Scattering for anisotropic potentials

SCA TTERING F OR ANISOTR OPIC POTENTIALS EV GENY K OROTY AEV Abstract. W e consider the scattering for the op erator H = H o + V , where the unp erturb ed op erator H o is not assumed to be elliptic and the p otential V is anisotropic. Under some conditions on H o and V w e show that the wa v e op erators for H o , H exist and are complete, H has no singular contin uous spectrum and the eigenv alues of H can accumulate only to zero. F or stronger conditions on V the op erator H has finite n umber of eigenv alues only . Moreo ver, these results are applied to the in v ariance principle and for time-dep endent p otentials. 1. Intr oduction and main resul ts 1.1. In tro duction. W e consider the scattering for an operator H = H o + V on L 2 ( R d ), where H o = P ( − i ∇ ) and V ( x ) , x ∈ R d is an anisotropic p oten tial satisfying (1.3). W e denote the co ordinate v ariable by x , and the conjugate momen tum v ariable by k . Here P ( k ) is a real function of k ∈ R d satisfying the condition Condition P . The function P ( k ) , k ∈ R d is r e al and for some a j > 1 and 0 ⩽ j − ⩽ j + ⩽ ν has the form P ( k ) = ν X 1 p j ( k j ) , k = ( k j ) ν j =1 ∈ ⊕ ν 1 R d j = R d , p j ( k j ) =      | k j | a j , j = 1 , 2 , .., j − | k j | a j sign k j j − < j ⩽ j + , d j = 1 −| k j | a j j + < j ⩽ ν . (1.1) There are a lot of results about the scattering under the condition V ( x ) = O ( | x | − q ) as | x | → ∞ , where q > 1 see e.g. [33]. W e call such p otential isotropic. The completeness of the wa v e op erators for anisotropic cases was prov ed by Deich-Korot y aev-Y afaev [8], where the main gradien t is imbedding theorems of Korot y aev-Y afaev [23]. Our goal is to study the scattering for H o , H , singular-con tinuous spectrum of H and its eigen v alues. Roughly sp eaking w e use ”mixed approac h”: the Enss method [9] for the first v ariable, Kato’s smooth metho d [13] to the second v ariable plus a priori estimates. W e shortly describe our main results: • the w a ve op erators for H o , H exist and are complete, singular-contin uous sp ectrum of H is absen t, its eigenv alues can accum ulate only to zero; • H has finite n um b er of eigen v alues, under stronger conditions on p otentials V , • the inv ariance principle for anisotropic case, • scattering for time-dep endent anisotropic p otentials. 1.2. Main results. Consider the self-adjoin t operators H o = P ( − i ∇ ) , H = H o + V acting on the Hilb ert space H = L 2 ( R d ), where P satisfies Condition P and the p otential V ∈ L q ∪ L ε , Date : March 24, 2026. 2020 Mathematics Subje ct Classific ation. 34A55, (34B24, 47E05). Key wor ds and phr ases. scattering, anisotropic p otential, time-dep endent p oten tials. 1 2 EV GENY KOR OTY AEV defined b elow. Here P ac ( H o ) = 1 1 is the identit y op erator. F or the op erators H o , H and an op en in terv al ω ⊂ R we introduce the w a ve op erators W ± ( H , H o , ω ) = s − lim e itH e − itH o E o ( ω ) as t → ±∞ , (1.2) where E o ( · ) is the sp ectral pro jector of H o . If W ± ( H , H o , ω ) H = E ( ω ) H ac , where E ( · ) is the sp ectral pro jector of H , then the w a ve op erators are called complete. F or the multi-index ε = ( ε j ) ν 1 , ε j ⩾ 0 we define the function ϱ ε ( x ) by ϱ ε ( x ) = ν Y j =1 ϱ ε j j ( x ) , ϱ j = ⟨ x j ⟩ := (1 + x 2 j ) − 1 2 , x = ( x j ) ν 1 ∈ ⊕ ν 1 R d j = R d . W e introduce the space L ε of anisotropic functions and the space L q of isotropic functions. Definition. i) L et L ε denote the sp ac e of r e al functions f ∈ L ∞ ( R d ) such that ϱ − ε f ∈ L ∞ ( R d ) , sup | x | = r | ϱ − ε ( x ) f ( x ) | → 0 as r → ∞ , (1.3) wher e the multi-index ε = ( ε j ) ν 1 b elongs to E ± or E o given by E ± = T j ∈ J ± E j , E o = { ε ∈ R ν + : r > 1 , r j > 1 2 for some j ∈ J } , J = J ν = { 1 , .., ν } , E j = n ε ∈ R ν + : ε j + r − r j > 1 o \ n ε ∈ R ν + : ε j ⩽ 1 2 , r m ⩽ 1 2 , for some m ∈ J \ { j } o , j ∈ J , r = P ν 1 r j , r j = 1 2 a j min { 2 ε j , d j } , J + = { 1 , 2 , .., j + } , J − = { j − − 1 , j − , .., ν } . ii) If ε = q > 1 , then we set L q = L q . L et L ε,q denote the sp ac e of r e al functions f = f ε + f q , wher e ( f ε , f q ) ∈ L ε × L q . Here and b elow the multi-index ε = ( ε j ) ν 1 ∈ R ν + . W e presen t the main results. Theorem 1.1. Consider H = H o + V on L 2 ( R d ) , d ⩾ 2 , wher e the op er ator H o = P ( − i ∇ ) . i) L et the p otential V ∈ L ε,q , ε ∈ E ± , q > 1 and the interval ω = R ± . Then the wave op er ators W ± ( ω ) in (1.2) exist and ar e c omplete, H has no singular c ontinuous sp e ctrum, its eigenvalues have finite multiplicity and c an ac cumulate only to zer o. ii) If in addition ε ∈ E o , then H has a finite numb er of eigenvalues, c ounte d with multiplicity. Remark. The existence of the w a v e op erators is prov ed by the stationary phase metho d under the condition ε 1 + .... + ε ν > 1, see e.g. [33]. The completeness of the wa v e op erators needs some additional assumptions [8], see Examples in Section 2. Our conditons on potentials is stronger, in general, than the conditions from [8], but for sp ecific dimensions d j , j ∈ N ν they coincide. Here all p oten tials are b ounded, how ev er, the results can b e extended to the case of p oten tials with lo cal singularities, see Examples in Section 2. There are many examples whic h are not co ve r by the previous considerations. W e formulate invarianc e principle for the following op erators T = T o + V , T o = f ( H o ) , H o = P ( − i ∇ ) , where the anisotropic p otential V ∈ L q or V ∈ L ε and the function h satisfies Condition IP . W e denote the range of P ( k ) , k ∈ R d b y Γ, where Γ = [0 , ∞ ) or Γ = R . Condition IP 1) The r e al function f ∈ L ∞ loc (Γ) and | f ( λ ) | → ∞ as | λ | → ∞ , λ ∈ Γ . 