arXiv:1704.04603v1 [math.SP] 15 Apr 2017
SCH’NOL’S THEOREM AND THE SPECTRUM OF LONG RANGE OPERATORS RUI HAN Abstract. We extend some basic results known for finite range operators to long range operators with off-diagonal decay. Namely, we prove an analogy of Sch’nol’s theorem. We also establish the connection between the almost sure spectrum of long range random operators and the spectra of deterministic periodic operators.
1. Introduction Long range operators on l2 pZd q arise naturally from the discrete Laplacian on half-space Zd`1 ` after dimension reduction (see e.g. [9, 6, 7]). However, compared to Schr¨odinger operators or finite range operators, little has been studied. Our goal is to extend some basic results known for finite range operators to long range setting with off-diagonal decay. The first part of this note concerns a generalization of Sch’nol’s theorem to the case of a long range normal operator. The second part provides a connection between the almost sure spectrum of long range self-adjoint random operators and the spectra of deterministic periodic operators. We hope these could serve as ready-to-use tools in the future for people who study discrete operators. The classical Sch’nol’s theorem for Schr¨odinger operators is well-known, see [15, 2, 16, 4] for the continuum case, from which the discrete version could be derived. It asserts that any spectral measure gives full weight to the set of energies with generalized eigenfunctions, moreover, the spectrum is the closure of this set. It has many applications, for example, relating the spectrum to non-uniform hyperbolicity of the corresponding cocycle, as well as providing a priori estimate which turns out to be crucial in the proofs of the almost-localization and localization. So far there have been various generalizations of Sch’nol’s theorem (see e.g. [13, 14, 3, 5]). In this note, we generalize this result to the long range case. Let us consider long range normal operators with polynomially decaying off-diagonal terms. pHuqpnq “
(1.1)
ÿ
jPZd zt0u
anj upn ´ jq ` an0 upnq,
with ´r |am for any j, m P Zd , j | ď Cp1 ` }j}q
(1.2)
where }j} is the Euclidean norm of j and C ą 0 is a constant. We assume ÿ ÿ n´j n´j anj an´m (1.3) a´j am´j for any m, n P Zd , j´m “ jPZd
jPZd
to ensure H is a normal operator. We introduce generalized eigenfunctions. 1
2
RUI HAN
Definition 1.1. (ǫ-generalized eigenvalue/eigenfunction) An energy z of H is called an ǫ-generalized eigenvalue if there is a formal solution to the equation Hφz “ zφz , with φz p0q “ 1 and |φz pnq| ď d C 1 p1 ` }n}q 2 `ǫ for some constant C 1 . We denote the set of ǫ-generalized eigenvalues by Gǫ . The spectral measures of H are defined by µn,mř pBq “ pen , χB pHqem q, for any Borel sets ř B Ă C and m, n P Zd . We denote µn,n :“ µn . Let ´d´2ǫ {p nPZd p1 ` }n}q´d´2ǫ q. Clearly, any spectral measure µ “ nPZd λn µn , where λn “ p1 ` }n}q is absolutely continuous with respect to µ. Our main theorem is: Theorem 1.1. Let H be a normal operator defined as in (1.1) with r ą 2d. Then for any 0 ă ǫ ă r ´ 2d, we have the following: paq. Gǫ Ď σpHq, pbq. µpσpHqzGǫ q “ 0, pcq. Gǫ “ σpHq. The proof of part (a) relies on Lemma 2.1. We will discuss both in Section 2. The proofs of (b), (c) are standard. We will include them in the appendix for completeness. Switching our attention to the long range self-adjoint case, we are able to cover operators with unbounded potentials an0 . Let H be a self-adjoint operator defined as follows. ÿ pHuqpnq “ (1.4) anj upn ´ jq ` an0 upnq, jPZd zt0u
with (1.5)
´r |am for any j ‰ 0, m P Zd , and anj “ an´j j | ď Cp1 ` }j}q ´j .
