BBethe-Sommerfeld conjecture in semiclassicalsettings * , โ Victor Ivrii โก February 4, 2019
Abstract
Under certain assumptions (including d โฅ we prove that thespectrum of a scalar operator in L ( โ d ) A ๐ ( x , hD ) = A ( hD ) + ๐ B ( x , hD ), covers interval ( ๐ โ ๐ , ๐ + ๐ ) , where A is an elliptic operator and B ( x , hD ) is a periodic perturbation, ๐ = O ( h ๐ ) , ๐ > .Further, we consider generalizations. This work is inspired by a paper [PS] by L. Parnovski and A. Sobolev, inwhich a classical Bethe-Sommerfeld conjecture was proven for operators ( โ ๐) m + B ( x , D ) with operator B of order < m . In this paper the cru-cial role was played by a (pseudodifferential) gauge transformation and * : 35P20. โ Key words and phrases : Microlocal Analysis, sharp spectral asymptotics, integrateddensity of states, periodic operators, Bethe-Sommerfeld conjecture. โก This research was supported in part by National Science and Engineering ResearchCouncil (Canada) Discovery Grant RGPIN 13827 a r X i v : . [ m a t h . SP ] F e b . Introduction . Now I would like to apply this gauge transform to Bethe-Sommerfeld conjecture in the semiclassical settings. The results obtainedare more general (except the smoothness with respect to ๐ assumptions in[PS] are more general than here) and the proofs are simpler.Consider a scalar self-adjoint h -pseudo-differential operator A h := A ( x , hD ) in โ d with the Weyl symbol A ( x , ๐ ) , such that | D ๐ผ x D ๐ฝ๐ A ( x , ๐ ) | โค c ๐ผ๐ฝ ( | ๐ | + ๐ฃ) m โ ๐ผ , ๐ฝ (1.1)and A ( x , ๐ ) โฅ c | ๐ | m โ C โ ( x , ๐ ) โ โ d . (1.2)Then A h is semibounded from below. Also we assume that it is ๐ -periodicwith the lattice of periods ๐ :(1.3) A ( x + ๐, ๐ ) = A ( x , ๐ ) โ x โ โ n โ ๐ โ ๐. We assume that ๐ is non-degenerate and denote by ๐ * the dual lattice :(1.4) ๐พ โ ๐ * โโ โจ ๐พ , ๐ โฉ โ ๐ โค โ ๐ โ ๐; since we use ๐ * and itโs elements in the paper much more often, than ๐ anditโs elements, it is more convenient for us to reserve notation ๐พ for elementsof ๐ * .Also let ๐ช = โ d / ๐ and ๐ช * = โ d / ๐ * be fundamental domains ; we identifythem with domains in โ d .It is well-known that ๐ฒ๐๐พ๐ผ( A ) has a band-structure . Namely, considerin L ( ๐ช ) operator A h ( ฮพ ) = A ( x , hD ) with the quasi-periodic boundarycondition :(1.5) u ( x + ๐) = e i โจ ๐, ฮพ โฉ u ( x ) โ x โ ๐ช โ ๐ โ ๐ The other components of the proof were not only completely different, but in theframework of the different paradigm. In fact, we consider A h := A ( x , hD , h ) . I.e. with the
๐ = { ๐ = n ๐ + ... + n d ๐ d , ( n , ... , n d ) โ โค d } with linearly independent ๐ , ... , ๐ d โ โ d . . Introduction ฮพ โ ๐ช * ; it is called a quasimomentum . Then ๐ฒ๐๐พ๐ผ( A h ( ฮพ )) is discrete(1.6) ๐ฒ๐๐พ๐ผ( A h ( ฮพ )) = โ๏ธ n ๐ n , h ( ฮพ ) and depends on ฮพ continuously. Further,(1.7) ๐ฒ๐๐พ๐ผ( A h ) = โ๏ธ ฮพ โ๐ช * ๐ฒ๐๐พ๐ผ( A h ( ฮพ )) =: โ๏ธ n ๐ n , h , with the spectral bands ๐ n , h := โ๏ธ ฮพ โ๐ช * { ๐ n , h ( ฮพ ) } .One can prove that the with of the spectral band near energy level ๐ is O ( h ) . Spectral bands could overlap but they also could leave uncoveredintervals, called spectral gaps . It follows from [Ivr3] that in our assumptions(see below) the width of the spectral gaps near energy level ๐ is O ( h โ ) .Bethe-Sommerfeld conjecture in the semiclassical settings claims that thereare no spectral gaps near energy level ๐ (in the corresponding assumptions,which include d โฅ ). We assume that(1.8) A h := A ( x , hD ) = A ( hD ) + ๐ B ( x , hD ), where A ( ๐ ) satisfies (1.1), (1.2) and B ( x , ๐ ) satisfies (1.1) and (1.3) and ๐ > is a small parameter. For A ( ๐ ) instead of ๐ n ( ฮพ ) we have(1.9) ๐ ๐พ ( ฮพ ) := A ( h ( ๐พ + ฮพ )) with ๐พ โ ๐ * . Recall that (as in [Ivr3]) B ( x , ๐ ) = โ๏ธ ๐พ โ ๐ b ๐พ ( ๐ ) e i โจ ๐พ , x โฉ (1.10)with ๐ = ๐ * where due to (1.1) | D ๐ฝ๐ b ๐พ ( ๐ ) | โค C L ๐ฝ ( | ๐พ | + ๐ฃ) โ L ( | ๐ | + ๐ฃ) m โ ๐ฝ โ ( x , ๐ ) โ โ d (1.11)with an arbitrarily large exponent L . . Introduction Theorem 1.1.