2) L et ω ⊂ Γ b e some interval and the mapping f : ω → Ω = f ( ω ) b e a bije ction, f ∈ C m ( ω ) , m > 3 and f ′ ( λ ) > c for any λ ∈ ω and some c > 0 . Our second main result is form ulated for the op erators T 0 = f ( H o ) and T = T 0 + V . SCA TTERING FOR ANISOTROPIC POTENTIALS 3 Theorem 1.2. ( Inv ariance principle. ) L et a function f satisfy Condition IP and let a p otential V ∈ L ε,q , wher e ε ∈ E ± , q > 1 and an interval ω ⊂ R ± . Then i) the wave op er ators W ± ( T , T o , Ω) exist and ar e c omplete, ii) σ sc ( T ) ∩ Ω = ∅ and eigenvalues of T , b elonging to Ω have finite multiplicities and c an ac cumulate only to the ends of the interval Ω . 1.3. Time-dep endent p otentials. Now we consider the scattering for the time-dep enden t Sc hr¨ odinger equation with the Hamiltonian H ( t ) = H o + V t on L 2 ( R d ) given by i ∂ u ( t ) ∂ t = H ( t ) u ( t ) , u (0) = u o ∈ L 2 ( R d ) , where H o = P ( − i ∇ ) is the unp erturb ed op erator as b efore and the p oten tial V t ( x ) is time- dep enden t, anisotropic. W e assume that the p otential V t ( x ) , t ∈ R satisfies Condition VT. The function V t ( x ) ∈ L ∞ ( R d × R ) is r e al, the mapping t → V t is the c ontin- uously differ entiable L ∞ ( R d ) –value d function. In order to discuss scattering we need to introduce the propagator for our Hamiltonian. A pr op agator is a t wo-parameter family of unitary op erators U ( t, s ), t, s ∈ R , acting on H and satisfying the following conditions: • U ( t, s ) U ( s, r ) = U ( t, r ) for al l t, s, r ∈ R , • U ( t, t ) = 1 1 for al l t ∈ R , wher e 1 1 is the identity op er ator in H , • U ( t, s ) is str ongly c ontinuous in t, s ∈ R . Our Hamiltonian H ( t ) = H o + V t , t ∈ T is a family of time-dep endent op erators on H and V t is a b ounded op erator. Suppose a propagator U ( t, s ) leav es D = D ( H o ) inv arian t and that d dt U ( t, s ) f = − iH ( t ) U ( t, s ) f , f ∈ D , (1.4) in the strong sense. Then the family U ( t, s ) is called the pr op agator for H ( t ). In this situation w e ha ve d ds U ( t, s ) f = iU ( t, s ) H ( s ) f , f ∈ D , (1.5) where the deriv ative has the strong sense. Due to well kno wn results ab out time-dep enden t Hamiltonians (see e.g. Theorems X.70, X.71 in [32]) for our Hamiltonian H ( t ) under the Condition VT there exists the propagator U ( t, s ). The propagator U o ( t, s ) for H o has the form U o ( t, s ) = e − i ( t − s ) H o . Theorem 1.3. L et H ( t ) = H o + V ( t ) , wher e H o = P ( − i ∇ ) and a p otential V t satisfy Con- ditions VT and the fol lowing | V t ( · ) | ⩽ g ( t ) ϱ ε , where ε ∈ E o , ⟨ t ⟩ − γ g ∈ L 2 ( R ) , (1.6) for some γ > 0 such that γ + r > 1 2 . Then ther e exist the unitary wave op er ators given by W ± = s − lim U (0 , t ) e − itH o as t → ±∞ . (1.7) No w we discuss the scattering for time p erio dic Hamiltonian H ( t ) = H o + V t , t ∈ T = R / Z , where H o = P ( − i ∇ ) is the op erator as b efore. W e assume that the p oten tial V t ( x ) , t ∈ T = R / Z is 1-p erio dic in time, i.e., V t +1 = V t for any t ∈ R and satisfies Condition VT. Here w e in tro duce the additional p erio dicity condition for the propagators: • U ( t, s ) is 1 –p erio dic, i.e. U ( t + 1 , s + 1) = U ( t, s ) for al l t, s ∈ R . It is w ell known that for our Hamiltonian H ( t ) there exists the propagator U ( t, s ), which is 1-p erio dic, i.e. it ob eys U ( t + 1 , s + 1) = U ( t, s ) for all t, s ∈ R , see [10], [36]. Introduce the mono drom y op erators M = U (1 , 0) and M o = e − iH o for H ( t ) , H o resp ectiv ely . W e present our main results ab out time p erio dic scattering. 4 EV GENY KOR OTY AEV Theorem 1.4. L et H ( t ) = H o + V ( t ) , wher e V t ( · ) is 1-p erio dic in time and ob eys Condition VT and the fol lowing | V t ( · ) | ⩽ F , where F ∈ L ε,q , and ε ∈ E − ∩ E + , q > 1 . Then the wave op er ators W ± = s − lim U (0 , t ) e − itH o as t → ±∞ , (1.8) exist and ar e c omplete, i.e., W ± H = H ac ( M ) , σ sc ( M ) = ∅ and eigenvalues of M have finite multiplicity and c an ac cumulate only to 1. If in addition, F ∈ L ε , ε ∈ E o , then M has a finite numb er of eigenvalues, c ounte d with multiplicity. 1.4. Related w orks. The case of isotropic p otentials and simply characteristic p olynomials P w as studied b y Agmon and Hormander [4]. They show that there exist the complete wa v e op erators, singular contin uous sp ectrum is absen t and the p oin t sp ectrum is in v estigated. In our anisotropic case the completeness of the wa v e op erators was prov ed b y Deic h, Korot yaev and Y afaev [8], where the smo oth tec hnique of Kato [13], La vine [24] and the im b edding theorems of Koroty aev and Y afaev from [23] were applied. Later the scattering for the isotropic p oten tials and simply characteristic p olynomials P was studied by Muth uramalingam [27], [28] and P ascu [30] by the Enss metho d. Scattering for Stark op erators p erturb ed by anisotropic p oten tials w as discussed by Koroty aev and Pushnitski [22]. Note that there exists a b o ok of P erry [31], where the Enss metho d are applied to v arious op erators. There are many results devoted to scattering mainly for self-adjoin t time-dep endent Hamil- tonians H ( t ) = − ∆ + V t ( x ), on R d , d ⩾ 1. Mainly articles are dev oted for time-p erio dic Hamiltonians and to the sp ectral analysis of the corresp onding mono dromy op erator. Com- pleteness of the wa v e op erators for H o , H ( t ) w as established b y Y a jima [36]. Later on it w as sho wn that the mono dromy op erator do es not hav e singular contin uous sp ectrum, see [10], [15], [16] and it has finite num b er of eigen v alues [16]. The case of Sc hro dinger op erators with time-p erio dic electric and homogeneous magnetic field w as discussed in [15], [17], [35], see also recent pap ers [1, 2, 3, 14, 25, 37]. Moreov er, scattering for three b o dy systems was considered in [17, 26, 29]. There are o lot of results ab out scattering for discrete Schr¨ odinger op erators, see [6], [11]. There are only few results ab out time p erio dic Hamiltonians on the graphs [12, 19, 20, 21]. 