Reflectionless discrete Schrödinger operators are spectrally atypical
aa r X i v : . [ m a t h . SP ] D ec REFLECTIONLESS DISCRETE SCHR ¨ODINGER OPERATORS ARESPECTRALLY ATYPICAL
TOM VANDENBOOM
Abstract.
We prove that, if an isospectral torus contains a discrete Schr¨odinger operator withnonconstant potential, the shift dynamics on that torus cannot be minimal. Consequently, wespecify a generic sense in which finite unions of nondegenerate closed intervals having capacity oneare not the spectrum of any reflectionless discrete Schr¨odinger operator. We also show that the onlyreflectionless discrete Schr¨odinger operators having zero, one, or two spectral gaps are periodic. Introduction and Main Results
A discrete Schr¨odinger operator (DSO) is a self-adjoint linear operator H on the Hilbert spaceof square-summable sequences ℓ ( Z ) which acts entrywise via( Hu ) n = u n +1 + u n − + V ( n ) u n , u ∈ ℓ ( Z ) , (1.1)where V is a bounded real potential funtion V : Z → R . A DSO H V is called almost-periodicif its potential function V : Z → R is almost-periodic; that is, if every sequence of translates V k = V ( · + n k ) has a uniformly convergent subsequence. A prominent example of an almost-periodic DSO is the Almost-Mathieu Operator given by H V with V λ,θ,α ( n ) = 2 λ cos(2 π ( θ + nα ))for α ∈ [0 , \ Q . The DSO is most naturally identified as a discretized model of the differentialSchr¨odinger operator, L = − ∆ + V . The Schr¨odinger operator and DSO tend to share many of thesame spectral characteristics, and as such the question of identifying the spectral characteristics of aDSO with a fixed almost-periodic potential V is thoroughly studied and reasonably well-understood.On the other hand, one can likewise ask the dual question: if one fixes certain spectral character-istics, what, if any, bounded self-adjoint operators demonstrate them? In this context, the Jacobioperator J = J ( a, b ) ( J u ) n = a n u n − + b n u n + a n +1 u n +1 , u ∈ ℓ ( Z )(1.2)arises in a very natural way. Namely, when one fixes a compactly supported probability measure µ on R , multiplication by the independent parameter x with respect to the basis of orthogonalpolynomials in L ( dµ ) is unitarily equivalent to operation by a half-line Jacobi operator J on ℓ ( N ). Under the assumption of absolute continuity and almost-periodicity, a whole-line Jacobioperator can be completely reconstructed from any of its half-line restrictions. In fact, for a finiteunion of nondegenerate disjoint closed intervals E ⊂ R (henceforth, a “finite-gap compact”), theset J ( E ) of all absolutely continuous, almost-periodic, whole-line Jacobi operators with spectrum E is naturally homeomorphic to a finite-dimensional torus [8, 24]. In this case we call J ( E ) theisospectral torus for E .For finite-gap compacts E , certain potential-theoretic properties are directly related to spectralproperties of elements of J ( E ). If M ( E ) denotes the set of Borel probability measures supportedon E , define the logarithmic capacity of E bycap( E ) := sup µ ∈M ( E ) exp (cid:18) − Z E Z E log (cid:0) | x − y | − (cid:1) dµ ( x ) dµ ( y ) (cid:19) . T.V. was supported in part by NSF grants DMS–1361625 and DMS–1148609.
The logarithmic capacity of a finite-gap compact E can be determined as the limit of the geometricmeans of the off-diagonal sequences of Jacobi operators in the isospectral torus [23, 24]:cap( E ) = lim n →∞ ( a a · · · a n ) /n , J ( a, b ) ∈ J ( E ) . (1.3)We denote by K nα the set of n -gap compact subsets of R of capacity α : K nα := { E ⊂ R : E n -gap compact , cap( E ) = α } . Similarly, we denote K n the set of all n -gap real compacts, and by K α the set of finite-gap compactsubsets of R of capacity α : K n = [ α> K nα , K α = ∞ [ n =1 K nα . Note that, by (1.3), if E = σ ( H V ) for some DSO H V , then E / ∈ K α for any α = 1.Comparing equations (1.1), (1.2), and (1.3) immediately reveals that DSOs are a special classof Jacobi operator with spectrum E of unit capacity cap( E ) = 1. Thus, an absolutely continuous,almost-periodic DSO with finite-gap spectrum E is a member of the isospectral torus J ( E ). How-ever, the isospectral torus is defined in terms of the much broader class of Jacobi operators. In thiscontext, the following question is quite natural: Question.
Consider E ∈ K . Does the isospectral torus J ( E ) contain a DSO?Our titular result says that the answer to this question is generically “no”: Theorem 1.1.
The set of finite-gap compacts E ∈ K for which there exists a DSO H ∈ J ( E ) ismeager and has measure zero. By a meager set, we mean a countable union of nowhere dense sets in the relevant topology.Here, the topology and measure on K is induced by a topology on each K nα ; namely, any E ∈ K n can be identified uniquely by its ordered 2( n + 1) distinct gap edges. The topology and measureare then induced by inclusion of the corresponding vector in R n +1) . We clarify precisely what wemean here in Section 2. Remark.