Let d โฅ and let operator A h be given by (1.8) with ๐ = O ( h ๐ ) with arbitrary ๐ > and with A h = A ( hD ) satisfying (1.1) , (1.2) and B ( x , ๐ ) satisfying (1.1) and (1.3) .Further, assume that the microhyperbolicity and strong convexity con-ditions on the energy level ๐ are fulfilled: (1.12) | A ( ๐ ) โ ๐ | + |โ ๐ A ( ๐ ) | โฅ ๐ and (1.13) ยฑ โ๏ธ j , k A ๐ j ๐ k ( ๐ ) ๐ j ๐ k โฅ ๐ | ๐ | โ ๐ : A ( ๐ ) = ๐ โ ๐ : โ๏ธ j A ๐ j ( ๐ ) ๐ j = ๐ข. Furthermore, assume that there exists ๐ โ ๐จ ๐ such that for every ๐ โ ๐จ ๐ , ๐ ฬธ = ๐ , such that โ ๐ A ( ๐ ) is parallel to โ ๐ A ( ๐ ) (1.14) ๐จ ๐ , intersected with some vicinity of ๐ and shifted by ( ๐ โ ๐ ) , coincidesin the vicinity of ๐ with { ๐ : ๐ k = g ( ๐ ฬ k } and ๐จ ๐ coincides in the vicinity of ๐ with { ๐ : ๐ k = f ( ๐ ฬ k } and โ ๐ผ ( f โ g )(๐ข) ฬธ = ๐ข for some ๐ผ : | ๐ผ | = ๐ค .Then ๐ฒ๐๐พ๐ผ( A h ) โ [ ๐ โ ๐ , ๐ + ๐ ] for sufficiently small ๐ > .Remark 1.2. (i) If ๐จ ๐ is strongly convex and connected then for every ๐ โ ๐จ ๐ there exists exactly one antipodal point ๐ โ ๐จ ๐ ; then ๐ < and assumption(1.14) is fulfilled. In particular, if A ( ๐ ) = | ๐ | m , then ๐ = โ ๐ and ๐ = โ .(ii) If ๐จ ๐ is is strongly convex and consists of p connected components, thenthe set Z ( ๐ ) = { ๐ โ ๐จ ๐ , ๐ ฬธ = ๐ : โ ๐ A ( ๐ ) โ โ ๐ A ( ๐ ) } contains exactly p โ elements, and for p of them ๐ < and assumption (1.14) is fulfilled for sure,while for ( p โ of them ๐ > . One needs to understand, how gaps could appear, why they appear if d = ๐ฃ and why it is not the case if d โฅ . Observe that ๐ n ( ฮพ ) can be identified I.e. ๐ A ( ๐ ) = ๐ โ ๐ A ( ๐ ) with ๐ ฬธ = ๐ข ; we call ๐ antipodal pont . With ๐ ฬ k meaning all coordinates except ๐ k . Obviously โ ( f โ g )(๐ข) = ๐ข . One can prove easily, that if this condition holds at ๐ with some ๐ผ : | ๐ผ | > , then changing slightly ๐ , we make it fulfilled with | ๐ผ | = ๐ค . . Introduction ๐ ๐พ ( ฮพ ) only locally, if ๐ ๐พ ( ฮพ ) is sufficiently different from ๐ ๐พ โฒ ( ฮพ ) forany ๐พ โฒ ฬธ = ๐พ .Indeed, in the basis of eigenfunctions of A ฮพ ( hD ) perturbation ๐ B ( x , hD ) can contain out-of-diagonal elements ๐ b ๐พ โ ๐พ โฒ ( ฮพ ) and such identification ispossible only if | ๐ ๐พ ( ฮพ ) โ ๐ ๐พ โฒ ( ฮพ ) | is larger than the size of such element.If d = ๐ฃ , A ( ๐ ) = ๐ and ๐ โค ๐ โฒ h with sufficiently small ๐ โฒ > and ๐ โ ,it can happen only if ๐พ โฒ coincides with โ ๐พ or with one of two adjacent pointsin ๐ * and | ฮพ โ ( ๐พ + ๐พ โฒ ) | = O ( ๐ h โ ) . This exclude from possible values ofeither ๐ ๐พ ( ฮพ ) or ๐ ๐พ โฒ ( ฮพ ) the interval of the width O ( ๐ h โ ) and on such intervalcan happen (and really happens for a generic perturbation) the realignment: ฮพ ๐ n ( ฮพ ) ๐ m ( ฮพ ) (a) ฮพ ๐ n ( ฮพ ) ๐ m ( ฮพ ) ฮพ ๐ n ( ฮพ ) ๐ m ( ฮพ ) (b) Figure 1: Spectral gap is a red intervalIf d โฅ the picture becomes more complicated: there are much moreopportunities for ๐ ๐พ ( ฮพ ) and ๐ ๐พ โฒ ( ฮพ ) to become close, even if ๐พ and ๐พ โฒ are notthat far away; on the other hand, there is a much more opportunities forus to select ๐ = h ( ๐พ + ฮพ ) โ ๐จ ๐ and then to adjust ฮพ so that ๐ = h ( ๐พ + ฮพ ) remains on ๐จ ๐ but ๐ = h ( ๐พ โฒ + ฮพ ) moves away from ๐จ ๐ sufficiently far away and then tune-up ฮพ once again so that ๐ โ ๐ฒ๐๐พ๐ผ( A h ( ฮพ )) .In fact, we prove the following statement which together with Theorem 1.1(which follows from it trivially) are semiclassical analogue of Theorem 2.1 of[PS]: Theorem 1.3.
