Eigenvalue bounds and spectral stability of Lamé operators with complex potentials
aa r X i v : . [ m a t h . SP ] J a n Eigenvalue bounds and spectral stability of Lam´eoperators with complex potentials
Biagio Cassano , Lucrezia Cossetti and Luca Fanelli Dipartimento di Matematica, Universit`a degli Studi di Bari Aldo Moro, Via Edoardo Orabona 4, 70125 Bari, Italy;[email protected] Fakult¨at f¨ur Mathematik, Institut f¨ur Analysis, Karlsruher Institut f¨ur Technologie (KIT), Englerstraße 2, 76131Karlsruhe, Germany; [email protected] Ikerbasque & Departamento de Matem´aticas, Universidad del Pa´ıs Vasco/Euskal Herriko Unibertsitatea (UPV/EHU),Barrio Sarriena s/n, 48940, Leioa, Spain; [email protected]
January 26, 2021
Abstract
This paper is devoted to providing quantitative bounds on the location of eigenvalues, both discrete and embedded , of non self-adjoint Lam´e operators of elasticity − ∆ ∗ + V in terms of suitable norms of thepotential V . In particular, this allows to get sufficient conditions on the size of the potential such thatthe point spectrum of the perturbed operator remains empty. In three dimensions we show full spectralstability under suitable form-subordinated perturbations: we prove that the spectrum is purely continuousand coincides with the non negative semi-axis as in the free case. This paper is devoted to the analysis of the spectrum of the perturbed Lam´e operator of elasticity − ∆ ∗ + V. The Lam´e operator − ∆ ∗ acts on smooth vector fields as − ∆ ∗ u := − µ ∆ u − ( λ + µ ) ∇ div u, u ∈ C ∞ ( R d ) d := C ∞ ( R d ; C d ) , where the material-dependent Lam´e parameters λ, µ ∈ R satisfy the standard ellipticity conditions ( cfr. [20,Sec. 2.2]) µ > , λ + 2 µ > . The Lam´e operator is self-adjoint on H ( R d ) d and σ ( − ∆ ∗ ) = σ ac ( − ∆ ∗ ) = [0 , + ∞ ); we refer the reader to [60,63]for a detailed exposition of the general theory of elasticity and to [5–8] and references therein for previous resultsin the topic. We consider the perturbation V : R d → C d × d to be a multiplication operator by a (possibly) non-hermitian matrix: this frames our study into a non-self-adjoint setting. Spectral analysis of non-self-adjointmodels has seen a huge development in the last decades and nowadays the literature in this direction is veryextensive, see [1,9,17,24,25,27–32,34,35,37–39,42–45,47–51,59,64,71,72] which is just a selection of the existingmaterial in the subject.The study of the discrete spectrum of the non-self-adjoint Lam´e operator − ∆ ∗ + V was started in [21]: in thispaper we extend these results to cover embedded eigenvalues. Moreover, we investigate the spectral stability ofthe Lam´e operator of elasticity and get in any dimension d ≥ d = 3 weshow that the whole spectrum is preserved under suitable form-subordinated perturbations.Adapting to the Lam´e operator new techniques introduced by Frank in [42] for the Laplacian, in [21] it isshown that every discrete eigenvalue z ∈ C \ [0 , ∞ ) of − ∆ ∗ + V lies in the closed disk of the complex plane1entered at the origin and with radius whose size depends on the Lebesgue, Morrey-Campanato or Kerman-Sawyer norm, according to the chosen class of potentials considered. More specifically, when the size of thepotential is measured with respect to the L p topology, [21, Theorem 1.2] shows that any eigenvalue z ∈ C \ [0 , ∞ )of − ∆ ∗ + V satisfies | z | γ ≤ C k V k γ + d L γ + d ( R d ) , (1)for some C >
0, with d ≥ ≤ γ ≤ / γ = 0 if d = 2).In order to cover potentials with stronger local singularities one considers the Morrey-Campanato class L α,p ( R d ), that is the class of functions W such that for α > ≤ p ≤ d/α the following norm k W k L α,p ( R d ) := sup x,r r α (cid:16) r − d Z B r ( x ) | W ( x ) | p dx (cid:17) p is finite. For example, 1 / | x | α / ∈ L d/α ( R d ) = L α,d/α ( R d ) but 1 / | x | α ∈ L α,p ( R d ) for α > ≤ p < d/α . Inparticular, the inverse-square potential of quantum mechanics V ( x ) = 1 / | x | , x ∈ R , at first ruled out by the L p type condition, can be recovered once the size of the potential is measured in terms of Morrey-Campanatonorms. In [21, Theorem 1.3], the analogous bound to (1) for potentials in the Morrey-Campanato class isprovided: any eigenvalue z ∈ C \ [0 , ∞ ) of − ∆ ∗ + V satisfies | z | γ ≤ C k V k