Note that the polynomial decay condition (1.5) does not involve j “ 0, allowing us to take unbounded potentials into account. Following exactly the same line of the proof of Theorem 1.1, we have Sch’nol’s theorem for long range self-adjoint operators (with unbounded potentials). Theorem 1.2. Let H be a self-adjoint operator defined as in (1.4) with r ą 2d. Then for any 0 ă ǫ ă r ´ 2d, the conclusions of Theorem 1.1 hold. The localization property of long range self-adjoint operators with polynomially decaying offdiagonal terms has been studied in the random potential case, when the potentials an0 “ Vω pnq are i.i.d. random variables. Some known results are pure point spectrum in the large disorder and high energy regime when r ą d [1], purely singular spectrum for typical ω when d “ 1 and r ą 4 [17], pure point spectrum for typical ω when d “ 1 and r ą 8 (conjectured to be r ą 2) under some conditions on the density of distribution [8]. We hope Theorem 1.2 could serve as a step towards the proof of the conjecture, as well as establishing localization in more general setting. The second part of this note is devoted to extending some basic results which were previously known for random Schr¨odinger operators to long range self-adjoint cases. Ş Let Γ be Zd such that Γ and ´ΓŤform a partition of Zd zt0u in the sense that Γ p´Γq “ Ť a subset of H and Γ p´Γq “ Zd zt0u. Denote Γ0 “ t0u Γ. Let tγ piq uiPΓ be a sequence of compactly supported probability measures on C and γ p0q be a probability measure on R. Similar to Theorem 1.2 we do not make compact support assumption on γ p0q . This enables us to cover unbounded operators. Let dκ “ Ś Ś d piq piq piq Zd and Ω “ iPΓ0 psupp γ piq qZ “ tpω piq qiPΓ0 | ω piq “ pωj qjPZd , ωj P supppγ piq qu. For iPΓ0 pdγ q piq . any n P Zd , we define the translations T˜ n on Ω as T˜n pω piq q “ pT n ω piq q, where pT n ω piq qj “ ω j`n
SCH’NOL’S THEOREM AND THE SPECTRUM OF LONG RANGE OPERATORS
3
Consider long range self-adjoint operator Hpωpiq q on l2 pZd q: ÿ (1.6) pHpωpiq q uqpnq “ an´j pT˜n pω piq qqupjq, jPZd
pkq
p´kq
where ak ppω piq qq “ ω0 for k P Γ0 , and ak ppω piq qq “ ω´k γ pkq have shrinking supports, namely, (1.7)
for k P p´Γ0 q. We will further assume
supp γ pkq Ď BpCp1 ` }k}q´r q for k P Γ,
for some constant C ą 0 and r ą d2 , with BpM q being the ball in C centered at 0 with radius M . Following a standard argument, one can show that there exists a non-random Σ, such that σpHpωpiq q q “ Σ for κ´almost every pω piq q P Ω [12]. Ź We say a set Ω1 is a dense subset of Ω, if for any pω piq q P Ω, 0 ă ξ P R and any finite sets 1 Ă Γ0 Ź Ź Ź piq piq ω piq q P Ω1 such that |˜ ωj ´ ωj | ă ξ for any i P 1 and j P 2 . It is and 2 Ă Zd , there exists p˜ clear that any subset Ω0 Ď Ω with κpΩ0 q “ 1 is dense in Ω. The following result is similar to Theorem 3 [11]. Theorem 1.3. Let Ω1 be a dense subset of Ω, then ď σpHpωpiq q q Σ“ pω piq qPΩ1
This theorem has a direct corollary which implies the almost sure spectrum is determined by the piq piq spectra of periodic operators. We say pω piq q is p´periodic if ωj “ ωj`p for any i, j P Zd . Corollary 1.4.