In the framework of Theorem 1.1 there exist n and ฮพ * suchthat ๐ n ( ฮพ * ) = ๐ and ๐ n ( ฮพ ) covers interval [ ๐ โ ๐ h , ๐ + ๐ h ] when ฮพ runs ball Consisting of ๐พ๐๐( i โจ x , ๐พ + ฮพ โฉ ) . This will happen if either โ ๐ A ( ๐ ) differs from ๐ โ ๐ A ( ๐ ) , or if coincides with it but(1.14) is fulfilled. . Reduction of operator ๐ก( ฮพ * , ๐ ) while | ๐ m ( ฮพ ) โ ๐ | โฅ ๐๐ h for all m ฬธ = n and ฮพ โ ๐ก( ฮพ * , ๐ ) . Here (1.15) ๐ = ๐ {๏ธ h ( d โ ๐๐๐(๐ฃ, ๐ โ d โ / h ( d โ ๐ ) d โฅ h ๐๐๐( | ๐ ๐๐ h | โ , ๐ โ / h ๐ ) d = ๐ค with arbitrarily small exponent ๐ > . Proof of Theorem 1.3 occupies two next sections. In Section 2 we reduceoperator in the vicinity of ๐จ ๐ to the block-diagonal form and study itsstructure. To do this we need to examine the structure of the resonant setof the operator. In Section 3 we prove Theorem 1.3 and thus Theorem 1.1.Finally, in Section 4 we discuss our results and the possible improvements. On this step we reduce A to the block-diagonal form in the vicinity of ๐จ ๐ (2.1) ๐ฎ ๐ := { ๐ : | A ( ๐ ) โ ๐ | โค C ๐ h โ ๐ฟ } . In what follows, we assume that ๐ โฅ h , i.e.(2.2) h โค ๐ โค h ๐ , ๐ > To do this we need just to repeat with the obvious modifications defi-nitions and arguments of Sections 1 and 2 of [Ivr3]. Namely, now
๐ := ๐ * is a non-degenerate lattice rather than the pseudo-lattice, as it was in thatpaper, and all conditions (A), (B), (C), (D), and (E), are fulfilled with ๐ โฒ := ๐ โฉ ๐ก(๐ข, ๐ ) with ๐ = h โ ๐ where we select sufficiently small ๐ > later and ๐ โฒ K = ๐ โฉ ๐ก(๐ข, K ๐ ) be an arithmetic sum of K copies of ๐ โฒ withsufficiently large K to be chosen later.We call point ๐ non-resonant if(2.3) |โจโ ๐ A ( ๐ ), ๐ โฉ| โฅ ๐ โ ๐ โ ๐ โฒ K โ with ๐ โ [ ๐ / h โ ๐ฟ , h ๐ฟ ] with arbitrarily small ๐ฟ > . Otherwise we call it resonant . More precisely(2.4) ๐ := โ๏ธ ๐ โ ๐ โฒ K โ ๐ ( ๐ ), . Reduction of operator ๐ ( ๐ ) is the set of ๐ , violating (2.3) for given ๐ โ ๐ โฒ K โ .It obviously follows from the microhyperbolicity and strong convexityassumptions (1.12) and (1.13) that(2.5) ยต ๐ -measure of ๐ โฉ ๐จ ๐ , does not exceed C r d โ ๐ and Euclidean measureof ๐ โฉ { ๐ : | A ( ๐ ) โ ๐ | โค ๐ } does not exceed C r d โ ๐๐ , where r = Kh โ ๐ .Indeed, ( d โ -dimensional measure of { x : | x | = ๐ฃ, |โจ x , ๐ โฉ| โค ๐ } doesnot exceed C | ๐ | โ ๐ and and due to microhyperbolicity and strong convexityassumptions maps ๐จ ๐ โ โ A ( ๐ ) / |โ A ( ๐ ) | โ ๐ d โ and { ๐ : | A ( ๐ ) โ ๐ | โค ๐ } โ ( โ A ( ๐ ) / |โ A ( ๐ ) | , A ( ๐ )) โ โ d are diffeomorphisms.Furthermore, according to Proposition 2.7 of [Ivr3] that on the non-resonant set one can โalmostโ diagonalize A ( x , hD ) . More precisely Proposition 2.1.