The restriction to E ∈ K is necessary here to make the theorem statement nontrivial;indeed, we will see later that K n is codimension 1 inside K n . However, if we were to extend ourdefinition of DSO to include any Jacobi operator with constant off-diagonal sequence a , all of ourtheorems would remain true without any restriction on the capacity of E .We also would be remiss not to mention the recent work [13], wherein the methods of canonicalsystems are used to prove a result on the sparsity of m -functions of DSOs amongst those for Jacobioperators.On the other hand, it is known in general that, if V ( · ) = V ( · + p ) is periodic for some p ∈ Z , theassociated DSO H V is purely absolutely continuous with spectrum consisting of a union of at most p closed intervals. Our second result says that, at least when the number of gaps in E is small, asort of converse holds: Theorem 1.2.
Suppose there exists an almost-periodic DSO H with σ ( H ) = E with n = 0 , , or open spectral gaps such that the essential support of the a.c. spectrum Σ ac ( H ) agrees with E up tosets of Lebesgue measure zero; that is, | E \ Σ ac ( H ) | = 0 . Then H is periodic. That this is true for the case with n = 0 gaps has been explored in-depth and is quite well-known [2, 5, 12, 14]. Similarly, while we have never seen the case n = 1 explicitly addressed, it hasprobably been observed before by virtue of trace formulas. The real novelty of this theorem is thatthe same holds when n = 2, which we will see is the first truly nontrivial case. Of course, this raisesthe following natural follow-up question: are there any aperiodic absolutely continuous DSOs with EFLECTIONLESS DISCRETE SCHR ¨ODINGER OPERATORS ARE SPECTRALLY ATYPICAL 3 finite-gap spectrum? While Theorem 1.2 is evidence that the answer could be negative, we arereluctant to make a conjecture one way or another. Theorem 1.2 is thoroughly algebro-geometricin nature, and it may be that the result is a consequence of the low dimension of the correspondingisospectral torus rather than an illustration of a universal principle.Theorems 1.1 and 1.2 are more precisely viewed as corollaries of broader statements whichcan be made in the context of reflectionless Jacobi operators, which we now define. Denote by r ( n, z, J ) = h δ n , ( J − z ) − δ n i the Green’s function for J . For each n ∈ Z the Green’s function r ( n, z, J ) has almost-everywhere defined radial limits on σ ( J ); that is, for almost-every x ∈ σ ( J ),lim ε ↓ r ( n, x + iε, J ) exists. When this limit exists, we denote it by r ( n, x + i , J ).Consider a positive-measure compact E ⊂ R . We say that the Jacobi operator J is reflectionlesson E when Re( r ( n, x + i , J )) = 0 for (Lebesgue) almost-every x ∈ E for all n ∈ Z . A Jacobioperator is called reflectionless if it is reflectionless on its spectrum σ ( J ). There is a subtle andintimate relationship between the reflectionless property, almost periodicity, and the geometry andabsolute continuity of the spectrum [1, 3, 9, 10, 11, 15, 20, 24, 28]. From a heuristic perspective, re-flectionless Jacobi operators are those whose dynamics lack reflection; that is to say, those operatorswhose states coming completely from −∞ in backward time proceed completely to + ∞ in forwardtime [3]. Periodic Jacobi operators are perhaps the initial example of reflectionless operators, but,more generally, almost-periodic Jacobi operators are reflectionless on the essential support of theira.c. spectrum [15, 20]. A partial converse likewise holds under certain geometric restrictions onthe spectrum [24]; however, the converse may fail in spectacular fashion without these restrictions[1, 28].Fix a positive-measure compact E ⊂ R , and define the isospectral torus for E as J ( E ) := { J : σ ( J ) ⊂ E and J reflectionless on E } . This is a compact, path-connected subset of the space of uniformly bounded Jacobi operators whenendowed with the topology of pointwise convergence [16, 21]. The action S of conjugation by theleft-shift S : δ n δ n +1 preserves both the spectrum and the reflectionless condition, and thus( J ( E ) , S ) is a discrete-time dynamical system. Theorem 1.1 is a straightforward consequence ofthe following dynamical characterization of those isospectral tori containing a DSO: Theorem 1.3.
Fix a positive-measure compact E ⊂ R , and suppose there exists a DSO H V ∈J ( E ) . Then either V is a constant potential V = C , or the dynamical system ( J ( E ) , S ) is notminimal. Remark.
Note that Theorem 1.3 makes no assumption on E beyond positive-measure and com-pact; in particular, the result applies to sets with infinitely many spectral gaps. Furthermore,Theorem 1.3 does not merely hold for the topology of pointwise convergence on J ( E ); it holds forany topology under which the limit of a DSO remains a DSO.The dichotomy in Theorem 1.3 raises a follow-up question of its own: under the assumption thatthere exists a DSO H ∈ J ( E ), to what extent can the shift dynamics fail to be minimal? Whilewe do not have a complete answer to this question, we can report the following progress: Theorem 1.4.