Σ“
ď
pPZ`
ď
p´periodic pω piq q
σpHpωpiq q q
This can be viewed as an extension of Theorem 3.9 [10] for Schr¨odinger operators. We also point out that in [5], the authors established a similar result for the extended CMV matrices. If we focus on one-dimensional random Jacobi matrices, it turns out we could get a better result ˜ “ than Corollary 1.4. Let γ be a compactly supported measure on R. Let d˜ κ “ pdγqZ and Ω Z ˜ psupp γq . For any ω P Ω, we consider
(1.8)
pHω uqpnq “ apT n ωqupn ` 1q ` apT n´1 ωqupn ´ 1q ` bpT n ωqupnq
where apωq “ ω1 , bpωq “ ω0 and T ωn “ ωn`2 , which ensures that all apT i ωq and bpT j ωq are independent. We have Theorem 1.5. Let Hω be a random Jacobi matrice defined as in (1.8). We have ď Σ“ σpHtαj u q. 2´periodic tαj u
Ť Remark 1.1. Unlike the random Schr¨odinger case, for Jacobi matrices, Σ “ constanttαu σpHconstanttαu q Ť is in general not true. For example, take suppγ “ t0, 1u, then constanttαu σpHconstanttαu q “ r´1, 3s, however Σ “ r´2, 3s. We organize the note as follows: a key lemma and proof of Theorem 1.1 will be presented in Section 2, proofs of Theorems 1.3 and 1.5 will be given in Section 3, the proof of our key lemma will be included in Section 4.
4
RUI HAN
2. Key lemma and the proof of Theorem 1.1 In order to prove Theorem 1.1, we need the following lemma. q for some ǫ ą 0 and q P Z` . Let Lemma 2.1. Let H be defined as in (1.1) with r ą p2d ` ǫq q´1 z P Gǫ and φ be a corresponding generalized eigenfunction. Let " φpjq ´N ď }j} ď N, (2.1) φN pjq “ 0 otherwise.
and ΦN “ pH ´ zqφN . We have lim inf
(2.2)
LÑ8
}ΦLq }l2 “ 0. }φLq }l2
Proof of Theorem 1.1. We only prove part (a) here, the proofs of part (b) and (c) are standard, we include them in the appendix for reader’s convenience. q . Then Lemma Since r ą 2d ` ǫ, we can choose q P Z` large enough such that r ą p2d ` ǫq q´1 2.1 directly implies dist pz, σpHqq “ 0. l 3. Proofs of Theorems 1.3 and 1.5 3.1. Proof of Theorem 1.3. Let Ω0 be a shift invariant set with κpΩ0 q “ 1 such that σpHpωpiq q q “ Σ Ź Ź for every pω piq q P Ω0 . Then given any pω piq q P Ω, ξ ą 0, and any finite sets 1 Ă Γ0 , 2 Ă Zd , Ź Ź pjq pjq there exists p˜ ω piq q P Ω0 such that |ωn ´ ω ˜ n | ă ξ for any j P 1 and n P 2 . The “ Ě ” direction. Take pω piq q P Ω1 and E P σpHpωpiq q q, by Weyl’s criterion, for any L ą 0 there exists φpLq P l2 pZd q such that }pHpωpiq q ´ EqφpLq } ă L1 }φpLq }. First, we show such φpLq can be taken with compact support. This is standard if Hpωpiq q is a bounded operator, but since Hpωpiq q could be unbounded here, we will work out the detail. pLq
Let us consider a cut-off function φk pnq “ χ}j}ďk pnqφpLq pnq. We split Hpωpiq q into two parts 0 1 0 0 Hpωpiq q ` Hpω piq q , where Hpω piq q contains only the diagonal multiplication, namely pHpω piq q uqpnq “ a0 pT˜n pω piq qqupnq, and H 1 piq is the off-diagonal part. Clearly, H 1 piq is a bounded operator while pω
q
pω
0 Hpω piq q could be unbounded. Now let’s consider pLq
q
pLq
pLq
1 pLq 0 pLq pHpωpiq q ´ EqpφpLq ´ φk q “ pHpω ´ φk q ` Hpω ´ φk q. piq q ´ Eqpφ piq q pφ pLq
pLq
1 pLq 1 pLq We know }pHpω ´ φk q}l2 ď }pHpω ´ φk q}l2 Ñ 0 as k Ñ 8. Also since piq q ´ Eqpφ piq q ´ Eq} }pφ pLq
0 pLq 1 pLq 0 pLq }Hpω } ď }pHpω } ` L1 }φpLq } ă 8, we have }Hpω ´ φk q}l2 Ñ 0, as k Ñ 8. piq q φ piq q ´ Eqφ piq q pφ pLq
Thus }pHpωpiq q ´ EqpφpLq ´ φk q} Ñ 0 as k Ñ 8. Taking kL large enough, we may assume pLq
}pHpωpiq q ´ EqφkL } ă
(3.1)
2 pLq }φ }. L kL
pLq
For simplicity, we still denote φkL by φpLq and we assume it is supported on a compact set t}n} ď Ku. 2r´d Now, we take an integer m such that pm ´ 1qŹ ą L2 K 2d´2r , this choice is Ş Ź large enough guaranteed by r ą d{2. Let 1 “ Γ0 t}j} ď pm ` 1qKu and 2 “ t}n} ď mKu. There exists
SCH’NOL’S THEOREM AND THE SPECTRUM OF LONG RANGE OPERATORS pjq
pjq
p˜ ω piq q P Ω0 such that |ωn ´ ω ˜n | ă
1 LK d md{2
for any j P
Ź
and n P
1
Ź
2.