Let assumptions (1.12) and (1.13) be fulfilled.(i) Then there exists a periodic pseudodifferential operator P = P ( x , hD ) such that (๏ธ e โ i ๐ h โ P Ae i ๐ h โ P โ ๐ )๏ธ Q โก (2.6) with ๐ = A ( hD ) + ๐ B โฒ ( hD ) + ๐ B โฒโฒ ( x , hD ) (2.7) modulo operator from H m to L with the operator norm O ( h M ) with M arbitrarily large and K = K ( M , d , ๐ฟ ) in the definition of non-resonant pointprovided Q = Q ( hD ) has a symbol, supported in { ๐ : | A ( ๐ ) โ ๐ | โค ๐ h โ ๐ฟ } .Here P ( x , hD ) , B โฒ ( hD ) and B โฒโฒ ( x , hD ) are operator with Weyl symbols ofthe same form (1.10) albeit such that | D ๐ผ๐ D ๐ฝ x P | โค c ๐ผ๐ฝ ๐ โ โ| ๐ผ | โ ๐ผ , ๐ฝ , (2.8) | D ๐ผ๐ D ๐ฝ x B โฒโฒ | โค c โฒ ๐ผ๐ฝ ๐ โ| ๐ผ | โ ๐ผ , ๐ฝ , (2.9) and symbol of B โฒ also satisfies (2.9) . ยต ๐ = d ๐ : dA ( ๐ ) is a natural measure on ๐จ ๐ . . Reduction of operator (ii) Further, (2.10) ๐ / โ ๐ ( ๐ ) = โ b โฒโฒ ๐ ( ๐ ) = ๐ข. and B โฒ ( ๐ ) coincides with b ( ๐ ) modulo O ( ๐๐ โ ) . In what follows(2.11) ๐ ( hD ) := A ( hD ) + ๐ B โฒ ( hD ) and โฌ := B โฒโฒ ( x , hD ). Remark 2.2. (i) Eigenvalues of ๐ are(2.12) ๐ ๐พ ( ฮพ ) = ๐ ( h ( ๐พ + ฮพ )). (ii) If ๐ = h ( ๐พ + ฮพ ) is non-resonant, then due to (2.10) in ๐ โฒ ๐ -vicinity of ๐ this ๐ ๐พ ( ฮพ ) is also an eigenvalue of ๐ .(iii) Without any loss of the generality one can assume that(2.13) | ๐ | โฅ ๐ h โ ๐ฟ = โ b โฒโฒ ๐ = ๐ข. We assume that this is the case.(iv) Let us replace operator ๐ defined by (2.7) by operator(2.14) ๐ โฒ = ๐ ( hD ) + ๐ โฌ โฒ ( x , hD ), โฌ โฒ ( x , hD ) = T ( hD ) โฌ T ( hD ) with T ( hD ) operator with symbol T ( ๐ ) which is a characteristic function of ๐ฎ ๐ defined by (2.1) with C = ๐จ . Then (2.6) holds.From now on ๐ := ๐ โฒ and โฌ := โฌ โฒ .It would be sufficient to prove Theorem 1.3 for operator ๐ . Indeed, Proposition 2.3. (i) For each point ๐ โ ๐ฒ๐๐พ๐ผ( A ( ฮพ )) โฉ {| ๐ โ ๐ | โค ๐ h โ ๐ฟ } ๐ฝ๐๐๐( ๐ , ๐ฒ๐๐พ๐ผ( ๐ ( ฮพ )) โค Ch M .(ii) Conversely, for each point ๐ โ ๐ฒ๐๐พ๐ผ( ๐ ( ฮพ )) โฉ {| ๐ โ ๐ | โค ๐ h โ ๐ฟ } ๐ฝ๐๐๐( ๐ , ๐ฒ๐๐พ๐ผ( A ( ฮพ )) โค Ch M .(iii) Furthermore, if ๐ โ ๐ฒ๐๐พ๐ผ( A ( ฮพ )) โฉ{| ๐ โ ๐ | โค ๐ h โ ๐ฟ } is a simple eigenvalueseparated from the rest of ๐ฒ๐๐พ๐ผ( A ( ฮพ )) by a distance at least h M โ , thenthere exists ๐ โฒ โ ๐ฒ๐๐พ๐ผ( ๐ ( ฮพ )) โฉ {| ๐ โฒ โ ๐ | โค Ch M } separated from the rest of ๐ฒ๐๐พ๐ผ( ๐ ( ฮพ )) by a distance at least h M โ . . Reduction of operator (iv) Conversely, if ๐ โฒ โ ๐ฒ๐๐พ๐ผ( ๐ ( ฮพ )) โฉ {| ๐ โ ๐ | โค ๐ h โ ๐ฟ } is a simple eigenvalueseparated from the rest of ๐ฒ๐๐พ๐ผ( ๐ ( ฮพ )) by a distance at least h M โ , thenthere exists ๐ โ ๐ฒ๐๐พ๐ผ( A ( ฮพ )) โฉ {| ๐ โฒ โ ๐ | โค Ch M } separated from the rest of ๐ฒ๐๐พ๐ผ( A ( ฮพ )) by a distance at least h M โ .Proof. Trivial proof is left to the reader.
Remark 2.4.
One can generalize Statements (iii) and (iv) of Proposition 2.3from simple eigenvalues to subsets of
๐ฒ๐๐พ๐ผ( A ( ฮพ )) and ๐ฒ๐๐พ๐ผ( ๐ ( ฮพ )) separatedby the rest of the spectra; these subsets will contain the same number ofelements. We start from the case d = ๐ค . Then We have only one kind of resonantpoints ๐ฃ = ๐ . If d โฅ then there are d โ kinds of resonant points. First,following [PS] consider lattice spaces V spanned by n linearly independentelements ๐ , ... , ๐ n โ ๐ * โฉ ๐ก(๐ข, r ) , where we take r = Kh โ ๐ . Let ๐ฑ n be theset of all such spaces.It is known [PS] that Proposition 2.5.
For V โ ๐ฑ n , W โ ๐ฑ m either V โ W , or W โ V or theangle between V and W is at least ๐ r โ L with L = L ( d ) and ๐ = ๐ ( d , ๐) . Fix < ๐ฟ < ... < ๐ฟ n arbitrarily small and for V โ ๐ฑ n let us introduce(2.15) ๐ ( V , ๐ n ) := { ๐ โ ๐ฎ ๐ : |โจโ ๐ A ( ๐ ), ๐ โฉ| โค ๐ n | ๐ | โ ๐ โ ๐ฑ} with ๐ n = ๐ h โ ๐ฟ n .We define ๐ฃ n by induction. First, ๐ฃ d = โ . Assume that we defined ๐ฃ d , ... , ๐ฃ n +๐ฃ . Then we define(2.16) ๐ฃ n := โ๏ธ V โ๐ฑ n , ๐ โ ๐ ( V ) โฉ ๐ฎ ๐ ( ๐ + V ) โฉ ๐ฎ ๐ . It follows from Proposition 2.5 that
This angle ฬ ( V , W ) is defined as the smallest possible angle between two vectors v โ V โ ( V โฉ W ) and w โ W โ ( V โฉ W ) . . Reduction of operator Proposition 2.6.
