the computation <.lone to prove Lemma 2.1 should 
be fami liar from the context of continuum operators. ln the discrete case, these are 
usually unnecessary, since everything is as smooth as one wants. However, in this 
work (2.6) is essential, since it, allows us to avoid getting extra eigenvalues inside of 
gaps of opera.tors, when restricting to some space interval. Using Lemma 2.1, we 
will show the following theorem. 
Theorem 2.2. Let H be a Schrodinger ope·rator·, c E (0, l), N 2: 1 large enmtgh, 
& ~ IR. c < c < d < d such that 
[c, d] = /\._ U Ac U A+ 
with A_ = [c, cj, Ac = [c + 1, d- 1], and A+ = [d, d] disjoint intervals. Assttme 
for E E & that theTe exist [a±, b± J ~ J\± and Schrodinger operator's H ± such that 
b± - a.± 2: N . 
1 1 l 
(2.7) u(H±) n [E- JVl-o; , E + JVl-g] = 0, IIV- V± lle.x([a± ,b± ]) s 2JV1- f:' 
Also assume 
(2 .8) 
Then there exists !3 ~ & such that 
:. 
(i) 1!31 S e- N 4 • 
( ii) For x E { c, d}, y E Ar;, and E E £ \ !3, we have 
(2.9) IG[c.dl (E,x,y)i::; e-N ~ . 
In words this theorem says local gaps imply decay of the Green's function. (2. 7) 
makes precise what I mean by a local gap. 
For the proof, we will need the Combes- Thomas estimate, [3]: 
Lemma 2.3. Lett: > 0. Then theTe exists "' = ~~:(t:) > 0, Oo = o0 (e) > 0 snch that 
the following hold: 
ForEE IR, 8 E (0, 80 ), H : f 2 (A) ~ £2 (/\.) a SchrodingeT opemtoT with 
(2.10) u(H) n [E- t5, E + 6] = 0, 
and x, yE A with ix- Vi 2: (p\,., we have 
(2.11) fG(E,x, y) l S e-~~;<l j x-y l_ 
Let E E & and a±, b± as in ( 2. 7) . Define 
(2.12) H± = H± + X[a±,b±J(V - V±)· 
T hen by assumption IIH±- H±ll $ 2NL. and thus 
~ 1 1 
(2.13) u (H±) n [E- 2l'P-€, E + ZNl-&] = 0. 
By the Combes-T homas estimate, we can conclude for E E I , 
y = l a± ; b± J , x E {a±, b±} 
SCliRODJNGER OPERATORS WITH POTJ:~NTJAJ, V(n) = ,. -.,. cos(Z1rr> 1') 113 
that 
(2.14) IG--;Ja±.b±i(E . )I< - KN' ± ,x,y _ e 
as long as N is large enough. Here, a±, b± depend on E, but 1'<, dues not. 
Lemma 2.4. Define B by 
(2.15) !3 = 
Then !BI ~ e-N ~ . 
PROOF. The numbex of po:.;sible choices for a, b is bounded by (d - c)2 and also 
#a(Hia.bJ) ::; (d- c) for all possible choices of a , b. Hence 
IBI ::; 2(d- c)3e-N~ . 
The claim follows. D 
PROOF OF THEOREM 2.2. Let E E 1 \ B. Then by applying (2.6) with A = 
[c, d] and [a, b] = [Y-, Y+], we can conclude that 
IGic,dJ(E,x,y) l ~ eN l . (IGic,dl (E,x,y_)l + IG{c,dJ(E,x,y+)i). 
Now using the result obtained by the Combes-Thomas estimate, and (2.6) once 
more, we obtain 
IG~c.d] (E, .1:, Y) i S 8e2N~ · e-~<N'. 
Choosing N large enough, the result follows. D 
3. P roof of Theore m 1.2 
Introduce for a irrational. w E [0 , 1]. and 1\ > 0 the Almost-:Mathieu operator 
fh,.a.w : €2 (Z) , €2 (2), 
(3.1) 
H>..,c. ,wu(n) = u(n + 1) + u(n - 1) + 2A cos(27r(w + na))u(n). 
We will need the following fact about the spectrum of this operator. 
Theorem 3.1. Let 6 > 0. There exists a constant /'i, = f'<,(b') > 0 and >.a = Ao(6) > 0 
such that for A E (0, Ao) , w E [0, 1], and a satisf'l}ing 
(3.2) 2 cos(1ra) E [-2 + 6, - oj U (o, 2- o], 
we have 
(3.3) dist(± 2cos(7ra), a(H>.,a,w)) ~AI'<,. 
PROOF. See [1], {5], [10], [12). 
For k ~ 2, we introduce the disjoint sets 
(3.4) A% = [2k I 2k+l], 
(3.5) AJ; = [2k-t, 2k - 1], 
(3.6) 
and Ak = AJ; U A'k U At. We also define 
1 
(3.7) c = 6(2- p- 2')'). 
0 
114 H.KR0GE;R 
If (1.2) holds, t.hen f > 0. Also note ; + 3.: = 1- ~ E (0. I). 
Proposition 3 .2. Let 8 > 0, k 2: 1.:1 (6) and o: E [0. l ] with 
(3.8) 2 cos( no:) E [-2 + 8, -8] U [6, 2- 6]. 
Then there exist intervals J:.r. ~ !'it satisfying 
( i) #(I±) ;::: 2 . 2(-r+Z£ )k + l. 
(ii) IIV...,..p - V>.,cc.w lle""(J:::) = 0( 2k,Lp1 ) forsomew and A E [ 2-,L 11 , 2~<!+21 ] . 
PROOF. The arguments for f _ and It are similar, so I restrict myself to L . 
Define c = l 2~·-• t2k - l J and o:m = pmP- l . It is easy to check that 
d p-1- 0( 1 ) -d px - -2-. 
X :J: -p 
Using this. one concludes that for k large enough there exists rh such that 
lrh - cl <_ 2£~ ., lpmP- 1 a:l < 4 - - z<k- 1 H2-p) · 
Let e = f2h+le-)kl and define L = [rh- £. 1n + P.]. By some computations the claim 
follows. 0 
B:v Theorem 2.2 with N = 2(")'_._Z<lk, we now obtain 
Corollary 3. 3. Let o > 0 and k 2: k2 ( o). There exists a set Eo.k such that 
(i) IEa,k I ::; k\ . 
(ii) Let y E Ak, x E {2k-l,2k+l +2k-l), and 
(3.9) E E ([-2 + 8, - oJ U [8, 2- 8]) \ Ea,k· 
We have 
(3.10) 
\Ve now proceed to derive Theorem 1.2. The stralegy of proof is often called 
spectral averaging. See for example Section 12.3. in the book [17] by Simon for 
another implementation of this strategy. Fix some/. p satisfying (1.2). For 8 > 0, 
introduce 
(3.11) Es = n ( LJ Es k) . 
t~k2(J> k~e ' 
The Borel- Cantelli argument shows that Eo I = 0. In particular E = Ui22 £:} also 
has zero measure. Jt, is well-known tha1 there exists a unique probability measure 
p,f:J chat satisfies 
(3.12) 
for Im(z) > 0. This measure is known a the spectral measure. 
Lemma 3.4. There exists a set B such that IR \ Bl = 0 and for f3 E B , we have 
J.LP( f ) = 0. 
SCHROf)[:-.IGEH OPERATORS WITH POTENTIAL t (t~ ) = ,, -~ cos(:.!1rnP) 115 
PROOF. By Tlworem 11.8. in [17], we haw that J J1 8 dJ is th<' Lebesgue mea-
sure. Thus 
j 11 "~(£)dJ = O. 
Since Jti\£) ~ 0. Lhe claim follows. 0 
P ROOF OF THEORE~I 1.2. Let :3 E B. For p!7 almost every E E ( -2, 2) \ {0} 
there exisrs a generalized eigenfunction (see Lemma 3.1. in [20J), that is a nonzero 
solut.ion v of H~,p11 = Eu satisfying l7t(n) l :S n for n ~ 1 and u(O) = 0. If we show 
that u is in f2(Z+). we obtain Lhat 11 8 almost every E is an eigenwuue. thus that 
f.L8 is pure point. 
By construction of £. we o1n choose 8 > 0 and e ;::: 1 sucll that 
E E [-2 + 6. - <5] U fo, 2- 6J. 
and fork ~ f 
E E Eo.k· 
For x E.'\~= [21.. 2k+ 1!, we h~we 
u(x) = -G~~p(E, x. 2k)u(2" 1 -1)- G~.~(E, x, 2k+l + 2" 1)u(2k+l + 2k-l + 1) . 
Corollary 3.3. we obtain that lu(:r) l $ ~ for k large enough. This implies that 
u E f 2 (Z-t) finishing the proof. 0 
Acknowledgements 
I am thanking to Daniel t'eltschi and Robert Sims for their kind invitation to 
the second Arizona School of Analysis wit h Applications, where the idea for this 
project originated. Furthermore, I thank ~vlilivoje Lukic for useful discussions. I 
also like ro thank Gunter Stolz for useful discussions. which lead to the choice of 
(1. 7). 
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MATHEMATICS 253- :37, CALTRCH, PASAOE\I A. CA 91125 
E-mail address: helge@calt ech . e du 
URL: http://-www. i ts .ca l tec h. edu;-helge / 