5
Now let us consider
}pHω˜ piq ´ Hωpiq qφpLq }2l2 ÿ ÿ “ | pan´j pT˜ n p˜ ω piq qq ´ an´j pT˜ n pω piq qqqφpLq pjq|2 nPZd }j}ďK
ď}φpLq }2l2 “}φpLq }2l2 ď}φpLq }2l2
ÿ
nPZd
ÿ
}j}ďK
ÿ
}j}ďK
ÿ
}j}ďK
|an´j pT˜n p˜ ω piq qq ´ an´j pT˜n pω piq qq|2
p
ÿ
p
ÿ
}n}ąmK
`
ÿ
}n}ďmK,n´jPΓ0
`
´2r
}n}ąmK
2p1 ` }n ´ j}q
ÿ
q|an´j pT˜ n p˜ ω piq qq ´ an´j pT˜ n pω piq qq|2
}n}ďmK,n´jP´Γ
`
ÿ
}n}ďmK,n´jPΓ0
|˜ ωnpn´jq ´ ωnpn´jq |2
` ďC}φpLq }2l2 p
ÿ
ÿ
}n}ěmK }j}ďK
}n ´ j}´2r ` K 2d md
ďC}φpLq }2l2 ppm ´ 1qd´2r K 2d´2r ` ď
1 q L2 K 2d md
ÿ
}n}ďmK,n´jP´Γ
|˜ ωnpj´nq ´ ωnpj´nq |2 q
1 q L2
C pLq 2 }φ }l2 L2
Thus }pHω˜ piq ´ EqφpLq } ď
? C`1 pLq }. L }φ
Taking L Ñ 8, we get E P σpHp˜ωpiq q q “ Σ.
The “ Ď ” direction. Note that since Ω1 is dense in Ω, the proof follows from that of “ Ě ” by interchanging the roles of Ω0 and Ω1 . l
3.2. Proof of Theorem 1.5. Let Hω “ Hω1 `Hω2 , where pHω1 uqpnq “ apT n ωqupn`1q`apT n´1ωqupn´ 1q and pHω2 uqpnq “ bpT n ωqupnq. Let Σ1 “ σpHω1 q a.s., Σ2 “ σpHω2 q a.s. and M “ supt|α|, α P supp γu. Clearly, (3.2)
ď
2´periodic tαj u
σpHtαj u q “ r´2M, 2M s ` supp γ.