2. Preliminaries Recall results from [8], devoted to the more general case. Belo w we assume that ε ∈ R ν + . Prop osition 2.1. i) L et H = H o + V , wher e H o = − ∆ and ω = R + . L et the p otential V b elongs to L q , q > 1 or L ε , ε ∈ K + , wher e ε ∈ K + is given by K + = \ j ∈ J K j , K j = { ε ∈ R ν + : ε j + e r − e r j > 1 } , e r j = 1 2 min { ε j , d j } , e r = X i ∈ J e r i . (2.1) Then the wave op er ators W ± ( H , H o , ω ) in (1.2) exist and ar e c omplete. ii) L et H = H o + V , wher e H o = − ∆ 1 + ∆ 2 and ω = R + . L et the p otential V b elongs to L q , q > 1 or L ε , ε ∈ K + , wher e ε ∈ K + given by K + = { ε ∈ R 2 + : ε 1 + e r 2 > 1 } . (2.2) Then the wave op er ators W ± ( H , H o , ω ) in (1.2) exist and ar e c omplete. SCA TTERING FOR ANISOTROPIC POTENTIALS 5 Remark that from Prop osition 2.1 and Theorem 1.1 we deduce that K + ⊆ E + , but b elo w w e sho w that K + = E + for sp ecific cases. 2.1. Examples. W e illustrate Theorem 1.1 b y few examples. Example 1. Let H = H o + V in L 2 ( R d ), where H o = − ∆ and the p oten tial V ∈ L ε . A) The general case: ν ⩽ d and ω = R + . Recall that a j = 2 and r j = 1 4 min { 2 ε j , d j } ⩽ d j 4 and ε j ⩾ 0 , j ∈ J ν and r = P r j . Here we ha ve set E + = ∩ ν 1 E j , where E j , j ∈ J ν is given by E j = n ε j + r − r j > 1 o \ n ε j ⩽ 1 2 , r m ⩽ 1 2 , for some m ∈ J ν \ { j } o , (2.3) so that r j ⩽ 1 2 if d j ⩽ 2 and r j > 1 2 if d j ⩾ 3 , ε j > 1. Comparing (2.3), (2.1) we ha v e that E + ⊆ K + . Thus if ε ∈ K + , then the wa v e op erators W ± ( H , H o ) are complete, if in addition ε ∈ E + , then σ sc ( H ) = ∅ and eigen v alues of H can accumulate only to zero. B) Discuss the simple case: ν = d . Consider K + = ∪ K j for the case ε j = ε 1 < 1 for all j ∈ J ν . Th us we hav e K j = { ε j + e r − e r j > 1 } = { ε j = ε 1 > 1 d } for all j ∈ J ν . By Prop osition 2.1, if ε ∈ K + , then the wa v e op erators W ± ( H , H o ) exist and are complete. F rom (2.3) we obtain r j ⩽ 1 4 , E j = { ε j + r − r j > 1 } \ { ε j ⩽ 1 2 } ⊂ { ε j = ε 1 > 1 2 } , j ∈ J ν , whic h yields r j = 1 4 , r = d 4 , E j = { 4 ε j > max { 2 , 5 − d }} . In particular, w e hav e E + = { ε j > 1 2 , ∀ j ∈ J ν } for d ⩾ 3. By Theorem 1.1, if ε ∈ E + , then σ sc ( H ) ∩ ω = ∅ and eigenv alues of H can accum ulate only to zero. Note that we ha v e E + ⊂ K + . C) F or simplification we consider the case ν = 2 , d = d 1 + d 2 . • If ε j ⩽ d j 2 , j = 1 , 2, then we ha v e E + = { ε 1 + ε 2 2 > 1 , ε 1 2 + ε 2 > 1 } = K + , E o = { ε 1 + ε 2 > 2 } . (2.4) • If d j > ε j > d j 2 , j = 1 , 2, then r j = d j 2 and we hav e E 1 = { ε 1 + d 2 4 > 1 } \ { ε 1 < 1 2 , d 2 4 ⩽ 1 2 } = { ε 1 + d 2 4 > 1 } , E 2 = { ε 2 + d 1 4 > 1 } , K + = { ε 1 + ε 2 2 > 1 , ε 1 2 + ε 2 > 1 } ⊂ E + = E 1 ∩ E 2 . (2.5) • If ε ∈ E o , then by Theorem 1.1, the num b er of eigenv alues of H is finite and here E o = { r 1 + r 2 > 1 } \ { r 1 ⩽ 1 2 or r 2 ⩽ 1 2 } = { r 1 + r 2 > 1 } . Then d = d 1 + d 2 ⩾ 5 and max { r 1 , r 2 } > 1 2 . If r 1 > 1 2 , then ε 1 > 1 , d 1 ⩾ 3 and we get E o = { r 1 + r 2 > 2 , ε 1 + d 2 2 > 2 , ε 2 + d 1 2 > 2 } . Example 2. Let H = H o + V , where H o = − ∆ 1 + ∆ 2 on L 2 ( R d ) and the p oten tial V ∈ L ε . Consider the case ν = 2 , d = d 1 + d 2 W e are interesting in the p ositive part of the op erator H and thus ω = R + . By Theorem 2.1, if ε ∈ K + = { ε 1 + e r 2 > 1 } , then the wa v e op erators W ± ( H , H o , R + ) exist and are complete. By Theorem 1.1, if ε ∈ E + , then σ sc ( H ) ∩ ω = ∅ and p ositiv e eigenv alues of H , b elonging to ω , ha ve finite multiplicit y and can accum ulate only to zero. Here we ha ve E + = E 1 = { ε 1 + r 2 > 1 } \ { ε 1 ⩽ 1 2 , r 2 ⩽ 1 2 } = { ε 1 + r 2 > 1 } . (2.6) Note that if ε 2 ⩾ d 2 , then we ha ve K + = E + . 6 EV GENY KOR OTY AEV Example 3. W e illustrate Theorem 1.2 by an example. Let H o = − ∆ , H = − ∆ + V on L 2 ( R d ) , d = d 1 + d 2 , where the p oten tial V ∈ L ε , ε ∈ E + and the set E + is defined in Example 1. Define a function and in terv als h ( λ ) = (1 + λ 4 ) r > 0 , λ ∈ ω = R + , r > 0 , Ω = (1 , ∞ ) . Then due to Theorem 1.2 and Example 1, for the op erators T 0 = h ( H o ) , T 1 = T 0 + V w e hav e: 1) the wa v e op erators W ± ( T 1 , T 0 , Ω) exist and are complete, 2) σ sc ( T 1 ) ∩ Ω = ∅ and eigen v alues of T 1 , b elonging to Ω can accum ulate only at 1 and ∞ . Example 4. Let H o = − ∆ 1 + ∆ 2 , H = H o + V on L 2 ( R d ) , d = d 1 + d 2 , where the p otential V ∈ L ε , ε ∈ E + and the set E + is defined by (2.6). Define the function and an in terv al h ( λ ) = sinh λ, λ ∈ R , Ω = ω = R + . Then due to Theorem 1.2 and Example 2, for the op erators T 0 = h ( H o ) , T 1 = T 0 + V w e hav e: 1) the wa v e op erators W ± ( T 1 , T 0 , Ω) exist and are complete, σ sc ( T 1 ) ∩ Ω = ∅ , 2) eigenv alues of T 1 , b elonging to Ω, hav e finite multiplicit y and can accumulate only at 1 and + ∞ . 2.2. Preliminary results. W e denote the class of all b ounded and compact op erators in a Hilb ert space H b y B ( H ) and B ∞ ( H ) resp ectively . Consider self-adjoin t op erators H o , H = H o + V acting on the Hilb ert space H , where V is a b ounded op erator and P ac ( H o ) = 1 1. W e form ulate conditions on H o and V and the interv al ω ⊂ R . Condition 1. The op er ator V ( H o − i ) − 1 ∈ B ∞ . 2. L et an interval ω ⊂ R . F or any φ ∈ C ∞ 0 ( R ) such that 0 ⩽ φ ⩽ 1 and supp( φ − 1) = ω ther e exist b ounde d op er ators ζ j , φ j , j ∈ J ν for some ν ∈ N such that φ ( H o ) = ν X 1 ζ j φ j , (2.7) wher e al l op er ators H o , ζ j , φ j , j ∈ N ν ar e c ommute. F or e ach j ∈ J ν ther e exist two b ounde d op er ators Q ± j such that Q + j + Q − j = 1 1 and the fol lowing holds true: s − lim Q ± j ∗ e ± itH o = 0 as t → ∞ . (2.8) 3. F or any ( j, f ) ∈ J ν × H and every r ⩾ 1 the fol lowing estimate holds true Z t>r ∥ V e ∓ itH o ζ j Q ± j f ∥ dt ⩽ C r ∥ f ∥ , wher e the c onstant C r do es not dep end on f and C r → 0 as r → ∞ . 