γ + d L α,p ( R d ) , (2)for some C > d ≥ , ≤ γ ≤ / γ = 0 if d = 2) and with ( d − γ + d ) / [2( d − γ )] < p ≤ γ + d/ α = 2 d/ (2 γ + d ) . We remark that for α > < p ≤ d/α the condition W ∈ L α,p ( R d ) ensures the L weighted boundednessof fractional integrals (see Fefferman [40] for the special case α = 2 and [69] for the more general result, seealso [4, Section 2.2]), that is the existence of a non-negative constant C ( W ) > k I α/ f k L ( R d ,W dx ) ≤ C ( W ) k f k L ( R d ) , for all f ∈ C ∞ c ( R d ) , (3)where d I α f ( ξ ) = | ξ | − α b f . If W ∈ L α,p ( R d ) the constant C ( W ) in (3) can be written more explicitly in terms ofthe Morrey-Campanato norm k W k L α,p,d ( R d ) of W, more specifically C ( W ) = C α,p,d k W k / L α,p,d ( R d ) , (4)for C α,p,d > W. The largest class of functions W such that this inequality is available is theKerman-Saywer space KS α ( R d ) (see [55, Theorem 2.3]), namely the set of all the functions W such that for0 < α < d the following norm k W k KS α ( R d ) := sup Q (cid:16) Z Q | W ( x ) | dx (cid:17) − Z Q Z Q | W ( x ) || W ( y ) || x − y | d − α dx dy is finite (the supremum is taken over all dyadic cubes Q in R d ). As a matter of fact the finiteness of this normis a necessary and sufficient condition for the validity of (3) and the best constant in it is C ( W ) = C α,d k W k / KS α ( R d ) , (5)for some constant C α,d > W . In particular this implies k W k KS α ( R d ) ≤ C k W k L α,p ( R d ) , for α >
0, 1 < p ≤ d/α and C >
0, which gives L α,p ( R d ) ⊆ KS α ( R d ). In the case α = 2 , (3) is equivalent to thevalidity of an Hardy-type inequality for the weight W, namely Z R d | W || f | dx ≤ a W Z R d |∇ f | dx, for all f ∈ C ∞ c ( R d ) , (6)where a W := C ( W ) and C ( W ) is the constant in (3). In the case d = 3 we have that a W = (cid:26) c F k W k L ,p ( R ) , if W ∈ L ,p ( R ) ,c KS k W k KS ( R ) , if W ∈ KS ( R ) , (7)2here we have set c F = c F ( p ) := C ,p, , with C ,p, as in (4) and c KS := C , , with C , as in (5).In [21, Theorem 1.4] it is shown that any eigenvalue z ∈ C \ [0 , ∞ ) of − ∆ ∗ + V satisfies | z | γ ≤ C Q ( | V | ) γ + d k| V | β k β ( γ + d ) KS α ( R d ) , (8)for C > d ≥ , / ≤ γ < / d = 2 and 0 ≤ γ < / d ≥ α = 2 dβ/ (2 γ + d ) and β = ( d − γ + d ) / [2( d − γ )] , under the additional assumption that | V | belongs to the A ( R d ) Muckenhouptclass of weights, i.e. , the set of measurable non-negative functions w such that the following quantity Q ( w ) := sup Q | Q | Z Q w ( x ) dx ! | Q | Z Q w ( x ) dx ! is finite. Here the supremum is taken over any cube Q in R d . We stress that, in the higher dimensional case d ≥
3, the validity of bounds (1), (2) and (8) providesconditions which guarantee the absence of non-embedded discrete eigenvalues depending on the size of thepotential, measured with respect to the corresponding norm. Indeed, once γ = 0 is fixed, for any eigenvalue z ∈ C \ [0 , ∞ ) of − ∆ ∗ + V one has 1 ≤ C k V k d , (9)where k V k denotes k V k L γ + d ( R d ) , k V k L α,p ( R d ) or Q ( | V | ) d k| V | d − k d − KS d − ( R d ) , respectively. If C k V k d < , (9)yields a contradiction and so σ d ( − ∆ ∗ + V ) = ∅ ( cfr . [21], Thm. 1.2, Cor. 1.1 and Cor. 1.2).Seeking for eigenvalue bounds like (1), (2) and (8) for perturbed Lam´e operators − ∆ ∗ + V with V possiblynon-hermitian was mainly motivated by the existence in the literature of the corresponding bounds for non-self-adjoint Schr¨odinger operators − ∆ + V and by the link between the two operators given by the Helmoltzdecomposition, see Lemma 2.1. More motivation come from the one-dimensional framework, where the Lam´eoperator becomes a constant multiple of the Laplacian, i.e. ∆ ∗ = ( λ +2 µ ) d /dx . As far as real-valued potentialsare considered, it comes merely as a consequence of Sobolev inequalities that the distance from the origin ofevery eigenvalue z of the Schr¨odinger operator lying in the negative semi-axis can be bounded in terms of L p norm of the potential, see [14, 54, 68]. The non-self-adjoint situation requires different tools. A key strategyin the subject was provided by Abramov, Aslanyan and Davies: in [1] they prove that for a possibly complex-valued V, every discrete eigenvalue z ∈ [0 , ∞ ) of the one-dimensional Schr¨odinger operator − d /dx + V lies inthe complex plane within a 1 / k V k L ( R ) distance from the origin. The generalization to the higher dimensionalcase d ≥ d ≥ z ∈ C of theSchr¨odinger operator − ∆ + V, with short-range potentials V ∈ L γ + d/ ( R d ) , γ ≤ / . In the same work [48]the authors investigated also the case of long-range potentials and showed that a bound of the form (1) couldnot hold for such a class: they construct a sequence of real-valued potentials V n with k V n k L γ + d/ ( R d ) → ,γ > / , such that − ∆ + V n has eigenvalue 1. A better understanding of the distribution of eigenvalues ofSchr¨odinger operators with slowly decaying potentials V ∈ L γ + d/ ( R d ) , γ > / , was led later by Enblom [34]and Frank [43]. In [43] it is proved that a bound of type (1) holds true with a correction which depends on thedistance of the eigenvalue z from the positive half-line, that is, defining δ ( z ) := dist( z, [0 , ∞ )) , one has δ ( z ) γ − / | z | / ≤ C γ,δ k V k γ + d/ L γ + d/ ( R d ) . (10)Notice that (10) is weaker than (1) since δ ( z ) ≤ | z | . As far as the size of the potential is measured in terms of L p norms, one requires p ≥ d/ d ≥ p > d = 2 in order to define − ∆ + V as an m -sectorial operator:this rules out the possibility to treat physically interesting classes of potentials which might display strongerlocal singularities and demands for enlarging the class of potentials considered. The analogous of bound (2)for Schr¨odinger operators with potentials in the Morrey-Campanato class can be found in [42], whereas theanalogous of bound (8) for potentials in the Kerman-Saywer class is proved by Lee and Seo in [65], see also [73].We observe that the bound obtained in [65] presents a constant which is independent of V , differently from (8) in3ur setting: this shows a pathological behavior of the Lam´e operator as compared to the Schr¨odinger operator,consequence of the non-uniform weighted boundedness properties of the Riesz transform with respect to theweight | V | , cfr. Lemma 2.4.The proofs of bounds (1), (2) and (8) ( cfr. [21, Theorems 1.2–1.4]) all display the same underlying structurestrongly based on the Birman-Schwinger principle ( cfr . [74], Thm. III.12, Thm. III.14). The usefulness of theBirman-Schwinger principle to localize eigenvalues of self-adjoint and non-self-adjoint Hamiltonians is by nomeans questionable, as a matter of fact an extensive bibliography on the subject has been produced adoptingthis methodology. Without any hope of completeness we refer to [38,42,48] for results on Schr¨odinger operatorsand [51] for an adaptation to the discrete setting, see also [59] where matrix-valued damped wave operators areconcerned. Lower order operators, such as Dirac or fractional Schr¨odinger models, are investigated in [16, 25,27, 32, 37] (see also [26, 41]) and in [17] respectively in the continuous and discrete scenario; as for higher orderoperators refer to [50]. Associated spectral stability results obtained with different techniques and related toolscan be found in [2, 11, 12, 15, 22, 36, 46].In our context, the Birman-Schwinger principle states that z ∈ C \ [0 , ∞ ) is an eigenvalue of − ∆ ∗ + V if andonly if − K z := | V | / ( − ∆ ∗ − z ) − V / on L ( R d ) d , where V / := | V | / sgn( V ) and sgn( V ) denotes the complex sign function. In particular, if − K z the norm of K z is at least one and then proving bounds (1), (2) and (8) descends from proving that k| V | / ( − ∆ ∗ − z ) − V / k γ + d L → L ≤ c | z | − γ k V k γ + d , where k V k = k V k L γ + d ( R d ) , k V k = k V k L α,p ( R d ) or k V k = Q ( | V | ) k| V | β k β KS α ( R d ) for (1), (2) or (8) respectively.Treating eigenvalues z ∈ C \ [0 , ∞ ), the Birman-Schwinger operator K z is well defined since σ ( − ∆ ∗ ) = [0 , ∞ ) . The natural strategy to cover also z ∈ [0 , ∞ ) is to study an approximating Birman-Schwinger operator, that is, K z + iε := | V | / ( − ∆ ∗ − z − iε ) − V / , for some ε >
0, retracing the proofs of (1), (2) and (8) valid for z + iε outside the spectrum and eventually passing to the limit ε → . Thanks to this approach, in the following weextend [21, Theorems 1.2–1.4] to the whole point spectrum of − ∆ ∗ + V .The following theorem extends [21, Theorems 1.2] to treat the whole point spectrum. Theorem 1.1.