The “ Ď ” direction. }Hω1 } ď 2M , we have Σ1 Ă r´2M, 2M s. This immediately implies Σ Ď Σ1 ` Σ2 Ď r´2M, 2M s ` suppγ. The “ Ě ” direction. Let Ω0 be the full κ ˜ -measure set so that σpHω q “ Σ for any ω P Ω0 . Since M “ supt|α|, α P supp γu, for any β P p´2M, 2M q, there exists a set F with γpF q ą 0 such that 1 there exists a sequence pαn , ξn q P Ω0 such that 2 |β| ă inft|α|, α P F u. Then taking any ξ P supp γ, Ť 1 |β| ă α and ξ Ñ ξ as n Ñ 8. Clearly, β ` ξ P n n n σpHt...αn ξn αn ξn αn ξn ...u q “ Σ. Thus we have 2 p´2M, 2M q ` supp γ Ď Σ, which implies the desired result after taking closure. l
6
RUI HAN
4. Proof of Lemma 2.1 ř j an´j φpjq. Clearly, ´ z by aj0 . Then ΦN pnq “
aj0
For simplicity, we denote
}j}ďN
ÿ
nPZd
|ΦLq pnq|2 “ “
ÿ
ÿ
|
ajn´j φpjq|2
nPZd }j}ďLq
ÿ
ÿ
|
}n}ąLq }j}ďLq ,}j´n}ąLq´1
` ď2
ÿ
ajn´j φpjq `
ÿ
|
ajn´j φpjq
}n}ďLq }j}ąLq ,}j´n}ąLq´1
ÿ
¨
}n}ąLq
`2
˝|
ÿ
ÿ
}j}ďLq ,}j´n}ąLq´1
¨
˝|
}n}ďLq
ÿ
}j}ďLq ,}j´n}ďLq´1
`
}j}ąLq ,}j´n}ąLq´1
ÿ
ajn´j φpjq|2
}j}ąLq ,}j´n}ďLq´1
ajn´j φpjq|2 ` |
ÿ
ajn´j φpjq|2
ÿ
}j}ďLq ,}j´n}ďLq´1
ajn´j φpjq|2 ` |
ÿ
ajn´j φpjq|2 ‚
}j}ąLq ,}j´n}ďLq´1
:“2pΣ1 ` Σ2 ` Σ3 ` Σ4 q.
(4.1)
˛
ajn´j φpjq|2 ‚
Estimate of Σ1 . ÿ
}n}ąLq
|
ď}φLq }2l2 ď}φLq }2l2
ÿ
ajn´j φpjq|2
}j}ăLq ,}j´n}ąLq´1
ÿ
}n}ąLq
ÿ
}j}ăLq ,}j´n}ąLq´1
|ajn´j |2
ÿ
ÿ
p1 ` }n ´ j}q´2r
}n}ąLq
}j}ăLq ,}j´n}ąLq´1
ďC}φLq }2l2 Lqp2d´2rq`2r´1 .
(4.2)
The proof of (4.2) will be included in the appendix. Estimate of Σ2 . ÿ
}n}ąLq
ď
ÿ
ajn´j φpjq|2
}j}ďLq ,}j´n}ďLq´1
ÿ
Lq `Lq´1 ě}n}ąLq
¨ ˝
ÿ
}m}ďLq´1
˛2
χ}n´m}ďLq |an´m φpn ´ mq|‚ m
˛ 12 ,2 / . ´r ˝ 2‚ ď p1 ` }m}q χ}n´m}ďLq |φpn ´ mq| / ’ %}m}ďLq´1 Lq `Lq´1 ě}n}ąLq ÿ p1 ` }m}q´r q2 . ďp}φpL`1qq }2l2 ´ }φpL´1qq }2l2 qp $ ’ &
(4.3)
|
ÿ
¨
ÿ
mPZd
˛
SCH’NOL’S THEOREM AND THE SPECTRUM OF LONG RANGE OPERATORS
7
Estimate of Σ3 . ÿ
|
ÿ
ajn´j φpjq|2
ÿ
p
ÿ
|ajn´j φpjq|q2
}n}ďLq }j}ąLq ,}j´n}ąLq´1
ď
ď
ď
}n}ďLq
}j}ąLq ,}j´n}ąLq´1
$ ’ & ÿ
¨
|φpjq| ˝
’ %}j}ąLq $ ’ & ÿ
ÿ
}n}ďLq
χ}j´n}ąLq´1 ¨
d
’ %Lq `Lq´1 ě}j}ąLq
}j} 2 `ǫ ˝
ÿ
}n}ďLq
˛ 21 ,2 / . j 2‚ |an´j | / -
˛ 21
χ}j´n}ąLq´1 p1 ` }n ´ j}q´2r ‚
˛ 21 ,2 / . d `ǫ ˝ ´2r ‚ 2 p1 ` }n ´ j}q ` }j} / }n}ďLq }j}ąLq `Lq´1 ¨
ÿ
˛ 12 ,2 / . d qp2d´rq`r`ǫq `ǫ ˝ ´2r ‚ 2 ď CL ` }j} p1 ` }n ´ j}q ’ / % }j}ąLq `Lq´1 }n}ďLq $ ’ &
(4.4)
ÿ
¨
ÿ
ÿ
1
ďpCLqp2d´rq`r`ǫq q 2 .