4. F or some m ∈ J ν and any f ∈ L 2 ( R d ) , r ⩾ 1 the fol lowing estimate holds true: Z t>r ∥ ϱ ε e ∓ itH o Q ± m f ∥ dt ⩽ C r ∥ f ∥ , (2.9) wher e the c onstant C r → 0 do es not dep end on f and C r → 0 as r → ∞ . F rom Condition 1 the standard arguments yield that for any φ ∈ C ∞ 0 ( R ) we hav e φ ( H ) − φ ( H o ) ∈ B ∞ , (2.10) see e.g., [32]. F rom Conditions 1 and 3 we obtain K ± j := φ ( H )( W ± ( ω ) − 1 1) ζ j Q ± j ∈ B ∞ , ∀ j ∈ J ν . (2.11) W e describ e scattering for H o , H and the sp ectrum of H . SCA TTERING FOR ANISOTROPIC POTENTIALS 7 Theorem 2.2. i) L et H = H o + V , wher e H o = P ( − ∇ ) and the p otential V ob eys Con- ditions 1-3 for some interval ω ⊂ R . Then the wave op er ators W ± ( H , H o , ω ) exist and ar e c omplete, σ sc ( H ) ∩ ω = ∅ and eigenvalues of H , b elonging to ω , have finite multiplicity and c an ac cumulate only at the ends of the interval ω . ii) L et ε ∈ E o , wher e ε m > 1 2 for some m ∈ J ν . L et Conditions 1,4 and (2.8) hold true for j = m . Then op er ators H has finite numb er of eigenvalues on any b ounde d interval. Pro of. i) Note that the existence of the wa v e op erators is a simple fact and it is established b y the stationary phase metho d [33]. In the pro of we use the known Enss approac h from [9], mo difed for our case. Let W + = W + ( ω ) for shortness. W e show that W + H = H ac , the pro of for W − is similar. The pro of is based on the con tradiction. Let f ∈ H ac ⊖ W + H and f  = 0. Th us w e can assume that E ( ω ) f = f for some small interv al ω ⊂ R + , the pro of for ω ⊂ R − is similar. Let 0 ⩽ φ ∈ C ∞ 0 ( R ) b e such that φ | ω = 1 , and supp φ ⊂ R + . (2.12) Let f n = e it n H f for some increasing sequence t n > 0 , n ∈ N suc h that t n → + ∞ as n → ∞ . Condition 1 and (2.10) yield (1 1 − φ ( H o )) f n = ( φ ( H ) − φ ( H o )) f n and φ ( H ) − φ ( H o ) ∈ B ∞ and then ∥ f ∥ 2 = ∥ f n ∥ 2 = ( f n , φ ( H o ) f n ) + ( f n , ( φ ( H ) − φ ( H o )) f n ) = ( f n , φ ( H o ) f n ) + o (1) (2.13) as n → ∞ . Due to (2.7) we hav e ( f n , φ ( H o ) f n ) = ( f n , P ν 1 ζ j φ j f n ), where ( f n , ζ j φ j f n , ) = X τ = ± ( f n , ζ j Q τ j φ j f n ) = X τ = ± h ( f n , W τ ζ j Q τ j φ j f n ) − ( f n , K τ j f n ) i = X τ = ± ( f n , W τ ζ j Q τ j φ j f n ) + o (1) = ( f n , W − ζ j Q − j φ j f n ) + o (1) , (2.14) since f ∈ H ac ( H ) ⊖ W + H and due to (2.11) we hav e K ± j ∈ B ∞ . Then Condition 2 implies ( f n , W − ζ j φ j Q − j f n ) = ( f , W − e it n H o ζ j Q − j φ j f n ) = o (1) as n → ∞ , which gives the contradiction. In order to sho w other results w e prov e that dim E ( ω )( H ⊖ H ac ) < ∞ . The pro of is based on the contradiction. Let f n ∈ E ( ω )( H ⊖ H ac ) , n ∈ N b e some orthonormal sequence . Thus w e can assume that E ( ω ) f n = f n . Using ab ov e arguments and f n ⊥ W ± H , we obtain 1 = ∥ f n ∥ 2 = X τ = ± ν X j =1 ( f n , W τ ζ j Q τ j φ j f n ) + o (1) = o (1) . ii) Let ε ∈ E 0 . W e show that the num b er of eigen v alues coun ting multiplicit y is finite. It is enough to consider the in terv al ω = ( − 1 , 1). Let 0 ⩽ φ ∈ C ∞ 0 ( R ) b e suc h that φ | ω = 1. F rom Condition 1, (2.10) we hav e φ ( H ) V = ( φ ( H ) − φ ( H o )) V + φ ( H o ) V ∈ B ∞ . Then from Condition 4, we obtain X ± := ± i Z ∞ 0 e ± itH φ ( H ) V e ∓ itH o Q ± m dt ∈ B ∞ . (2.15) 8 EV GENY KOR OTY AEV Let f n ∈ E ( ω ) H , n ∈ N b e some orthonormal sequence of eigen v alues of H . If n → ∞ , then from (2.15) we ha ve the contradiction, since 1 = ∥ f n ∥ 2 = X τ = ± h ( f n , W τ Q τ m f n ) − ( f n , X τ f n ) i = − X τ = ± ( f n , X τ f n ) = o (1) . W e consider the eigenv alues on the in terv al ω = ( s, ∞ ) for large s ≫ 1. W e will show that under Condition 3( ∞ ), the op erator H has not eigen v alues greater than s . The pro of of the case ( −∞ , − s ) is similar. W e in tro duce the smo oth function ζ ∈ C ∞ ( R ) by ζ ( λ ) = ( 1 λ ⩾ 1 0 λ ⩽ 1 2 . (2.16) W e present conditions for the case ω = ( s, ∞ ), where s ⩾ 1. Condition 3( ∞ ). F or any ( j, f ) ∈ J ν × H and s ⩾ 1 lar ge enough the fol lowing estimate holds true: Z R + ∥ V e ∓ itH o ζ ( h j /s ) Q ± j f ∥ dt ⩽ C ( s ) ∥ f ∥ , wher e the c onstant C ( s ) do es not dep end on f and C ( s ) → 0 as s → ∞ . Theorem 2.3. L et ther e exist the wave op er ators W ± ( H , H o , ω ) for some ω = ( s, ∞ ) and let Conditions 3( ∞ ) hold true for some s ⩾ 1 lar ge enough. Then the op er ator H has no eigenvalues gr e ater than s . Pro of. Let f ∈ E ( H , ( b, b + 1)) H b e an eigenv alues of H and ∥ f ∥ = 1, where b ⩾ 2 s and z = b + i . Then ∥ ( H − z ) f ∥ ⩽ 2 ∥ f ∥ and we ha v e ∥ (1 1 − ζ ( H o /s )) f ∥ ⩽ ∥ (1 1 − ζ ( H o /s ))( H − z ) − 1 ∥ 2 ∥ f ∥ ⩽ 2 ∥ f ∥ /s. F or any s large enough the function ζ ( P /s ) has a decomp osition ζ ( P /s ) = ν X 1 ζ ( p j /s ) φ j ( s ) , (2.17) for some b ounded functions φ j ( s, k ) , ( j, k ) ∈ J ν × R d . Thus using (2.17) and rep eating the argumen ts from Theorem 2.2 we obtain 1 = ∥ f ∥ = ( ζ ( H o /s ) f , f ) + O (1 /s ) = ν X 1 ( ζ j ( s ) φ j ( s ) f , f ) + O (1 /s ) , ( ζ j ( s ) φ j ( s ) f , f ) = X τ = ± h ( W τ ζ j ( s ) Q τ j φ j ( s ) f , f ) + ((1 1 − W τ ) ζ j ( s ) Q τ j φ j ( s ) f , f ) i = o (1) , whic h yields a contradiction for s large enough. 3. Pr oof of Theorem 1.1 W e prov e Theorem 1.1 chec king Conditions 1,2,3, and 3 ( ∞ ) for the op erator H o = P ( − i ∇ ) and the p otential V ∈ L ε,q , where ε ∈ E ± , q > 1. W e define op erators Q ± j , j ∈ J ν . F or the case ν = 1 we define the operator T ± from [35] suc h that T − + T + = 1 1 and describ e their prop erties. Let an op erator H o act on L 2 ( R d ) by H o = | ∆ | a 2 , d ⩾ 1 , or H o = | ∂ | a − 1 ∂ , ∂ := − i d dx , d = 1 , (3.1) SCA TTERING FOR ANISOTROPIC POTENTIALS 9 where a > 1. W e denote the F ourier transform of a function f ( x ) , x ∈ R d b y b f ( k ) = (Φ d f )( k ) = (2 π ) − d 2 Z R d e − ikx f ( x ) dx, k ∈ R d . Define the op erator T ± in the sp ectral representation of the space H , when the op erator H o acts on L 2 ( R d ) as a multiplication op erator b y an indep endent v ariable. Definition of T ± . • Consider H o = | ∆ | a 2 , d ⩾ 1 fr om (3.1). The sp e ctr al r epr esentation of H o is given by Ψ = U Φ d , wher e U : L 2 ( R d ) → L 2 ( L 2 ( S d − 1 ) , R + , dt ) has the form ( U f )( λ, µ ) = a − 1 2 λ d − a 2 a f ( λ 1 /a , µ )) , ( λ, µ ) ∈ R + × S d − 1 . The op er ator Ψ H o Ψ ∗ is the multiplic ation op er ator by λ . L et J b e the natur al imb e dding of L 2 ( R + ) into