Suppose E ∈ K n for n = 0 , , or . If there exists a DSO H ∈ J ( E ) , the dynamicalsystem ( J ( E ) , S ) is periodic. Of course, without the assumptions of finite-gap and reflectionless, the conclusion of Theorem1.4 fails completely: one can construct limit-periodic DSOs with pure point spectrum and finitelymany gaps [18], and the subcritical AMO and Pastur-Tkachenko class of DSOs are reflectionlesswith homogeneous infinite-gap spectra [4, 7]. But it would be very interesting to construct anexample of a reflectionless Jacobi operator with periodic off-diagonal, aperiodic main diagonal, andfinite-gap spectrum.
T. VANDENBOOM
This paper will be structured as follows: Section 2 contains a brief crash-course on the requisitebackground material. Section 3 contains the majority of the proofs of the above theorems, save fora proposition demonstrating the genericity with which Theorem 1.1 holds. This proposition (whichmay be well-known, although we could not find it in the literature) is proven in Section 4. Section5 can be viewed as an appendix; it includes some basic facts about the Toda hierarchy that weimplement in our proofs.
Acknowledgments.
We would like to thank Daniel Bernazzani and Vitalii Gerbuz for a numberof helpful conversations, and offer a special thanks to Jake Fillman for pointing out the applicationof these methods to off-diagonal Jacobi operators.2.
Background
A Jacobi operator is a linear operator J on sequences u ∈ C Z parametrized by a pair of bounded,real-valued sequences a, b ∈ ℓ ∞ ( Z ). The operator is defined termwise by( J u ) n = a n u n − + b n u n + a n +1 u n +1 . In this paper, we will deal only with those Jacobi operators whose off-diagonal sequences a haveentries which are positive and bounded uniformly away from 0. We make no notational distinctionbetween J and its restriction to ℓ ( Z ).The resolvent set ρ ( J ) of a Jacobi operator J is defined as the set of complex energies z forwhich the operator ( J − zI ) − exists and is bounded on ℓ ( Z ). The complement of the resolventset in the plane is called the spectrum of J , denoted σ ( J ) = C \ ρ ( J ). Because J is boundedand self-adjoint, σ ( J ) is a compact subset of R . The absolutely continuous spectrum σ ac ( J ) is thecommon topological support of the absolutely continuous parts of all spectral measures for J .Define the Green’s function r ( n, z, J ) := h δ n , ( J − zI ) − δ n i , z ∈ ρ ( J ) , n ∈ Z . For each n ∈ Z , the Green’s function r ( n, z, J ) is Herglotz; that is, a function which is analytic fromthe upper half plane to itself. As such, r ( n, z, J ) has Lebesgue almost-everywhere defined radiallimits on σ ( J ): for almost-every x ∈ σ ( J ), lim ε ↓ r ( n, x + iε, J ) =: r ( n, x + i , J ) exists in C ∪ {∞} .We say J is reflectionless on a positive-measure set E whenRe( r ( n, x + i , J )) = 0for (Lebesgue) almost-every x ∈ E for every n ∈ Z . If J is reflectionless on its spectrum σ ( J ), wesimply call J reflectionless.Recall also that we define the isospectral torus for E as J ( E ) := { J : σ ( J ) ⊂ E and J reflectionless on E } . We endow J ( E ) with the topology of pointwise convergence of the parametrizing sequences ( a, b ).Conjugation by the left-shift S : δ n δ n +1 preserves the reflectionless condition and the spectrum:if J ∈ J ( E ), so is S J := SJ S − .Almost-periodic Jacobi operators J are reflectionless on the essential support Σ ac ( J ) of their a.c.spectrum: Theorem 2.1 (Theorem 1.4, [20]) . An almost-periodic Jacobi operator J is reflectionless on Σ ac ( J ) . Under certain geometric restrictions on the spectrum, one can likewise conclude almost-periodicityfrom reflectionlessness:
Theorem 2.2 (Theorem C, [24]) . Suppose E ∈ K n . Then the dynamical system ( J ( E ) , S ) isconjugate to (( R / Z ) n , T ω ) , where T ω ( α ) = α + ω , and ω is given by ω j = ω E (( −∞ , E − j ]) . EFLECTIONLESS DISCRETE SCHR ¨ODINGER OPERATORS ARE SPECTRALLY ATYPICAL 5
Furthermore, every J ∈ J ( E ) is almost-periodic and purely absolutely continuous. The values ω j are called the harmonic frequencies of E . Here, ω E is the harmonic measure forthe domain Ω = ˆ C \ E with pole at infinity.Consider the set M ( E ) of Borel probability measures supported on E . For µ ∈ M ( E ), defineits potential energy E ( µ ) as E ( µ ) = Z E Z E log (cid:0) | x − y | − (cid:1) dµ ( x ) dµ ( y )If there exists µ ∈ M ( E ) for which E ( µ ) < ∞ , then E ( µ ) has a unique minimizer called theequilibrium measure dρ E . When E is finite-gap, the harmonic measure ω E ( dx ) agrees with theequilibrium measure [19, Theorem 4.3.14]. In this sense, we define the logarithmic capacity of E cap( E ) := sup µ ∈M ( E ) exp( −E ( µ ))= exp( −E ( ω E ( dx ))) . The capacity scales linearly; that is, if E is a compact set, α >