Let ๐ > in the definition of ๐ โฒ and ๐ฟ > in thedefinition of ๐ฎ ๐ be sufficiently small . Then for sufficiently small h (i) ๐ฃ n โ โ๏ธ V โ๐ฑ ๐ ( V , ๐ค ๐ n ) .(ii) If ๐ / โ ๐ฃ n +๐ฃ and ๐ โ ๐ โฒ + V , ๐ โ ๐ โฒโฒ + W for ๐ โฒ โ ๐ ( V ) , ๐ โฒโฒ โ ๐ ( W ) with V , W โ ๐ฑ n , then V = W . Corollary 2.7.
Let ๐ > in the definition of ๐ โฒ and ๐ฟ > in the definitionof ๐ฎ ๐ be sufficiently small . Let h be sufficiently small.Then for each ๐ โ ๐ฃ n โ ๐ฃ n +๐ฃ is defined just one V = V ( ๐ ) such that ๐ โ ๐ โฒ + V for some ๐ โฒ โ ๐ ( V ) . (2.17) We slightly change definition of ๐ฃ n : ๐ = h ( ๐พ + ฮพ ) โ ๐ฃ n ,๐๐พ๐ iff h ๐พ โ ๐ฃ n .From now on ๐ฃ n := ๐ฃ n ,๐๐พ๐ .Consider ๐ โฒ , ๐ โฒโฒ โ ๐ฃ n โ ๐ฃ n +๐ฃ . We say that ๐ โฒ โผ = ๐ โฒโฒ if there exists ๐ โ V , V โ ๐ฑ such that ๐ โฒ , ๐ โฒโฒ โ ๐ + V and if ๐ โฒ โ ๐ โฒโฒ โ ๐ . In virtue of above(2.18) This relation is reflexive, symmetric and transitive.For ๐ โ ๐ฃ n we define X ( ๐ ) = { ๐ โฒ : ๐ โฒ โผ = ๐ } . (2.19)Then ๐ฝ๐๐บ๐( X ( ๐ )) โค C ๐ d โ . (2.20) ๐ For ๐ โ ๐ฃ n โ ๐ฃ n +๐ฃ denote by H ( ๐ ) the subspace L ( ๐ช ) consisting of functionsof the form(2.21) โ๏ธ ๐ โฒ โ X ( ๐ ) c ๐ โฒ e i โจ x , ๐ โฒ โฉ . In virtue of the properties of ๐ and โฌ and of resonant sets we arrive to They depend on ๐ and ๐ฟ , ... , ๐ฟ n . . Proof of Theorem 1.3 Proposition 2.8.
Let ๐ > in the definition of ๐ โฒ and ๐ฟ > in thedefinition of ๐ฎ ๐ be sufficiently small . Let h be sufficiently small.Then for ๐ โ ๐ฃ n โ ๐ฃ n +๐ฃ operators โฌ and ๐ transform H ( ๐ ) into H ( ๐ ) . Let us denote by ๐ ๐พ ( ฮพ ) and โฌ ๐พ ( ฮพ ) restrictions of ๐ and โฌ to H ( h ( ๐พ + ฮพ )) .Here for n = ๐ข we consider ๐ฃ to be the set of all non-resonant points and X ( ๐ ) = { ๐ } for ๐ โ ๐ฃ .Then due to Propositions 2.5 and 2.8 we arrive to Proposition 2.9. (i) For each point ๐ โ ๐ฒ๐๐พ๐ผ( A ( ฮพ )) โฉ {| ๐ โ ๐ | โค ๐ h โ ๐ฟ } exists ๐พ โ ๐ * such that ๐ = h ( ๐พ + ฮพ ) โ ๐ฎ ๐ and ๐ฝ๐๐๐( ๐ , ๐ฒ๐๐พ๐ผ( ๐ ๐พ ( ฮพ )) โค Ch M .(ii) Conversely, for each point ๐ โ ๐ฒ๐๐พ๐ผ( ๐ ๐พ ( ฮพ )) โฉ {| ๐ โ ๐ | โค ๐ h โ ๐ฟ } and ๐ = h ( ๐ + ๐พ ) , ๐ฝ๐๐๐( ๐ , ๐ฒ๐๐พ๐ผ( A ( ฮพ )) โค Ch M .(iii) Further, if ๐ โ ๐ฒ๐๐พ๐ผ( A ๐พ ( ฮพ )) โฉ {| ๐ โ ๐ | โค ๐ h โ ๐ฟ } is a simple eigenvalueseparated from the rest of ๐ฒ๐๐พ๐ผ( A ( ฮพ )) by a distance at least h M โ , then thereexist ๐พ and ๐ โฒ , such that for ๐ = h ( ๐พ + ฮพ ) , ๐ โฒ โ ๐ฒ๐๐พ๐ผ( ๐ ( ๐ )) โฉ{| ๐ โฒ โ ๐ | โค Ch M } ,separated from the rest of ๐ฒ๐๐พ๐ผ( ๐ ๐พ ( ๐ )) by a distance at least h M โ and from โ๏ธ ๐พ โฒ โ ๐ * , ๐พ โฒ ฬธ = ๐พ ๐ฒ๐๐พ๐ผ( ๐ ๐พ โฒ ( ฮพ ) by a distance at least h M โ as well.(iv) Conversely, if ๐ โฒ โ ๐ฒ๐๐พ๐ผ( ๐ ๐พ ( ฮพ )) โฉ{| ๐ โ ๐ | โค ๐ h โ ๐ฟ } is a simple eigenvalueseparated from the rest of ๐ฒ๐๐พ๐ผ( ๐ ๐พ ( ฮพ )) by a distance at least h M โ , andalso separated from โ๏ธ ๐พ โฒ โ ๐ * , ๐พ โฒ ฬธ = ๐พ ๐ฒ๐๐พ๐ผ( ๐ ๐พ โฒ ( ฮพ )) by a distance at least h M โ ,then there exists ๐ โ ๐ฒ๐๐พ๐ผ( A ( ฮพ )) โฉ {| ๐ โฒ โ ๐ | โค Ch M } separated from the restof ๐ฒ๐๐พ๐ผ( A ( ฮพ )) by a distance at least h M โ .Proof. Proof is trivial. ๐พ * The first approximation is ๐ * โ ๐จ ๐ satisfying (1.14). Any ๐ โ ๐จ ๐ in ๐ โฒ -vicinityof ๐ * also fits provided ๐ โฒ > is sufficiently small.