L 2 ( R ) . Thus J ∗ is the natur al r estriction map fr om L 2 ( R ) into L 2 ( R + ) . L et χ ± b e the char acteristic function of the set R ± . We define op er ators T ± and F ± by T ± = F ∗ ± F ± , F ± =  χ ± Φ ∗ 1 ⊗ 1 1  J ∗ U Φ d : L 2 ( R d ) → H ± = L 2 ( L 2 ( S d − 1 ) , R ± , dσ ) . (3.2) L et b σ ± b e the multiplic ation op er ator by the variable σ in H ± . • Consider the op er ator H o = | ∂ | a − 1 ∂ on L 2 ( R ) fr om (3.1). Its sp e ctr al r epr esentation is given by Ψ = U Φ 1 , wher e U : L 2 ( R ) → L 2 ( L 2 ( S 0 ) , R + , dt ) , S 0 = {− 1 , 1 } and ( U f )( λ, ν ) = a − 1 2 | λ | d − a 2 a f ( λ 1 /a ) , λ 1 /a = | λ | 1 /a sign λ. Her e Ψ H o Ψ ∗ is the multiplic ation op er ator by λ . Define the op er ator T ± in terms F ± by T ± = F ∗ ± F ± , F ± = χ ± Φ ∗ 1 U Φ 1 : L 2 ( R ) → H ± = L 2 ( L 2 ( S 0 ) , R ± , dσ ) . (3.3) Let b σ ± b e the multiplication op erator by the v ariable σ in H ± . Below w e need Lemma 3.1. L et an op er ator H o = | ∆ | a 2 on L 2 ( R d ) , d ⩾ 1 or H o = | ∂ x | a − 1 ∂ x on L 2 ( R ) , a > 1 . Then for any δ, t > 0 , s ⩾ 1 the fol lowing estimates hold true ∥⟨ x ⟩ δ e ∓ itH o T ± ∥ ⩽ C ⟨ t ⟩ r , r = 1 a min { δ, d 2 } , (3.4) ∥⟨ x ⟩ δ e ∓ itH o ζ ( H o /s ) T ± ∥ ⩽ C ⟨ t ⟩ δ , (3.5) ∥⟨ x ⟩ δ e ∓ itH o ζ ( H o /s ) T ± ∥ ⩽ C ⟨ ts τ ⟩ δ , τ = a − 1 a , (3.6) wher e the function ⟨ x ⟩ = (1 + | x | ) − 1 , x ∈ R d and ζ is given by (2.16). Pro of . The estimates (3.4), (3.5) were prov ed in [35]. W e shall prov e (3.6) for the case H o = | ∆ | a 2 and the sign ” + ”. The pro of of other cases is similar. Let F ( σ, µ ) = ( F + f )( σ, µ ) , f ∈ H . F rom the definition of T + = F ∗ + F + w e obtain that g ( x ) := ( e − itH o ζ ( H o /s ) T + f )( x ) has the form g ( x ) = C d Z ∞ 0 dσ Z S d − 1 F ( σ , µ ) dµ Z ∞ 0 e i ( x,ν ) λ 1 /a − iλ ( t + σ ) ζ ( λ/s ) λ γ dλ, where γ = d − a 2 a . Integrating b y parts with resp ect to λ we obtain g ( x ) = C d Z ∞ 0 dσ Z S d − 1 dµ Z ∞ 0 e i ( x,µ ) λ 1 /a − iλ ( t + σ ) ζ ( λ/s ) λ 1 − γ  ζ ′ ( λ/s ) iλ − iζ ( λ/s ) γ + ( x, µ ) λ 1 /a a  dλ t + σ . W e rewrite this iden tity in the form g ( x ) = − i  e − itH o  ζ ′ ( H o /s ) + γ ζ ( H o /s ) H o /s  1 s + 1 s τ d X 1 x n e − itH o ( ∇ n / | ∇ | ) ζ ( H o /s ) ( aH o /s )  F ∗ + 1 b σ + + t F + , 10 EV GENY KOR OTY AEV where x = ( x 1 , ..., x d ) ∈ R d , ∇ = ( ∇ 1 , ..., ∇ d ). F rom here we obtain ∥⟨ x ⟩ e − itH o ζ ( H o /s ) T + ∥ ⩽ C ⟨ ts τ ⟩ . In tegrating b y parts m = 0 , 1 , .... times we obtain ∥⟨ x ⟩ m e − itH o ζ ( H o /s ) T + ∥ ⩽ C ⟨ ts τ ⟩ m . Then using the interpolation theorem w e get (3.6). • Definition of T ± j and Q ± j . F or the op er ator h j = p j ( − i ∇ j ) on L 2 ( R d j ) fr om (1.1) we define the op er ator T ± j , j ∈ J + = { 1 , ..., j + } by (3.2), (3.3) in terms of c o or dinate x j ∈ R d j . Lemma 3.1 gives that for the op erator h j = p j ( − i ∇ j ), the function ϱ j = (1 + | x j | 2 ) − 1 2 and for b ounded op erators T ± j , j ∈ J + suc h that T − j + T + j = 1 1 there exist estimates: ∥ ϱ ε j j e ∓ ith j T ± j ∥ ⩽ C ⟨ t ⟩ r j , r j = 1 a j min { ε j , d j / 2 } , (3.7) ∥ ϱ ε j j e ∓ ith j ζ ( h j /s ) T ± j ∥ ⩽ C ⟨ t ⟩ ε j , (3.8) ∥ ϱ ε j j e ∓ ith j ζ ( h j /s ) T ± j ∥ ⩽ C ⟨ ts τ j ⟩ ε j , (3.9) for any ε j , t > 0 , s ⩾ 1, where τ j = a j − 1 a j and some constant C . Define the op erators Q ± j b y ( Q ± j = T ± j , if ε j > 1 2 Q ± j = T ± m f or some m ∈ J + \ { j } if ε j < 1 2 , r m > 1 2 . W e chec k main Condition 3 for our op erator H o and the p otential V . Lemma 3.2. i) L et V ∈ L q , q > 1 or V ∈ L ε , ε ∈ E j for some j ∈ J + . Then for any r , s ⩾ 1 , ϑ > 0 and any f ∈ L 2 ( R d ) the fol lowing estimate holds true: Z ∞ r ∥ V e ∓ itH o ζ ( h j /s ) Q ± j f ∥ dt ⩽ C 1 s − ϑ r − ϑ ∥ f ∥ , (3.10) wher e the c onstant C 1 = C ( V ) dep ends on V only. ii) L et ε ∈ E o . Then for some j ∈ J ν and any f ∈ L 2 ( R d ) the estimate (2.9) holds true. Pro of. i) Discuss the case when V ∈ L ε , ε ∈ E j and the case ” + ”, the pro of of other cases is similar. Define a function ϱ ε ( j ) = ϱ ε ϱ − ε j j , where ϱ ε = Q ν 1 ϱ ε j j . If ε j > 1 2 , then we hav e Q + j = T + j . F rom (3.8), (3.9) for ε j = 1 2 + γ + 2 δ, γ > 0 , 1 > δ > 0 and τ j = a j − 1 a j and F ( t ) := ∥ ϱ ε ( j ) e − itH o f ∥ we obtain Z ∞ r ∥ ϱ ε e − itH o ζ ( h j /s ) Q + j f ∥ dt ⩽ Z ∞ r ∥⟨ x j ⟩ ε j e − ith j ζ ( h j /s ) T + j ∥ F ( t ) dt ⩽ C Z ∞ r t δ − ε j ( ts τ j ) − δ F ( t ) dt ⩽ C r − δ s − δ τ j Z ∞ r t − 1 2 − δ [ t − γ F ( t )] dt. Then using the Sch w artz inequality and (5.1) we get (3.10) under the condition γ + r − r j > 1 2 . The last condition holds true if ε j > 1 2 , ε j + r − r j > 1, i.e., when ε ∈ E j . Note that the pro of for V ∈ L q , q > 1 is similar. If ε j < 1 2 , then w e hav e Q + j = T + m and r m > 1 2 for some m  = j . F or r m = 1 2 + γ + 2 δ, γ > 0 , δ > 0 and F ( t ) := ∥ ϱ ε ( m ) e − ith j ζ ( h j /s ) f ∥ using (3.7), we obtain: Z ∞ r ∥ ϱ ε e − itH o ζ ( h j /s ) Q + j f ∥ dt ⩽ Z ∞ r ∥ ϱ ε m m e − ith m T + m ∥ · F ( t ) dt SCA TTERING FOR ANISOTROPIC POTENTIALS 11 ⩽ C Z ∞ r t − r m F ( t ) dt ⩽ C r − δ Z ∞ r t − δ − 1 2  ⟨ t ⟩ γ F ( t )  dt. F rom here, using the Sc hw artz inequality and (5.2) we get (3.10) under the condition γ + ε j + r − r j − r m > 1 2 . The last condition holds true if ε m > 1 2 , ε j + r − r j > 1, i.e., when ε ∈ E j . ii) Let ε ∈ E 0 . Then r > 1 and some r m > 1 2 , m ∈ N . Th us we hav e r m = 1 2 + γ + 2 δ for some γ > 0 , δ > 0. It follows from (3.7) that for any r ⩾ 1 and F ( t ) := ∥ ϱ ε ( m ) e − itH o f ∥ : Z ∞ r ∥ ϱ ε e − itH o T + m f ∥ dt ⩽ Z ∞ r ∥ ϱ ε m m e − ith m T + m ∥ · F ( t ) dt ⩽ C Z ∞ r ⟨ t ⟩ − r m F ( t ) dt ⩽ C ⟨ r ⟩ − δ Z ∞ r ⟨ t ⟩ − δ − 1 2  ⟨ t ⟩ − γ F ( t )  dt. Then using the Sc h wartz inequalit y and (5.1) w e obtain (2.9) under the condition γ + r − r m > 1 2 and r m = 1 2 + γ + 2 δ . The last condition holds true if r m > 1 2 , r > 1, i.e., when ε ∈ E 0 . Pro of of Theorem 1.1 i) Due to Theorem 2.2 we need to c heck Conditions 1,2,3 for the op erator H o = P ( − i ∇ ), the p otential V ∈ L ε , ε ∈ E + and an interv al ω ⊂ R + . The pro of of other cases is similar. 