0, and αE is the α -rescaling of E ,one has [19, Theorem 5.1.2] cap( αE ) = α cap( E ) , α > . (2.1)The genericity of our theorems holds in the following sense: fix a finite union of nondegenerateclosed intervals E ∈ K n , and write it as a closed interval with a finite union of maximal open gapsremoved: E = [ E +0 , E − ] \ n [ j =1 ( E − j , E + j ) . (2.2)To avoid degenerate spectral bands, we assume E + j < E − j +1 for each 0 ≤ j ≤ n , with the convention E − n +1 := E − . For n ∈ N , denote T n = { ( x , · · · , x n ) ∈ R n : x < x < · · · < x n } . Any compact E ∈ K n written as (2.2) can be parametrized by a vector in T n +1) by reading offendpoints from left to right: E ( E +0 , E − , E +1 , E − , · · · , E + n , E − ) ⊤ . This parametrization naturally induces a topology and measure on K n via the subspace topologyof T n +1) ⊂ R n +1) .The topology on K n induces a projective topology on K nα in the following way. By (2.1), wehave that, with the identification K n with T n +1) above, we can topologize K nα with the subspacetopology and pushforward measure from the projection from K n ∼ = K nα × R + (see Proposition 4.4below). With these conventions, we have the following: Proposition 2.3.
For each positive integer n and real α > , the set of E ∈ K nα with rationallydependent harmonic frequencies is meager and has Lebesgue measure zero. Though we could not explicitly find a result of this nature in the literature, this result could beconcluded as a consequence of previous work (e.g., [26]). We offer a self-contained proof of thisproposition in Section 4.As a consequence of Theorem 2.2 and Proposition 2.3, in the finite-gap regime one genericallyhas minimality of the shift on the isospectral torus. In particular, it is typically the case that thehull of a given almost-periodic J ∈ J ( E ) is the entire isospectral torus. To prove our theorems, weutilize this fact in conjunction with a family of integral curves on J ( E ) called the Toda Hierarchy. T. VANDENBOOM
Consider a bounded linear operator A on ℓ ( Z ). Denote by A ± the restrictions of A to ℓ ( Z ± ) ֒ → ℓ ( Z ), where the inclusion map is given by assigning zeros to the left- or right-half line. Fix apolynomial P of degree n + 1 ≥
1. The n th Toda flow (for P ) is the integral curve J ( t ) of Jacobioperators satisfying the Lax pair ∂ t J = [ P ( J ) + − P ( J ) − , J ] . (2.3)There exist unique solutions to (2.3) for any bounded Jacobi initial condition J [25, Theorem 12.6].When there exists a monic polynomial P so that[ P ( J ) + − P ( J ) − , J ] = 0 . (2.4)we say J is stationary for P .For particular choices of polynomial P , the Toda flow induces a system of differential equationson the parametrizing sequences a, b ∈ ℓ ∞ . The critical facts about the Toda flow that we employare summarized in Proposition 2.4.
For any non-constant polynomial P :(1) [16, Corollary 1.3] Suppose J ( t ) is the unique solution to (2.3) with J (0) = J ∈ J ( E ) ,where E = σ ( J ) . Then J ( t ) ∈ J ( E ) for all t ∈ R .(2) [25, Theorem 12.8] The stationary solutions ∂ t J = 0 of (2.3) are finite-gap reflectionlessJacobi operators.(3) [25, Corollary 12.10] If J is stationary for P , then the spectrum of J has at most deg( P ) − (open) spectral gaps. Furthermore, for each E ∈ K n , there exists a polynomial P of degree n + 1 so that every J ∈ J ( E ) is stationary for P . Remark.
We note that property (3) above is not stated verbatim from its cited source; we offerits proof in Section 5. 3.
Proofs of the Main Theorems
We now have all of the requisite tools to prove our theorems.
Proof of Theorem 1.3.
Consider the Toda lattice, given by (2.3) with P ( z ) = z . The associatedsystem of differential equations is ∂ t a n = a n ( b n +1 − b n ) , (3.1) ∂ t b n = 2( a n − a n − ) . (3.2)Suppose there exists a DSO H ∈ J ( E ) and that ( J ( E ) , S ) is minimal. Then, by minimality ofthe shift and the topology on J ( E ), every J = J ( a, b ) ∈ J ( E ) has a n = 1 for all n . Consider anarbitrary J ∈ J ( E ), and let J ( t ) solve (2.3) with J (0) = J . By Proposition 2.4, J ( t ) ∈ J ( E ) forall t , and thus a n ( t ) = 1 for all t ∈ R and all n ∈ Z . But then ∂ t a n = 0, and by (3.1) b n +1 = b n forall n ∈ Z . (cid:3) From here, the proof of Theorem 1.1 is a matter of some bookkeeping:
Proof of Theorem 1.1.