(3.1) One can select ๐ * ๐๐พ๐ โ ๐จ ๐ such that | ๐ * ๐๐พ๐ โ ๐ * | โค h ๐ฟ and ๐ * ๐๐พ๐ satisfies(1.14) with ๐ = ๐พ := h ๐ฟ . Here ๐ฟ > is arbitrarily small and ๐ = ๐ ( ๐ฟ ) . . Proof of Theorem 1.3 ๐ * := ๐ * ๐๐พ๐ .Then, according to Proposition 2.1 we can diagonalize operator in ๐พ -vicinity of ๐ * and there ๐ = ๐พ . Then there |โ ๐ผ ( ๐ โ A ) | โค C ๐ผ ( ๐ + ๐ ๐ โ โ| ๐ผ | ) (3.2)and in particular |โ ๐ผ ( ๐ โ A ) | โค Ch ๐ฟ for | ๐ผ | โค (3.3)Let ๐จ โฒ ๐ = { ๐ : ๐ ( ๐ ) = ๐ } . (3.4)Then in the non-resonant points we are interested in functions ๐ ๐พ ( ฮพ ) = ๐ ( h ( ๐พ + ฮพ )) rather than in ๐ ๐พ ( ฮพ ) = A ( h ( ๐พ + ฮพ )) . One can prove easily thefollowing statements: Proposition 3.1. (i) One can select ๐ * := ๐ * ๐๐พ๐ โ ๐จ โฒ ๐ satisfying (1.14) andnon-resonant with ๐ = ๐พ .(ii) Further, all antipodal to ๐ * points ๐ * ,. . . , ๐ * p โ have the same properties. Let ๐ * =: h ( ๐พ * + ฮพ * ) , ๐พ * โ ๐ * and ฮพ * โ ๐ช * . Then values in the nearbypoints are sufficiently separated:(3.5) | ๐ ๐พ ( ฮพ ) โ ๐ ๐พ * ( ฮพ ) | โฅ ๐ h ๐ฟ โ ๐พ : | ๐พ โ ๐พ * | โค Kh โ ๐ โ ฮพ โ ๐ช * . Indeed, | ๐พ โ ๐พ * | โค Kh โ ๐ implies that ( ๐พ โ ๐พ * ) โ ๐ โฒ K and then |โจโ๐ ( ๐ * ), ๐พ โ ๐พ * โฉ| โฅ ๐พ while | ๐ ๐พ ( ฮพ ) โ ๐ ๐พ * ( ฮพ ) โ h โจโ๐ ( ๐ * ), ๐พ โ ๐พ * โฉ| โค Ch โ ๐ . Consider other non-resonant points (with ๐ = ๐ / h โ ๐ฟ ). Let us determinehow ๐ ๐พ ( ฮพ ) changes when we change ฮพ . Due to (3.3)(3.6) ฮด ๐ ๐พ := ๐ ๐พ ( ฮพ + ฮดฮพ ) โ ๐ ๐พ ( ฮพ ) = h โจโ A ( ๐ ), ฮดฮพ โฉ + O ( h | ฮดฮพ | ). To preserve ๐ ๐พ * ( ฮพ ) = ๐ in the linearized settings we need to shift ฮพ by ฮดฮพ which is orthogonal to โ ๐ ๐ ( ๐ * ) . . Proof of Theorem 1.3 ฮดฮพ = t ๐ (3.7) โ : | ๐ | = ๐ฃ, โจโ๐ ( ๐ * ), ๐ โฉ = ๐ข. Then in all non-resonant ๐ the shift will be โจโ ๐ ๐ ( ๐ ), ฮดฮพ โฉ with an absolutevalue |โจโ ๐ ๐ ( ๐ ), ๐ โฉ| ยท | t | . Case d = ๐ค . Let us start from the easiest case d = ๐ค . Without any lossof the generality we assume that ฮพ * is strictly inside ๐ช * (at the distance atleast C ๐ * from the border). Then there is just one tangent direction ๐ and(3.8) |โจโ ๐ ๐ ( ๐ ) | ๐ = h ๐พ , ๐ โฉ| โ | ๐๐๐ ๐ ( ๐พ * , ๐พ ) | , โ h ๐๐๐ โค k โค p | ๐พ โ ๐พ * k | where ๐ ( ๐พ * , ๐พ ) is an angle between โ ๐ ๐ ( ๐ ) | ๐ = h ๐พ * and โ ๐ ๐ ( ๐ ) | ๐ = h ๐พ , and ๐ * , ... , ๐ * p โ are antipodal points, and ๐ * p = ๐ * .As long as ๐๐๐ โค k โค p | ๐พ โ ๐พ * k | โณ h โ ๐ we may replace here ๐ = h ( ๐พ + ฮพ ) by ๐ = h ๐พ and ๐ by ๐ . In the nonlinear settings to ensure that(3.9) ๐ ๐พ * ( ฮพ * + ฮดฮพ ( t )) = ๐ we need to include in ฮดฮพ ( t ) a correction: ฮดฮพ ( t ) = t ๐ + O ( t ) but