1 . In [8] it w as prov ed that V ( H o − i ) − 1 ∈ B ∞ . 2 . Let a function φ ∈ C ∞ 0 ( R ) b e such that φ | ω = 1 and supp φ ⊂ R + . F rom the prop erties of P w e get the existence of the smo oth functions φ j , j ∈ J + and the num ber s > 0 such that φ ( P ( k )) = X j ∈ J + ζ ( p j ( k j ) /s ) φ j ( k , s ) . (3.11) Using (3.7), (3.11) w e obtain that Condition 2 holds true for the op erator ζ j = ζ ( h j /s ). 3 . The Condition 3 follows from Lemma 3.2. W e consider the sp ectrum of H at high energy and chec k Conditions 3( ∞ ). F rom the prop erties of P we obtain the decomp osition ζ ( P ( k ) /s ) = X j ∈ J + ζ ( p j ( k j ) /s ) φ j ( k , s ) , | φ j ( k , s ) | ⩽ 1 , for some smo oth functions φ j ( k , s ) , j ∈ J + . F or an y r ⩾ 1 we ha v e V = V 1 + V 2 , where ( V 1 = χ ( | x | < r ) V ( x ) , | V 1 ( x ) | ⩽ C r 2 d ⟨ x ⟩ − D | V 2 ( x ) | ⩽ C 2 ( r ) ⟨ x ⟩ − ε , C 2 ( r ) → 0 as | x | → ∞ , (3.12) and the multi-index D = 2( d j ) d 1 ∈ R d . Here χ ( A ) = 1 , x ∈ A and χ ( A ) = 0 , x / ∈ A for some set A . Then for the function f ( t, s ) = e − itH o ζ ( h j /s ) T + j f from (3.8), (3.9) w e obtain Z ∞ 0 ∥ V f ( t, s ) ∥ dt ⩽ Z ∞ 0  r 2 d ∥⟨ x ⟩ − D f ( t, s ) ∥ + C 2 ( r ) ∥⟨ x ⟩ − ε f ( t, s ) ∥  dt ⩽ C ( s ) ∥ f ∥ , where C ( s ) = C r 2 d s − ϑ j + C 2 ( r ) , ϑ j = 2 d j τ j > 0. T aking s = r 3 d/δ , δ = min ϑ j , we deduce that C ( s ) is small. ii) Lemma 3.2 gives Condition 4. F rom Theorems 2.2, 2.3 we obtain the pro of of ii). 12 EV GENY KOR OTY AEV 4. The inv ariance principle and time periodic potentials 4.1. The inv ariance principle. In order to pro ve the in v ariance principle we need some estimates. Let A, B , V b e self-adjoint op erators in the Hilb ert space H , where V is b ounded. F or a function F ∈ L 1 ( R 2 ) ∩ C ( R 2 ) we define an op erator Z ( F ) = Z R 2 e isB V e iτ A F ( s, τ ) dsdτ . (4.1) If F ( s, τ ) = δ ( s ) F 1 ( τ ), where F 1 ∈ L 1 ( R ) ∩ C ( R ), then the corresp onding op erator is giv en by Z ( F ) = √ 2 π V b F 1 ( − A ) . (4.2) Lemma 4.1. L et ⟨ τ ⟩ − b F ( s, τ ) ∈ L 1 ( R 2 ) ∩ C ( R 2 ) or ⟨ τ ⟩ − b F 1 ( τ ) ∈ L 1 ( R ) ∩ C ( R ) for some b ⩾ 0 . L et η ( t ) = ∥ V e − itA f ∥ for ( t, f ) ∈ R × H . Then for any r ⩾ 0 the fol lowing estimate holds true: Z ∞ r ∥ Z ( F ) e − itA f ∥ dt ⩽ C Z 0 −∞ ⟨| t | + r ⟩ b η ( t ) dt + ⟨ r ⟩ b Z t> 0 η ( t ) dt + Z 2 t>r η ( t ) dt. (4.3) Pro of. Consider F ∈ L 1 ( R 2 ) ∩ C ( R 2 ), the pro of for another case is similar. It follo ws from (4.1) that ∥ Z ( F ) e − itA f ∥ = ∥ Z R 2 F ( s, τ ) e isB V e i ( τ − t ) A dsdτ f ∥ ⩽ Z R 2 | F ( s, τ ) | η ( t − τ ) dsdτ = Z R g ( τ ) η ( t − τ ) dτ , where g ( τ ) := R R | F ( s, τ ) | ds . F rom here w e obtain G r = Z ∞ r ∥ Z ( F ) e − itA f ∥ dt ⩽ Z ∞ r dt Z R g ( τ ) η ( t − τ ) dτ = Z R ϕ ( u ) Z ∞ r g ( t − u ) dt, (4.4) where w ( z ) := Z ∞ r g ( t − u ) dt = Z ∞ 0 g ( t + z ) dt = Z ∞ 0 dt Z R | F ( s, t + z ) | ds, z := r − u. F rom the prop erties of F we obtain w ( z ) ⩽ C ( ⟨ z ⟩ b , z > 0 1 , , z < 0 . (4.5) F rom (4.4), (4.5) w e get G r ⩽ C Z r −∞ η ( u ) ⟨ r − u ⟩ b du + C Z ∞ r η ( u ) du ⩽ C Z 0 −∞ η ( u ) ⟨ r + | u |⟩ b du + C Z r 0 η ( t ) ⟨ r − u ⟩ b du + C Z ∞ r η ( u ) du. Using this and the estimate Z r 0 η ( u ) ⟨ r − u ⟩ b du ⩽ Z r 2 0 η ( u ) ⟨ r − u ⟩ b du + Z r r 2 η ( t ) ⟨ r − u ⟩ b du ⩽ C ⟨ r ⟩ b Z ∞ 0 η ( t ) dt + C Z ∞ r 2 η ( t ) dt SCA TTERING FOR ANISOTROPIC POTENTIALS 13 w e obtain (4.3). W e prov e the imp ortant result ab out the op erators T 0 = f ( H o ) and T = T 0 + V . Theorem 4.2. ( In v ariance principle. ) L et op er ators T o = f ( H o ) , T = T 0 + V , wher e a function f satisfy Condition IP and let H o , V satisfy Condition 1-3. Define op er ators A, A o by A = ϕ ( T ) , A o = ϕ ( T o ) = H o E ( H o , ω ) (4.6) wher e ϕ is the inverse function for f : ω → Ω . Supp ose that the wave op er ators W ± ( T , T o , Ω) and W ± ( A, A o , ω ) exist and satisfy W ± ( T , T o , Ω) = W ± ( A, A o , ω ) Then the wave op er ators W ± ( T , T o , Ω) ar e c omplete, σ sc ( T ) ∩ Ω = ∅ and eigenvalues of T , b elonging to Ω c an ac cumulate only at the ends of the interval Ω . Pro of. W e chec k Conditions 1-3 for T , T o . 1) The op erator V ( H o − i ) − 1 is compact and by assumption, h ( λ ) → ∞ as | λ | → ∞ , λ ∈ Γ. This gives V ( T o − i ) − 1 ∈ B ∞ . (4.7) Using (2.10) and (4.7) w e obtain η ( A ) − η ( A o ) = η ( ϕ ( T )) − η ( ϕ ( T o )) ∈ B ∞ , (4.8) for any η ∈ C ∞ 0 ( R ), which yields Condition 1. 2) Recall that ω 1 ⊂ ω is a close in terv al and smo oth function φ satisfies: φ | ω 1 = 1. Using Lemma 2.7 and (4.6) we obtain φ ( A o ) = φ ( H o ) = ν X 1 ζ j φ j . (4.9) The op erator H o satisfies Condition 2. Then from (4.9) w e deduce that the op erator A o satisfies Condition 2. 3) W e chec k the main Condition 3 for A, A o , ω . In tro duce the sufficiently smo oth functions w ( λ, µ ) = u ( λ ) u ( µ ) v ( µ ) − v ( λ ) µ − λ , u ( λ ) = φ ( ϕ ( λ )) , v ( λ ) = η ( ϕ ( λ )) , λ, µ ∈ ω , and an op erator X = u ( T )  v ( T ) − v ( T o )  u ( T o ) . By the theory of double op erator-v alued in tegrals [5], w e obtain X = Z Z w ( λ, µ ) dE µ V E o λ , (4.10) where E µ , E o µ are the sp ectral pro jector for the op erators T , T o . Introduce smo oth functions g ( λ, µ ) = φ ( y 2 ) φ ( y 1 ) η ( y 2 ) − η ( y 1 ) f ( y 2 ) − f ( y 1 ) , y = ( y 1 , y 2 ) ∈ Ω 2 . W e hav e the iden tit y w ( µ, λ ) = g ( y ) , y 1 = ϕ ( λ ) , y 2 = ϕ ( λ ) , λ, µ ∈ ω , y ∈ Ω 2 . F rom the assumption f ∈ C 3+ δ ( ω ) , δ > 0 we obtain g ∈ W 2 ϑ ( R 2 ) , ϑ = ( ϑ 1 , ϑ 2 ) ∈ R 2 , ϑ 1 , ϑ 2 ⩾ 0 , ϑ 1 + ϑ 2 = 2 + δ. (4.11) 14 EV GENY KOR OTY AEV Here W 2 ϑ ( R 2 ) is the Sob olev space of functions f ( y ) suc h that ⟨ τ ⟩ − ϑ b f ( τ ) ∈ L 2 ( R 2 ) , τ ∈ R 2 . In (4.11) we take g such that ⟨ τ 1 ⟩ − b b g ( τ ) ∈ L 2 ( R 2 ) , b = ϑ 2 − ϑ 1 > 1 . (4.12) Substituting the F ourier integral g ( y ) = 1 2 π R R e iy τ b g ( τ ) dτ in to the integral (4.10) w e obtain X = Z ( b q ) = 1 2 π Z Z b g ( τ ) e iA 1 τ 2 V e iτ 1 A 0 dτ . (4.13) F or