Fix an E ∈ K n , n ≥
1, and suppose there exists a DSO H ∈ J ( E ). Since E has at least one spectral gap, H = H V does not have constant potential V , so the dynamics( J ( E ) , S ) are not minimal by Theorem 1.3. But by Theorem 2.2, the dynamical system ( J ( E ) , S )conjugates to (( R / Z ) n , T ω ). Since ( J ( E ) , S ) is not minimal, some entries of ω are rationallydependent. This condition is meager and measure zero by Proposition 2.3. (cid:3) We extrapolate the ideas used above to prove Theorems 1.2 and 1.4.
Proof of Theorem 1.4.
Fix a compact E with zero, one, or two open spectral gaps. We prove bycase analysis on the number of gaps: EFLECTIONLESS DISCRETE SCHR ¨ODINGER OPERATORS ARE SPECTRALLY ATYPICAL 7 (1) If E has no spectral gaps, J ( E ) is a singleton set. If J ( E ) contains a DSO H V , H V mustbe stationary for (3.1) with a n = 1 for all n ; in particular, H V = S + S − + C is 1-periodic.(2) Suppose E has one spectral gap. Then if J ( E ) contains a DSO H V , V cannot be constant,and the shift dynamics thus cannot be minimal by Theorem 1.3. Thus, the lone harmonicfrequency must be rational, and by Theorem 2.2, the dynamical system ( J ( E ) , S ) is peri-odic, as claimed.(3) Suppose E has two spectral gaps, and suppose there is a DSO H V ∈ J ( E ). Then byProposition 2.4, H V is stationary for some monic degree 3 polynomial P ( z ) = z + c z + c z .Noting that (2.4) is linear in the choice of polynomial P , it is a long but straightforwardcalculation (cf. [25, Equation (12.44)]) to see that, from ∂ t b n = 0 and a n = 1, we have( b n +1 − b n − )( b n +1 + b n + b n − ) = − c ( b n +1 − b n − ) . Thus, for some constant C = − c , at least one of b n +1 + b n + b n − = C (3.3) b n +1 = b n − (3.4) must hold for every n ∈ Z . We claim that, if one of (3.3) or (3.4) holds for b n for all n ∈ Z ,then b n can assume only finitely many distinct values.If b n +1 = b n − for every n , b is 2-periodic and we are done. Otherwise, there exists some n for which b n +1 = b n − . By shifting a finite number of times, we may assume this n = 1. Thus, we have b = C − ( b + b ) . (3.5) We proceed inductively. Denote by B := { ( b i , b j ) , ≤ i = j ≤ } the set of all possible distinct pairs taken from { b , b , b } , listed with multiplicity. Here, wehave passed to considering distinct pairs ( b i , b j ) because it is more challenging to proceedinductively via our relations (3.3), (3.5) when multiplicities are not taken into account.Suppose that the pair ( b n , b n − ) ∈ B , and consider the pair ( b n +1 , b n ). If (3.4) holds, weare clearly done by assumption; otherwise, b n +1 = C − ( b n + b n − ). But by our assumptionthat ( b n , b n − ) ∈ B , we have b n + b n − ∈ { C − b , C − b , b + b } . In any case, b n +1 = C − ( b n + b n − ) implies b n +1 ∈ { b , b , b } \ { b n } , and hence ( b n +1 , b n ) ∈ B . This shows that b n takes finitely many values for n ≥
0. One can clearly apply the identical argument inreverse by the symmetry of (3.3) and (3.4) to conclude the same for n < , b ) is stationary, b n can assume only finitely manyvalues. But because H V ∈ J ( E ) and E is finite-gap, b n is an almost-periodic sequenceby Theorem 2.2. An almost-periodic sequence taking finitely many values is periodic, so H V ∈ J ( E ) is periodic.So, in each case, the existence of a DSO H ∈ J ( E ) implies that H is periodic. Thus, the compact E must be of periodic type; that is, ω j ∈ Q for each j = 1 , · · · , n (see, e.g., [23, Corollary 5.5.19]).By Theorem 2.2, it follows that the dynamical system ( J ( E ) , S ) is periodic. (cid:3) Of course, this makes the proof of Theorem 1.2 quite simple:
Proof of Theorem 1.2.
Fix a finite-gap compact E with n = 0 , , or 2 gaps, and suppose there existsan almost-periodic DSO H with σ ( H ) = E such that the essential support of the a.c. spectrumΣ ac ( H ) = E agrees with E up to sets of Lebesgue measure zero. Then by Theorem 2.1, H ∈ J ( E ),and by Theorem 1.4, H is periodic. (cid:3) T. VANDENBOOM
Remark.