The proof of (4.4) will be included in the appendix. Estimate of Σ4 . ÿ
ÿ
|
ajn´j φpjq|2
}n}ďLq }j}ąLq ,}j´n}ďLq´1
ď
ÿ
}n}ďLq
˝
ÿ
}m}ďLq´1
˛2
χ}n´m}ąLq |an´m φpn ´ mq|‚ m
˛ 12 ,2 / . ´r ˝ 2‚ p1 ` }m}q ď χ}n´m}ąLq |φpn ´ mq| / ’ %}m}ďLq´1 }n}ďLq ÿ ďp}φpL`1qq }2l2 ´ }φpL´1qq }2l2 qp p1 ` }m}q´r q2 . $ ’ &
(4.5)
¨
ÿ
¨
ÿ
mPZd
Eventually combining the estimates (4.2), (4.3), (4.4), (4.5) with our choice of r, we conclude that (4.6) Then lim inf LÑ8
lim inf lÑ8
}ΦLq }2l2 }φLq }22 l
}φpL`1qq }2l2 ´ }φpL´1qq }2l2 }ΦLq }2l2 ď C lim inf as L Ñ 8. LÑ8 }φLq }2l2 }φLq }2l2 ą κ ą 0 would imply that }φpL`1qq }2l2 ě p1 `
to exponential growth of φ, contradiction.
κ 2 q C q}φpL´1q }l2 ,
which leads l
8
RUI HAN
Appendix The proofs of (4.2) and (4.4) are in the same spirit. Proof of (4.2). ÿ }n}ąLq
“
ÿ
}j}ăLq ,}j´n}ąLq´1
ÿ
ďCLqp2d´2rq`2r´1 ` CLqd ďCL
ÿ
p1 ` }j ´ n}q´2r `
ż8
xd´1 dx px ´ Lq q2r
}j}ăLq ,}j´n}ąLq´1
Lq `Lq´1 ě}n}ąLq
qp2d´2rq`2r´1
p1 ` }j ´ n}q´2r
Lq `Lq´1
.
ÿ
}n}ąLq `Lq´1
ÿ
}j}ăLq
p1 ` }j ´ n}q´2r
l Proof of (4.4). d
ÿ
}j}ąLq `Lq´1
ďL „L
dq 2
ÿ
p1 ` }j}q 2 `ǫ ˝
}j}ąLq `Lq´1 dq 2
ďC L
¨
ż8
Lq `Lq´1
qp2d´rq`r`ǫq
p1 ` }j}q
ÿ
}n}ăLq
d 2 `ǫ
˛ 21
p1 ` }n ´ j}q´2r ‚
p}j} ´ Lq q´r
3d
x 2 `ǫ´1 dx px ´ Lq qr . l
Part (b). The proof is standard, we present it here for readers’ convenience and completeness. dµn,m Since µn,m ! µ, there exists a Fn,m pEq “ dµ pEq defined for µ-a.e. E. We will show that for any fixed n, for µ-a.e. E, Fn,m pEq is an ǫ-generalized eigenfunction, namely (1) Fn,m pEq is a solution to Hu “ Eu, d (2) |Fn,m pEq| ď p1 ` }m}q 2 `ǫ . Proof. (1).pH ´ EqFn,m pEq “ 0, µ-a.e. E, is equivalent to ż ż pHFn,m qpEqgpEq dµpEq “ σpHq
σpHq
EFn,m pEqgpEq dµpEq
for any compactly supported continuous function g on σpHq. For simplicity, we denote aj0 ´ E by aj0 . Then for any such g, gpHq is a bounded operator, moreover, ş EFn,m pEqgpEq dµpEq σpHq “ pen , HgpHqem q “ pgpHq˚ en , He ř mq m ˚ “ pgpHq e , ř n mkPZd ak´m ek q “ pe kPZd ak´m gpHqek q ş n, ř “ σpHq kPZd am k´m Fn,k pEqgpEq dµpEq
SCH’NOL’S THEOREM AND THE SPECTRUM OF LONG RANGE OPERATORS
“
|
ż
ş
σpHq
pHFn,m qpEqgpEq dµpEq.