f ∈ H , ∥ f ∥ = 1, we in tro duce the function f j ( t ) = ∥ V e − itH o φ ( H o ) ζ j ( s ) Q + j f ∥ , j ∈ J , t ∈ R . Using (4.12), (4.6), (4.3) w e obtain Z ∞ r ∥ V e − itA 0 φ ( A 0 ) ζ j ( s ) Q + j f ∥ dt ⩽ Z ∞ r f j ( t ) dt ⩽ C Z R ⟨| t | + r ⟩ b f j ( t ) dt + C ⟨ r ⟩ b Z ∞ 0 f j ( t ) dt + C Z ∞ r f j ( t ) dt. Th us it is enough to estimate the first integral from the RHS: Z R ⟨| t | + r ⟩ b f j ( t ) dt ⩽ C Z R ⟨| t | + r ⟩ b dt ⩽ C ⟨ r ⟩ b − 1 . Let A b e a self-adjoint operator and let E ( λ ) b e the corresp onding sp ectral pro jector. Denote b y M ( A ) the set of all f ∈ H such that d ( E ( λ ) f , f ) = | p ( λ ) | 2 dλ for some p ∈ L ∞ ( R ). Belo w w e need the well known inv ariance principle from [7] and [33] (Theorem XI.23). Lemma 4.3. L et T o , T 1 b e self-adjoint op er ators acting in the Hilb ert sp ac e H . L et a function f ∈ C 2 (Ω) for some b ounde d interval Ω ⊂ R satisfy: 1) f ′ ⩾ α on Ω for some c onstant α > 0 . 2) L et I ⊂ Ω b e any close interval and let D b e a dense set in E ( T o , I ) P ac ( T o ) H , D ⊂ M ( T o ) . F or any u ∈ D the function w ( t ) = e itT e − itT o u, t ∈ R is str ongly-differ entiate d and ∥ w ′ ( · ) ∥ ∈ L 2 ( R ) , ⟨ t ⟩ − δ ∥ w ′ ( · ) ∥ ∈ L 1 ( R ) , for some δ > 0 . (4.14) Then ther e exist wave op er ators W ± ( ϕ ( T ) , ϕ ( T o )) u = s lim e itϕ ( T ) e − itϕ ( T o ) u as t → ±∞ , and W ± ( ϕ ( T ) , ϕ ( T o )) u = W ± ( T , T o ) u . Pro of of Theorem 1.2. Consider the op erator H o = P ( −∇ ) acting on H = L 2 ( R d ) , d > 1 and a p otential V ∈ L ε , ε ∈ E ± , the pro of of other cases is similar. Recall that a real function f satisfies Condition IP and the interv al Ω = f ( ω ) where the interv al ω ⊂ P ( R d ). W e apply Lemma 4.3 to the follo wing case: the op erator T o = f ( H o ) , T = T o + V and ϕ is the inv erse function of f ( λ ) , λ ∈ ω . Using the stationary phase metho d we pro ve (4.14) for the pair T o , T . F rom here and from Lemma 4.3 w e obtain the existence and the identit y W ± ( ϕ ( T ) , ϕ ( T o )) = W ± ( T , T o ). Then from Theorem 4.2 w e get the pro of of Theorem 1.2. SCA TTERING FOR ANISOTROPIC POTENTIALS 15 4.2. Time dep ending p otentials. W e discuss time-deca ying p otentials. Pro of of Theorem 1.3. Let ε ∈ E o . Note that the existence of the wa v e op erators is a simple fact due to the stationary phase metho d [33]. F rom (1.6), (5.1) we obtain ∥ U (0 , t ) e − itH o f − f ∥ 2 ⩽  Z t 0 ∥ V s e − isH o f ∥ ds  2 ⩽  Z t 0 g ( s ) ∥ ϱ ε e − isH o f ∥ ds  2 ⩽ Z ∞ 0 ⟨ s ⟩ − 2 γ g 2 ( s ) ds Z ∞ 0 ⟨ s ⟩ 2 γ ∥ ϱ ε e − isH o f ∥ 2 ds ⩽ C 1 G (0) ∥ f ∥ 2 , where C 1 do es not dep end on f and G ( t ) = R ∞ t ⟨ s ⟩ − 2 γ g 2 ( s ) ds . Similar arguments imply for a large time ∥W + f − U (0 , t ) e − it ∆ f ∥ 2 ⩽  Z ∞ t g ( s ) ∥ ϱ ε e − isH o f ∥ ds  2 ⩽ G ( t ) Z ∞ 0 ⟨ s ⟩ 2 γ ∥ ϱ ε e − isH o f ∥ 2 ds ⩽ C 1 G ( t ) ∥ f ∥ 2 , where G ( t ) = o (1) as t → ∞ . This gives the norm con v ergence as t → ∞ and then the wa v e op erator W + is unitary . The pro of for W − is similar. W e discuss time-p erio dic p otentials. Pro of of Theorem 1.4. Consider V ∈ L ε , ε ∈ E + ∩ E − , the pro of of other cases is similar. Note that the existence of the wa v e op erators is a simple fact and it is established b y the stationary phase metho d [33]. W e hav e W ± ⊂ H c ( M ), where M = U (1 , 0). W e show that W + = H c ( M ), the pro of for ” − ” is similar. The pro of is based on the contradiction. Thus w e can assume that E ( u , M ) f = 0 for some small in terv al u ⊂ S := {| λ | = 1 } , such tha t 1 ∈ u , including its small neigh borho o d and let u ⋐ u 1 ⊂ S where for some u 1 . Let 0 ⩽ ϑ ∈ C ∞ ( S ) be suc h that ϑ | u = 1 and ϑ | S \ u 1 = 0. Let η ∈ C ∞ o ( R ), where η = 1 including small neighborho o d of the p oint 0. Define functions η 1 and η 2 b y η 1 = ϑ ( e iλ ) η ( λ ) ∈ C ∞ o ( R ) , η 2 = 1 − η 1 , λ ∈ R . (4.15) W e recall the standard fact: for any ( ϑ, φ ) ∈ C ( S ) × C ∞ 0 ( R ), we obtain φ ( H o )( ϑ ( M ) − ϑ ( M o )) ∈ B ∞ , (4.16) see e.g. [35]. Recall the well known fact: for an y finite num ber of compact op erators G j ∈ B ∞ , j ∈ J ν there exist sequences of integers n p suc h that n p → ±∞ as p → ±∞ : ν X j =1 ∥G j M n p f ∥ = o (1) as p → ±∞ , (4.17) see [9], [35]. Then for some subsequence n := n p → ∞ and for f n = M n f we ha ve ∥ f ∥ 2 = ∥ f n ∥ 2 = ( f n , η 1 ( H o ) f n ) + ( f n , η 2 ( H o ) f n ) , η 1 ( H o ) f n = η ( H o ) ϑ ( H o ) f n = η ( H o )( ϑ ( M o ) − ϑ ( M )) f n = o (1) , (4.18) since E ( u , M ) f = ϑ ( M ) f = 0 and due to (4.16) w e obtain η ( H o )( ϑ ( M ) − ϑ ( M o )) ∈ B ∞ , for some sequences of integers n p suc h that n p → ±∞ as p → ∞ . W e hav e the decomp osition η 2 = ϕ + ϕ − , ϕ = χ + η 2 ∈ C ∞ ( R ) , and ϕ − = χ − η 2 , where χ + ( t ) = 1 , t > 0 and χ + ( t ) = 0 , t < 0 and χ + + χ − = 1. Here we hav e a problem with ϕ , since V t ϕ is not a compact op erators in L 2 ( R d ). W e need some mo dification. W e consider the first term ϕ ( H o ) f n , the pro of for ϕ − ( H o ) f n is similar. W e hav e the decomp osition 16 EV GENY KOR OTY AEV ϕ = ζ 1 ϕ + b 1 ϕ , where ζ 1 = ζ ( h 1 /s ) and b 1 = 1 − ζ 1 . Let A ± 1 := ( W ± − 1 1) ζ 1 Q ± 1 . Due to (3.9), (3.10) we hav e ∥ A ± 1 ∥ = o (1) as s → ∞ . Then we hav e ( f n , ϕf n ) = ( f n , ζ 1 ϕf n , ) + ( f n , b 1 ϕf n ) , ( f n , ζ 1 ϕf n ) = X τ = ± [( f n , W τ ζ 1 Q τ 1 ϕf n ) − ( f n , A τ 1 f n )] = X τ = ± ( f n , W ν ϕ j Q ν j f n ) + o (1) = ( f n , W − ϕ j Q − j f n ) + o (1) as n = n p → ±∞ , (4.19) since f ∈ H ac ( M ) ⊖ W + H and from (3.7) and M m W ν = W ν M m o for all m ∈ Z w e deduce that ( M n f , W − ϕ j Q − j f n ) = o (1). Thus w e apply the same pro cedure for function ζ 2 and b 2 need to consider the sequence ( f n , b 1 ϕf n ). W e ha v e ( f n , b 1 ϕf n ) = ( f n , ζ 2 b 1 ϕf n ) + ( f n , b 2 b 1 ϕf n ) = ( f n , b 2 b 1 ϕf n ) + o (1) . Rep eating procedure w e obtain ( f n , ϕf n ) = ( f n , b ϕf n ) + o (1), where b = Q ν 1 b j . Then the op er- ator b ϕ ∈ B ∞ ( L 2 ( R d ), since the functions ϕ, b are compactly supp orted and ab ov e arguments yield ( f n , b ϕf n ) = o (1). 