As noted in the Acknowledgments, we are thankful to Dr. Jake Fillman for pointingout that some of our methods extend beyond the class of DSOs. Namely, we call J = J ( a ) anoff-diagonal Jacobi operator (ODJO) if it is of the form( J u ) n = a n u n − + a n +1 u n +1 . Theorem 1.3 holds with “DSO”, “ H V ”, and “ V ” replaced by “ODJO”, “ J ( a )”, and “ a ”, respec-tively, by noting that (3.2) must be zero under the assumption of minimality of ( J ( E ) , S ). Con-sequently, the natural analogue of Theorem 1.1 likewise holds for the ODJO class. However, ourmethods do not immediately extend to the proofs of Theorems 1.2 and 1.4, because the assumptionthat b n = 0 leads to certain degeneracies in the Toda hierarchy.4. Frequencies of Finite-gap Compacts are Generically Independent
In this section, we prove Proposition 2.3.Fix a compact E ∈ K n , and write it as a closed interval without a finite union of maximal opengaps as in (2.2): E = [ E +0 , E − ] \ n [ j =1 ( E − j , E + j ) . Associated to the compact E is a degree 2( n + 1) polynomial Q given by Q E ( z ) = n Y j =0 ( z − E + j )( z − E − j )(4.1)It is not hard to see that this polynomial is positive in R \ E .The following is classical, but we reproduce a proof (found in this form in [22]) for the sake ofcompleteness: Proposition 4.1.
Fix a compact E ∈ K n . There exists a unique monic critical polynomial P E ( z ) of degree n so that Z E + j E − j P E ( x ) p Q E ( x ) dx = 0 , j = 1 , , ..., n. (4.2) If we write P E ( z ) = z n − c z n − − ... − c n − z − c n , then the vector ~c with ( ~c ) j = c j is the unique solution to the linear system A~c = ~b , where ( A ) jk = Z E + j E − j x n − k p Q E ( x ) dx ( ~b ) j = Z E + j E − j x n p Q E ( x ) dx. Proof.
By definition, if there exists a unique ~c solving A~c = ~b as claimed, then the critical polynomialexists. Thus, it suffices to show the linear equation has a unique solution, i.e. that the matrix A isnon-singular.Suppose otherwise. Then there exists a nonzero polynomial P of degree at most n − Q is positive on each gap ( E − j , E + j ), P must change sign on the interior of eachgap ( E − j , E + j ), j = 1 , , ..., n . But then P has at least n zeroes, contradicting our assumption onits degree. (cid:3) Via a straightforward calculus exercise, it isn’t hard to check that
EFLECTIONLESS DISCRETE SCHR ¨ODINGER OPERATORS ARE SPECTRALLY ATYPICAL 9
Lemma 4.2.
The value I k ( a ) := Z a x k p x ( a − x ) dx is given by I k ( a ) = Γ( k + 1 / k + 1) √ πa k where Γ is the typical gamma function. In particular, I k ( a ) is analytic in a . With this in hand, one has that
Lemma 4.3.
The coefficients c j above depend analytically on the gap edges of E .Proof. By nonsingularity of the matrix A , if we show that terms like m j ( E ) := Z E + j E − j x n p Q E ( x ) dx (4.3)are analytic in each E ± k , we are done, because the c j are linear combinations of entries of A − and ~b above. Since Q E is positive and bounded away from zero on ( E − j , E + j ) when varying E ± k , k = j ,(4.3) is clearly analytic for all except E ± j . By symmetry in ± , it suffices to prove that the terms(4.3) are analytic in E + j .Fix a positive integer n ≥
1, and denote by U the following open subset of R n +1)+1 : U = ( v, t ) ∈ T n +1) × R : t ∈ n [ j =1 ( v j − , v j ) Define a function f j : U → R by f j ( E, x ) := x n r(cid:12)(cid:12)(cid:12)Q k = j ( E + k − x )( E − k − x ) (cid:12)(cid:12)(cid:12) . This function is analytic for (
E, x ) ∈ B δ ( E ) × ( E − j − δ, E + j + δ ) for δ smaller than the shortest bandand gap lengths. In this region, we expand f j ( E, x ) = ∞ X k =0 a jk ( E ) x k . where the coefficients a jk ( E ) are analytic in each E ± i , i = j . Then we have m j ( E ) = Z E + j E − j f j ( E, x ) q ( E + j − x )( x − E − j ) dx = ∞ X k =1 a jk ( E ) Z E + j E − j x k q ( E + j − x )( E − j − x ) dx = X ~m ∈ [1 , n +1)] n c ~m E ~m , with multi-index notation in the last equality. Above, the first equality is by definition, the secondby analytic expansion of f j and the Fubini theorem, and the last by Lemma 4.2 and analyticity ofthe coefficients a jk ( E ). (cid:3) Thus, for each compact E ⊂ R , we find an associated probability measure given by dρ E ( x ) = 1 π n X j =0 | P E ( x ) | p | Q E ( x ) | χ [ E + j ,E − j +1 ] ( x ) dx (4.4)where here we interpret E − n +1 := E − . In fact, (4.4) is the equilibrium measure for E [29]; that is, dρ E is the unique probability measure µ supported on E minimizing E ( µ ) = Z E Z E log( | x − y | − ) dµ ( x ) dµ ( y ) . For finite-gap sets of the form (2.2) with non-degenerate bands, recall that the equilibrium measureagrees with harmonic measure (e.g., [19, Theorem 4.3.14]). Thus, we may equivalently define thefrequency vector ω ∈ R n from Theorem 2.2 by ω j = ρ E (( −∞ , E − j ]) , j = 1 , , · · · , n. Because the pushforward of harmonic measure via a conformal bijection is the harmonic measure,the frequency vector ω ∈ R n is scaling-invariant; that is, ω ( αE ) = ω ( E ) , α > . (4.5)One could likewise check this fact by a straightforward calculation using Proposition 4.1 and (4.4).Fix a number n ∈ N , and denote T n = { ( x , · · · , x n ) ∈ R n : x < x < · · · < x n } . Any compact E ∈ K n written (2.2) can be parametrized by a vector in T n +1) by reading offendpoints from left to right: E ( E +0 , E − , E +1 , E − , · · · , E + n , E − ) ⊤ We identify K n to T n +1) in this way, and note that K n can be naturally viewed as a R + bundleover K n : Proposition 4.4.