still(3.10) ddt ๐ ๐พ ( ฮพ * + ฮดฮพ ( t )) โ h โจโ ๐ ๐ ( ๐ ) | ๐ = h ๐พ , ๐ โฉ โ Then the set ๐ฏ ( ๐ ) := { t : | t | โค ๐ , |๐ ( ๐ ( t )) โ ๐ | โค ๐ h } is an interval of thelength โ ๐ and then the union of such sets over ๐ = h ๐พ + ฮพ with indicated ๐พ does not exceed R ๐ with(3.11) R := โ๏ธ ๐พ |โจโ ๐ ๐ ( ๐ ) | ๐ = h ๐พ , ๐ โฉ| โ , where we sum over set { ๐พ : ๐พ โ ๐พ * | โณ h โ ๐ & | ๐ ๐พ ( h ๐พ ) โ ๐ | โค Ch } . Thelast restriction is due to the fact that ๐ฏ ( ๐ ) ฬธ = โ only for points with | ๐ ๐พ ( h ๐พ ) โ ๐ | โค Ch .One can see easily that R โ h โ | ๐ ๐๐ h | . Then, as R ๐ โค ๐ โฒ the set [ โ ๐ , ๐ ] โ โ๏ธ ๐พ ๐ฏ ( h ( ๐พ + ฮพ )) contains an interval of the length โ = ๐ and forall t , belonging to this interval,(3.12) | ๐ ๐พ ( h ( ๐พ + ฮพ + ฮดฮพ ( t ))) โ ๐ | โฅ ๐๐ h . . Proof of Theorem 1.3 ๐ = ๐ R โ = ๐ h | ๐ ๐๐ h | โ and for d = ๐ค as far asnon-resonant are concerned, Theorem 1.3 is almost proven . Case d โฅ . In this case we need to be more subtle and to make ( d โ steps. We start from the point ๐ * = h ( ๐พ * + ฮพ * ) ; again without any loss ofthe generality we assume that ฮพ * is strictly inside ๐ช * (at the distance atleast C ๐ * from the border). Then after each step below it still will be thecase (with decreasing constant). Step I . On the first step we select ๐ = ๐ and consider only ๐พ such that(3.8) holds; more precisely, the left-hand expression needs to be greater thanthe right-hand expression, multiplied by ๐ . Then R โ h โ d and thereforeexists ฮพ * such that ๐ ๐พ * ( ฮพ * ) = ๐ and | ๐ ๐พ ( ฮพ * ) โ ๐ | โฅ ๐๐ h with ๐ = ๐ h d โ forall ๐พ indicated above. Step II . On the second step we select ๐ = ๐ perpendicular to ๐ . Topreserve inequality (3.12) (with smaller constant ๐ ) for ๐พ , already coveredby Step I , we need to take | ฮด ๐ | โค ๐ โฒ ๐ and consider ฮดฮพ = t ๐ + O ( t ) .Then the same arguments as before results in inequality (3.12) with ๐ := ๐ = ๐ R โ ๐ for a new bunch of points. Then for d = ๐ฅ as far asnon-resonant are concerned, Theorem 1.3 is almost proven . Next steps . Continuing this process, on k -th step we select ๐ k orthogonalto ๐ , ... , ๐ k โ . Then we get ๐ k = ๐ R โ ๐ k โ and on the last ( d โ -th stepwe achieve a separation at least ๐ d โ = ๐ R โ d . Remark 3.2.
In Subsection 4.1 we discuss how to increase ๐ for d โฅ . Almost antipodal points.
We need to cover points with | ๐ โ ๐ * k | โค h โ ๐ for k = ๐ฃ, ... , ๐ค p โ and as we already know for each k (and fixed ฮพ ) thereexists no more than one such point ๐ = h ( ๐พ + ฮพ ) with | ๐ ๐พ ( ฮพ ) โ ๐ | โฒ h ๐ฟ .We take care of such points during Step I . Observe that during this stepwe automatically take care of any point with(3.13) |โ ๐ ๐ ( ๐ ), ๐ โฉ| โฅ ๐ h , We need to cover almost antipodal points and it will be done in the end of thissubsection. We need to consider resonant points and as well, and it will be done in thenext subsection.