5. Appendix Lemma 5.1. L et γ > 0 and let ψ ∈ L 2 ( R d ) and r = P ν 1 r j , r j = 1 a j min { ε j , d j / 2 } , ε j ⩾ 0 . i) If γ + r > 1 2 , then Z R ⟨ t ⟩ 2 γ ∥ ϱ ε e − itH o ψ ∥ 2 dt ⩽ C ∥ ψ ∥ 2 . (5.1) ii) If ε ℓ + r − r ℓ + γ > 1 2 for some ℓ ∈ J ν . Then for al l s ⩾ 1 and some ϑ > 0 Z R ⟨ t ⟩ 2 γ ∥ ϱ ε χ ( | k ℓ | > s ) e − itH o ψ ∥ 2 dt ⩽ C s − ϑε ℓ ∥ ψ ∥ 2 . (5.2) Pro of. F rom strict inequality for γ , r it is enough to prov e the estimates for 0 < 2 γ < 1 , 0 < 2 ε j < d j , j ∈ J ν and for 2 r < 1 in the case i) and ε ℓ + r − r ℓ < 1 2 in the case ii). Then since w e ha v e r j = ε j a j , j ∈ J ν and we define the v ector A = ( A j ) ν 1 ∈ R ν + b y A ℓ = ( 1 a ℓ , if r + γ > 1 2 1 if ε ℓ + r − r ℓ + γ > 1 2 . Th us the inequality for ε w e rewrite in the form ( ε, A ) > 1 2 − γ , 0 < 2 ε j < d j , j ∈ J . F rom these stric k inequality we get that enough to prov e (5.1), (5.2) for ε = ε ′ + δ d ′ , d ′ = ( d j ) ν 1 , γ > δ, ε ′ ∈ { t ∈ R ν + : ( t, A ) = 1 2 − γ , 0 < 2 t j < d j , ∀ j ∈ J ν } for δ > 0 small enough. Roughly sp eaking we pro ve (5.1), (5.2) for ε b elonging to the la yer 1 < 2( ε, A ) + 2 γ < 1 + 2 δ ( d ′ , A ) , δ > 0 . W e introduce functions f ( k , d ) = e −| k | | k | d , F d ( k , b ) = (2 π ) − d 2 Z R d e − ikx ⟨ x ⟩ − b dx, k ∈ R d , b > 0 , SCA TTERING FOR ANISOTROPIC POTENTIALS 17 F or the function F d there exists the following estimate (see [5]) | F d ( k , b ) | ⩽ C | k | b f ( k , b ) , k ∈ R d , d > b > 0 . (5.3) Let η b e the m ultiplication op erator by the function η ( k ) ⩾ 0. W e hav e Y ( η ) = Z R ⟨ t ⟩ 2 γ ∥⟨ x ⟩ ε e − itH o η ψ ∥ 2 dt = Z R ⟨ t ⟩ 2 γ ( ⟨ x ⟩ 2 ε e − itH o η ψ , e − itH o η ψ ) dt. = Z R 2 d F d ( k − p, 2 ε ) η ( k ) η ( p ) b ψ ( k ) ˆ ψ ( p ) Z R ⟨ t ⟩ 2 γ e − it ( P ( k ) − P ( p )) dk dpdt = Z R 2 d F d ( k − p, 2 ε ) η ( k ) η ( p ) b ψ ( k ) ˆ ψ ( p ) F 1 ( P ( k ) − P ( p ) , 2 γ ) dk dp. In tro duce the function G ( k , p ) = | y 0 | 2 γ f ( y 0 , 1) ν Y 1 | y j | 2 ε j f ( y j , d j ) , y = k − p, y j = k n − p j , j ∈ J ν . Th us from (5.3) we obtain Y ( η ) < C Z R 2 d G ( k , p ) η ( k ) η ( p ) | ψ ( k ) ψ ( p ) | dk dp. (5.4) Supp ose that Z R 2 d G ( k , p ) η ( k ) η ( p ) | ψ ( k ) ψ ( p ) | dk dp < C 1 < ∞ . (5.5) Then using the well kno w estimates of the integral op erators we obtain Y ( η ) < C 1 ∥ ψ ∥ 2 . (5.6) F rom here w e get (5.1) for η = 1 and (5.2) for η = χ ( | k n | > s ) and C 1 = C s − ϑε ℓ . W e shall prov e (5.5). W e introduce the num b er r ′ = ( ε ′ , A ) = P ν 1 r j , where r ′ j = ε ′ j A j and the functions b n ( σ ) = ( d j / 2) − ε j − ( d j r ′ − r ′ j ) σ γ − σ , β n ( σ ) = r ′ j a j − d j r ′ + ( d j r ′ − r ′ j ) σ γ − σ , σ ∈ R . Note that b j (0) > 0 , β j (1) > 0 , (5.7) and b j + β j − d j =      0 if j  = ℓ 0 if j = ℓ, r + γ > 1 2 ε ′ ℓ ( a ℓ − 1) γ − σ if j = ℓ, r + γ + ε ℓ − r j > 1 2 . (5.8) W e show that for any ℓ ∈ J ν there exists ϑ j ∈ (0 , 1) such that b j = b j ( ϑ n ) ∈ (0 , d j ) , β j = β j ( ϑ j ) > 0 . (5.9) Let σ j b e the solution of the equation b j ( σ ) = 0. There tw o cases: 1) 0 < σ j ⩽ 1. Then there exists some ϑ j < σ j suc h that the iden tit y (5.9) holds true, since β j ( σ j ) ⩾ d j . 2) σ j > 1. Then b j (1) > 0 and due to β j (1) > 0 there exists some ϑ j < (0 , 1) suc h that the iden tit y (5.9) holds true. 18 EV GENY KOR OTY AEV Let r j = | k j | and we in tro duce functions G 0 ( k , p ) = ν Y 1 | y j | − b j r − β j n e −| y j | , G n ( k , p ) = f ( y 0 , 1) | y j | ϑ n r a j − ϑ j j ν Y 1 | y n | − d n ϑ n r d n ( ϑ n − 1) n e −| y n | . W e hav e the iden tit y G ( k , p ) = η ( k ) η ( p ) G 2( γ − δ ) 0 ( k , p ) ν Y 1 G 2 r ′ j ( k , p ) e − δ | y j | . (5.10) F or the function G 0 w e obtain the estimate: G 0 ( k , p ) ⩽ | y ℓ | − b ℓ s − β ℓ ℓ e −| y ℓ | ν Y j  = ℓ | y j | − b j r − β j j e −| y j | , if | k ℓ | ⩾ s. (5.11) W e hav e the simple estimate sup p ∈ R d R R d G τ 0 ( k , p ) dk < ∞ if r + γ > 1 2 , τ ∈ (0 , 1) , (5.12) and using (5.11) w e ha ve for r + γ + ε ℓ − r ℓ > 1 2 sup p ∈ R d Z R d χ ( | k ℓ | > s ) G τ 0 ( k , p ) dk < ∞ if r + γ > 1 2 , τ ∈ (0 , 1) . (5.13) F rom Lemma 5.2 w e obtain sup n sup p ∈ R d Z R d G τ n ( k , p ) dk < ∞ if r + γ > 1 2 , τ ∈ (0 , 1) . (5.14) F rom (5.10) w e get Z R d G ( k , p ) dk < Z R d η ( k ) η ( p ) G 2( γ − δ ) 0 ( k , p ) ν Y 1 G 2 r ′ j j ( k , p ) dk . (5.15) F rom (5.12)-(5.14) and from an estimate 2( γ − δ ) + 2 ν X 1 r j = 2( γ − δ + r ′ ) = 1 − 2 δ < 1 , w e see that we can apply the Holder inequality to the right hand side of (5.15) with η = 1 in the case (5.15) and η = χ ( | k ℓ | > s ) in the case (5.2). Lemma 5.2. L et k , p ∈ R d , u ∈ S d − 1 and r = | k | , k = r u . 1) L et 0 < b < d − 1 . Then X ( r ) = Z S d − 1 e −| ru − p | | r ν − p | du ⩽ C e − 1 2 | r −| p || r b + r d − 1 , (5.16) wher e C do es not dep end on p . 2) L et b + β ⩽ d − 1 , b, β ⩾ 0 , a ⩾ 1 , and let the function G ( k , p ) = e −| r a − y |−| k − p | | r a − y || r − p | b r 1+ β − α , y ∈ R . Then sup p,y ∈ R d Z R d G τ ( k , p ) dk < ∞ ∀ τ ∈ (0 , 1) . (5.17) SCA TTERING FOR ANISOTROPIC POTENTIALS 19 Pro of. Let ϑ b e an angle b et w een v ectors k , p . Then | r − p | ⩾ r sin ϑ, 0 ⩽ ϑ ⩽ π , (5.18) σ ϑ ⩽ sin ϑ ⩽ ϑ σ , 0 ⩽ ϑ ⩽ π 2 , (5.19) for some σ > 0. F rom (5.18), (5.19) we obtain X ( r ) ⩽ C e −| r −| p || / 2 Z π 0 e − r 2 sin ϑ sin d − 2 ϑ | r sin ϑ | b dϑ ⩽ C r − b e −| r −| p || / 2 Z π 0 e − rσ ϑ 2 ϑ d − 2 − b dϑ. This yields X ( r ) ⩽ C r − b e −| r −| p || / 2 , 0 < r ⩽ 1 , and X ( r ) ⩽ C r 1 − d e −| r −| p || / 2 Z ∞ 0 e − t t d − 2 − b dt ⩽ C r 1 − d e −| r −| p || / 2 . W e shall pro ve (5.17). W e ha v e Y ( p, y ) = Z R d G τ ( k , p ) dk = Z ∞ 0 e − τ | r a − y | | r a − y | τ r τ (1+ β − α ) Z S d − 1 e − τ | r u − p | | r u − p | τ du r d − 1 dr . Using (5.16) we obtain Y ( p, y ) ⩽ Z ∞ 0 e − τ | r a − y | | r a − y | τ r τ (1+ β − α ) r d − 1 dr r τ b + r d − 1 ⩽ C ⩽ Z ∞ 0 e − τ | r a − y | | r a − y | τ r τ (1+ β − α ) r d − 1 dr r τ b + r d − 1 . 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