Fix an α > . The space K n is homeomorphic to K nα × R + .Proof. Consider the map h α : K n → K nα × R + given by h α ( E ) := (cid:18) α cap( E ) E, cap( E ) (cid:19) . By (2.1), h α is a bijection, with continuous inverse h − α ( E, β ) = βα E.
The capacity cap( E ) is continuous and positive on K n by Proposition 4.1 and (4.4). Thus, h α islikewise continuous, and h α is a homeomorphism, as claimed. (cid:3) Define a map ω : K n → T n which outputs the frequency vector ω ( E ) ∈ T n of a compact E ∈ K n .By (4.5), this map is constant on the R + fibres over K nα . We wish to investigate the “typicality” ofrational independence of the frequency vector with respect to the input compact E . With this inmind, we explore some properties of the map ω : Proposition 4.5.
The map ω : K n → T n is real analytic. Furthermore, if B ⊂ K n has measurezero, ω − ( B ) likewise has measure zero. EFLECTIONLESS DISCRETE SCHR ¨ODINGER OPERATORS ARE SPECTRALLY ATYPICAL 11
Proof.
We begin by proving analyticity.Recall the open set U defined above, and denote by g j ( E, t ) : U → R the function g j ( E, t ) = | P E ( t ) | r(cid:12)(cid:12)(cid:12)Q k = j ( t − E + k − )( t − E − k ) (cid:12)(cid:12)(cid:12) By Lemma 4.3 and because the zeros of P E lie in the gaps of E , for each E ∈ K n there exists avalue δ > g j ( E, t ) is bounded and analytic on B δ ( E ) × ( E + j − − δ, E − j + δ ). Notice that,as defined, we have ω ( E ) j = 1 π X k ≤ j Z E − k E + k − g k ( E, t ) q ( t − E + k − )( E − k − t ) dt. Each term in this sum is analytic by an argument exactly analogous to that in the proof of Lemma4.3 above. Consequently, ω : K n → T n is analytic, as claimed.Consider now the Jacobian Dω of ω . By Sard’s theorem, the critical set X ⊂ K n of values forwhich the Jacobian is not surjective has zero measure image under ω , that is, | ω ( X ) | = 0.Define a new analytic function g : K n → R given by g ( E ) = X ~m ∈ [1 , n +1)] n det(( Dω ( E )) ~m ) where ( Dω ) ~m is the matrix formed by the n columns of ( Dω ) indexed by ~m . The Jacobian issurjective (and thus ω a submersion) for all those E for which g ( E ) = 0. By analyticity of g , either { E : g ( E ) = 0 } is (Lebesgue) measure zero in K n or g is constantly zero. If g is constantly zero,then the set of critical points X is the whole domain K n . Thus, if g were constantly zero, then theimage of ω must be of zero measure. But this is certainly false; the bands of n -gap compacts canhave arbitrary harmonic measures [6, 26].Thus, { E : g ( E ) = 0 } has (Lebesgue) measure zero, and ω is almost-everywhere a submersion.It follows that the preimage of a measure zero set has measure zero [17]. (cid:3) Let q ∈ Z n , k ∈ Z , and denote B q,k = { v ∈ R n : q · v = k } ∩ T n . Of course, B q,k is a translationof a closed proper subspace of T n , and thus is nowhere dense. Denote by B = S q ∈ Z n S k ∈ Z B q,k .This set is a countable union of nowhere dense sets, i.e. a meager set. We claim that Lemma 4.6. ω − ( B q,k ) is nowhere dense and has measure zero.Proof. B q,k is a closed subset of T n of positive codimension, and thus of measure zero. By continuity, ω − ( B q,k ) is likewise closed, and by Lemma 4.5 ω − ( B q,k ) has measure zero. A closed, measurezero subset of Euclidean space is nowhere dense. (cid:3) We can now address the
Proof of Proposition 2.3.