One can see easily, that the opposite holds. . Proof of Theorem 1.3 | t | โค ๐ with sufficiently small ๐ = ๐ ( ๐ ) .Let us select ๐ so that on ๐ quadratic forms at points ๐ * , ... , ๐ * p โ incondition (1.14) are different from one at point ๐ * by at least ๐ . Then foreach j = ๐ฃ, ... , ๐ค p โ the the measure of the set { t : | t | โค ๐ , | ๐ ๐พ j ( ฮพ + ฮดฮพ ( t )) | โค ๐ h } does not exceed Ch โ ( ๐ h ) , and then the measure of the union of such sets(by j ) also does not exceed and therefore for ๐ = ๐ h d โ (for d โฅ ) and ๐ = ๐ h | ๐ ๐๐ h | โ (for d = ๐ค ) with sufficiently small ๐ we can find t : | t | โค ๐ so that condition (3.8) is fulfilled for all non-resonant points. Next on this step we need to separate ๐ ๐พ * ( ฮพ ) from all ๐ n ( ฮพ ) (save one,coinciding with it) by the distance at least ๐ h by choosing ฮพ . We canduring the same steps as described in the previous section: let ๐ ๐พ , j ( ฮพ ) denoteeigenvalues of ๐ ๐พ ( ฮพ ) with j = X ( ๐พ h ) .Observe that both ๐ ๐พ ( ฮพ ) and X ( ๐พ h ) depend on the equivalency class [ ๐พ ] of ๐พ rather than on ๐พ itself and that(3.14) โ๏ธ [ ๐พ ] X ( ๐พ h ) = โ๏ธ โค n โค d โ n ) = O ( h โ d + ๐ โฒ + ๐ / h โ d โ ๐ ), where on the left [ ๐พ ] runs over all equivalency classes with ๐พ โ โ๏ธ โค n โค d โ ๐ฃ n .We also observe that for resonant points(3.15) | ๐๐๐ ๐ ( ๐ , ๐ * ) | โฅ ๐ h ๐ฟ and therefore for ๐ โฒ ๐พ , which are eigenvalues of ๐ ( h ( ๐พ + ฮพ )) (3.10) holdsand signs are the same for ๐พ in the same block. On the other hand,(3.16) | ddt โฌ ( h ( ๐พ + ฮพ * + ฮดฮพ ( t )) | โค C ๐ h โช h ๐ฟ โฒ and therefore for ๐ ๐พ , j ( t ) which are eigenvalues of ๐ [ ๐พ ] ( ฮพ ) (3.10) sill holds.Therefore the arguments of each Step I , Step II etc extends to resonantpoints as well. However the number of new points to be taken into account
Recall, that ๐ is diagonal matrix. . Discussion R needs to be redefined(3.17) R := h โ d + ๐ / h โ d โ ๐ . This leads to the final expression (1.15) for ๐ . Theorem 1.3 is proven. ๐ Can we improve (increase) expression for ๐ given by (1.15)? I do not knowif one can do anything with the restriction ๐ โค ๐๐ โ ( d โ h d ( d โ which isdue to resonant points, but restriction ๐ โค ๐ h ( d โ could be improved for d โฅ , which makes sense only if(4.1) h โค h โค h โ ๐ . Indeed, on Step n , n โฅ , we need to take into account only non-resonantpoints belonging to the set(4.2) ๐ฅ := h (๐ * + ฮพ ) โฉ { ๐ : |๐ ( ๐ ) โ ๐ | โค C ๐ n โ h } . Determination of upper estimate for such number falls into realm of theNumber Theory. I am familiar only with the estimate(4.3)
๐ฅ โค Ch โ d ๐ n โ + Ch โ d โ / ( d +๐ฃ) , which follows from Theorem at page 224 of [Gui]. Probably it was improved,but those improvement have no value here.The second term in the right-hand expression of (4.2) is larger, however,the second term in the right-hand expression of (3.17) is still larger andtherefore on each Step n โฅ we have R := ๐ / h โ d โ ๐ , and this leads us tothe following improvement of Theorem 1.3: Theorem 4.1.
In the framework of Theorem 1.1, under additional assump-tions d โฅ and (4.1) , the statement of Theorem 1.3 holds with (4.4) ๐ = ๐๐ โ d โ / h d โ d โ โ ๐ with arbitrarily small exponent ๐ > . In particular, ๐ = ๐ h d โ d / โ ๐ for ๐ = h . . Discussion Remark 4.2.
One can try to improve further (4.4) for ๐ โค h . In this caseresonant points become the main obstacle. In the definition of resonantpoints we need to take ๐ = h / โ ๐ฟ ; however only resonant points ๐ with X ( ๐ ) โฉ { ๐ โฒ : |๐ ( ๐ โฒ ) โ ๐ | โค C ( ๐ n โ h + ๐ ) } ฬธ = โ should be taken into account on n -th step. We know that for connected component ๐จ ๐ this condition (1.14) is fulfilledautomatically.On the other hand, let ๐จ ๐ = โ๏ธ โค j โค p ๐จ ( j ) ๐ with connected ๐จ ( j ) ๐ . Let p = ๐ค and condition (1.14) be violated at each point of ๐จ ๐ . Does it mean that ๐จ (๐ค) ๐ = ๐จ (๐ฃ) ๐ + ๐ (i.e. ๐จ (๐ฃ) ๐ shifted by ๐ )?Next, let ๐จ ( j ) ๐ = ๐จ (๐ฃ) ๐ + ๐ j for j = ๐ค, ... , q . Then for these componentsinstead of condition (1.14) one can impose the similar condition involvinglevel surfaces ๐ ๐ , ๐ of ๐ ( ๐ ) + ๐ B ( ๐ ) . This would affect only Step I of ouranalysis, leading to ๐ := ๐๐๐( ๐ h , ๐ ) with ๐ defined without taking intoaccount of antipodal points. Since ๐ โค h d โ anyway, under assumption ๐ โฅ h we get the same formulae for ๐ for d โฅ as stated in Theorems 1.3and 4.1, while for d = ๐ค we get(4.5) ๐ = h ๐๐๐( ๐ , ๐ โ / h ๐ ). Definitely our result would follow from the asymptotics of the density ofstates ๐ญ โฒ h ( ๐ ) := d ๐ญ h ( ๐ ) d ๐ = ( ๐ โฒ ( ๐ ) + o (๐ฃ)) h โ d as h โ +๐ข, (4.6)where ๐ญ h ( ๐ ) = โซ๏ธ ๐ช * ๐ญ h ( ฮพ , ๐ ) d ฮพ (4.7)is an integrated density of states: ๐ญ h ( ฮพ , ๐ ) = { ๐ < ๐ , ๐ is an eigenvalue of A h ( ฮพ ) } . (4.8) ibliography ๐ญ h ( ฮพ , ๐ ) has a complete asymptotics (see f.e. [Ivr3]), wedo not know anything about asymptotics of ๐ญ โฒ h ( ๐ ) (even ๐ญ โฒ h ( ๐ ) โ h โ d isunknown). Bibliography [Gui] V. Guillemin,
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