Absolutely continuous spectrum for one-dimetMHmal Schrodmger operators u.-ith slowly decaymg potenttals: some optimal results.

Asymptotic behavicur of eigenfunctions for multiparticle SchrOdmger opemtors.

Jacobi Operators and Comple1.ely Integrable Nonlinea'r

Localization for Schrodinge1· operators with effectwe barrieTs,

Modified Prufe7' and BFGP transforms and the spectral analysiS of one-dtmenstonal Schrodmg('r operators.

Orthogonal polynomials with mcursion coefficients of gene1·alized bounded variatzon.

Posttwe Lyapovrwv expon<:nls for mo.~t energtes, Geometric a-spects of functional analysis,

Some Jacobt matrr.ces with decaying potential and dense point spectrum,

Spectral theory for slowly oscillating potenti.al.s. f. Jacobi matrice.~-

The a/n;uluLely cuntin·uuus spectrum of one-dimensional Sch.ri:idinger openztor·s with deca.ying potentials.

The one-dimensional Schrdmger operator unth unbounded potenttal: the pttN! 1JOint spectrum.

Trace Ideals and Thei1· Applications. Second edition,

Uber merkwiir-dige diskrete Etgenwerte.

http://authors.library.caltech.edu/31618/1/Kruger2011p18144.pdf

Schrödinger operators with potential V(n)=N^(-y) cos (2πn^p)

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