(2). For any Borel set B Ă C,
B
9
Fn,m pEq dµpEq| “ |µn,m pBq| ď d
a d d µpBq ď CµpBqp1 ` }n}q 2 `ǫ p1 ` }m}q 2 `ǫ µn pBqµm pBq| ď ? λn λm
Thus |Fn,m pEq| ď Cp1 ` }m}q 2 `ǫ for some constant C and µ-a.e. E.
l
Part (c). Part (c) follows directly from (a), (b) and the fact that spectrum of H is the smallest closed set which supports every spectral measure of H. l Acknowledgement I would like to thank Svetlana Jitomirskaya for suggesting this problem and for many useful discussions. This research was partially supported by the NSF DMS-1401204. References 1. Aizenman, M. and Molchanov, S., 1993. Localization at large disorder and at extreme energies: An elementary derivations. Communications in Mathematical Physics, 157(2), pp.245-278. 2. Berezanski˘ı, I.M., 1968. Expansions in eigenfunctions of selfadjoint operators (Vol. 17). American Mathematical Soc. 3. Boutet de Monvel, A., Lenz, D. and Stollmann, P., 2009. Schnols theorem for strongly local forms. Israel Journal of Mathematics, 173(1), pp.189-211. 4. Cycon, H.L., Froese, R.G., Kirsch, W. and Simon, B., 2009. Schrdinger operators: With application to quantum mechanics and global geometry. Springer. 5. Damanik, D., Fillman, J., Lukic, M. and Yessen, W., 2015. Uniform Hyperbolicity for Szeg˝ o Cocycles and Applications to Random CMV Matrices and the Ising Model. International Mathematics Research Notices, 2015(16), pp.7110-7129. 6. Grinshpun V.: Dopovidi Akademii Nauk Ukrainy (Proc. Acad. Sci. Ukraine) 8 (1992), 18; 9 (1993) 7. Jak˘si´ c, V. and Molchanov, S., 1998. On the surface spectrum in dimension two. 8. Jak˘si´ c, V. and Molchanov, S., 1999. Localization for one dimensional long range random hamiltonians. Reviews in Mathematical Physics, 11(01), pp.103-135. 9. Jak˘si´ c, V., Molchanov, S. and Pastur, L., 1998. On the propagation properties of surface waves. In Wave propagation in complex media (pp. 143-154). Springer New York. 10. Kirsch, W., 2007. An invitation to random Schrdinger operators. Panor. Synth` eses 25, 1119, Soc. Math. France, Paris, 2008. 11. Kirsch, W. and Martinelli, F., 1982. On the spectrum of Schr¨ odinger operators with a random potential. Communications in Mathematical Physics, 85(3), pp.329-350. 12. Kirsch, W. and Martinelli, F., 1982. On the ergodic properties of the spectrum of general random operators. J. reine angew. Math, 334(141-156), pp.102-112. 13. Klein, A., Koines, A. and Seifert, M., 2002. Generalized eigenfunctions for waves in inhomogeneous media. Journal of Functional Analysis, 190(1), pp.255-291. 14. Kuchment, P., 2005. Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs. Journal of Physics A: Mathematical and General, 38(22), p.4887. 15. Sch’Nol, I., 1957. On the behavior of the Schr¨ odinger equation. Mat. Sb, 42, pp.273-286. (in Russian) 16. Simon, B., 1981. Spectrum and continuum eigenfunctions of Schr¨ odinger operators. Journal of Functional Analysis, 42(3), pp.347-355. 17. Simon, B. and Spencer, T., 1989. Trace class perturbations and the absence of absolutely continuous spectra. Communications in Mathematical Physics, 125(1), pp.113-125. Department of Mathematics, University of California, Irvine E-mail address:
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