Consider the set A ⊂ K n of n -gap compacts whose frequency vectors arerationally dependent; that is, A = ω − ( B ). That A is meager follows from the definition andLemma 4.6. Because B is a countable union of zero measure sets, B likewise has measure zero. ByProposition 4.5, ω − ( B ) has measure zero in K n . Finally, because ω is fibre-wise constant, for each α > ω − ( B ) = ( ω − ( B ) ∩ K nα ) × R + , and it follows that ω − ( B ) ∩ K nα is meager and has measure zero under the pushforward measureinduced by h α . (cid:3) The Toda Hierarchy
Recall the definition of the Toda hierarchy from Section 2. Namely, for a polynomial P of degree n + 1 ≥
1, recall that the n th Toda flow (for P ) is the integral curve of Jacobi operators satisfyingthe Lax pair (2.3) ∂ t J = [ P ( J ) + − P ( J ) − , J ] . When there exists a monic polynomial P and Jacobi operator J such that P ( J ) + − P ( J ) − and J commute, [ P ( J ) + − P ( J ) − , J ] = 0 , we say that J is stationary for P .The Toda flow exists uniquely for any bounded Jacobi initial condition J [25, Theorem 12.6].Denote this flow by T P ( t ), that is, the Jacobi matrix T P ( t ) J should solve the Lax pair (2.3) with T P (0) J = J . An important fact about the Toda flow which we did not mention above is thefollowing: Proposition 5.1 (Theorem 12.5, [25]) . Fix a polynomial P . Then T P ( t ) J is unitarily equivalentto J for all t ∈ R . Thus, we see that the flow T P preserves spectral properties. In fact, this flow also preservesthe reflectionless condition [16, Corollary 1.3], and thus, for fixed polynomial P , descends as acontinuous-time dynamical system to J ( E ). Amazingly, under the assumption that E is homoge-neous, the homeomorphism from Theorem 2.2, which linearized the shift dynamics, simultaneouslylinearizes the Toda flow. Theorem 5.2 (Theorem 1.3, [27]) . Suppose E is a homogeneous compact with g gaps, ≤ g ≤ ∞ , and let P be a polynomial. Then the dynamical system ( J ( E ) , S , T P ( t )) is conjugateto (( R / Z ) g , T ω , T tξ P ) , where T tξ P ( α ) = α + tξ P , ξ P ∈ R g . We remark that this result was well-known in the finite-gap regime [8], but we state this moregeneral result for completeness.Fix a compact E ∈ K n and consider the polynomial Q E ( z ) defined as in (4.1) above: Q E ( z ) = n Y j =0 ( z − E + j )( z − E − j ) . Apply the square root with the typical branching and expand Q / E for | z | > max {| E +0 | , | E − |} as Q / E ( z ) = − z n +1 ∞ X j =0 c j ( E ) z − j . (5.1)We intend to address the proof of property (3) from Proposition 2.4, given the following statementfrom [25]: Proposition 5.3 (Corollary 12.10, [25]) . Let J ∈ J ( E ) . Then J is stationary for a Toda polyno-mial P ( z ) = z m +1 − c z m − · · · − c m z of degree m + 1 if and only if there exists a polynomial Q m − n of degree m − n such that c j = c j ( ˜ E ) ,where Q ˜ E is given by Q ˜ E ( z ) = Q m − n ( z ) Q E ( z ) . EFLECTIONLESS DISCRETE SCHR ¨ODINGER OPERATORS ARE SPECTRALLY ATYPICAL 13
Proposition 5.4 (Proposition 2.4, (3) ) . If J is stationary for P , then the spectrum of J has atmost deg( P ) − (open) spectral gaps. Furthermore, for each E ∈ K n , there exists a polynomial P of degree n + 1 so that every J ∈ J ( E ) is stationary for P .Proof. Let E = σ ( J ), and suppose J is stationary for P of degree m + 1. Since J is stationary, J ∈ J ( E ) with E finite-gap by Proposition 2.4 (2) . Suppose E has n gaps. Then, by Proposition5.3, we have P ( z ) = z m +1 − c z m − · · · − c m z, where c j = c j ( ˜ E ) are determined as in (5.1) for some polynomial Q ˜ E ( z ) of degree 2( m + 1).Furthermore, we have that Q E ( z ), a polynomial of degree 2( n + 1), is a factor of Q ˜ E ( z ). But forthis to be true, it must be the case that 2( m + 1) ≥ n + 1); that is, that m = deg( P ) − ≥ n .Now, suppose E ∈ K n , and define a sequence of coefficients c j = c j ( E ) as in (5.1). Then taking Q m − n ( z ) = 1 in Proposition 5.3 shows that J is stationary for P . (cid:3) While we have introduced all of these results, we feel it necessary, while not particularly relevantto the matter at hand, to note a simple consequence that we have not observed in the literature:
Theorem 5.5.
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