Let d ≥ , < γ ≤ / if d = 2 and ≤ γ ≤ / if d ≥ and V ∈ L γ + d ( R d ; C d × d ) . Then thereexists a universal constant c γ,d,λ,µ > independent on V such that σ p ( − ∆ ∗ + V ) ⊂ (cid:26) z ∈ C : | z | γ ≤ c γ,d,λ,µ k V k γ + d L γ + d ( R d ) (cid:27) . (11)As a corollary, the previous theorem provides a sufficient condition on the size of the potential to guarantee total absence of eigenvalues in the higher dimensional case d ≥ . Corollary 1.1. If d ≥ and c ,d,λ,µ k V k d L d ( R d ) < , then − ∆ ∗ + V has no eigenvalues. Furthermore, for d = 3 the constant c , ,λ,µ is explicitly given by c , ,λ,µ := (cid:18) / (1 + 6 cot ( π/ π / min { µ, λ + 2 µ } (cid:19) . Remark . In the context of Schr¨odinger operators, seeking for optimal conditions on both local integrabilityand asymptotic decay of the potentials under which absence of embedded eigenvalues is guaranteed has yieldeda considerable bibliography. Ionescu and Jerison in [52] obtained absence of embedded eigenvalues for V ∈ L d/ (or V ∈ L p , p > d = 2). We stress that as long as local integrability conditions are investigated, thisresult is optimal, indeed Koch and Tataru in [57] constructed non trivial compactly supported solutions of the0-eigenvalue equation ∆ u = V u with V ∈ L p loc , p < d/ d ≥ V ∈ L for d = 2). Later, Kochand Tataru in [58] proved the same result as in [52] for potentials V with the least possible decay at infinity,including V ∈ L ( d +1) / . The exponent ( d + 1) / , d ≥ V ∈ L p , p > ( d +1) / γ in Theorem 1.1 are rathernatural.For potentials in the Morrey-Campanato class we prove the next results, counterpart of [21, Theorem 1.3,Corollary 1.1]. Theorem 1.2.
Let d ≥ , ( d − γ + d ) / d − γ ) < p ≤ γ + d/ with < γ ≤ / if d = 2 and ≤ γ ≤ / if d ≥ and assume V ∈ L α,p ( R d ; C d × d ) with α = 2 d/ (2 γ + d ) . Then there exists a universal constant c γ,p,d,λ,µ > independent on V such that σ p ( − ∆ ∗ + V ) ⊂ n z ∈ C : | z | γ ≤ c γ,p,d,λ,µ k V k γ + d L α,p ( R d ) o . Corollary 1.2. If d ≥ and c ,p,d,λ,µ k V k d L ,p ( R d ) < , then − ∆ ∗ + V has no eigenvalues. Furthermore, for d = 3 the constant is explicitly given by c ,p, ,λ,µ := (cid:16) c F (1 + 6 C )min { µ, λ + 2 µ } (cid:17) , with c F = c F ( p ) as in (7) and C > independent on V. Remark . We recall that, thanks to the H¨older inequality, k V k L α,p ( R d ) ≤ V p − αd d k V k L dα ( R d ) , for α > ≤ p ≤ d/α and where V d denotes the volume of the unit d -dimensional ball. As a consequence,Theorem 1.1 follows from Theorem 1.2 for c γ,d,λ,µ in (11) equal to c γ,p,d,λ,µ ( V /p − α/dα ) γ + d/ , with α, p and c γ,p,d,λ,µ as in Theorem 1.2. Nonetheless, we decided to state and also give an alternative proof of Theorem 1.1as it is of interest in its own right. As a matter of fact, in dimension d = 3 and for γ = 0 , this alternative directproof provides an explicit bound on the constant c γ,d,λ,µ in (11) and, in turn, on the smallness of the size of thepotential in order to guarantee absence of eigenvalues.Finally, the following theorem is the counterpart of [21, Theorem 1.4], treating potentials in the Kerman-Saywer class. Theorem 1.3.
Let d ≥ , / ≤ γ < / if d = 2 and ≤ γ < / if d ≥ and assume | V | β ∈ KS α ( R d ) with α = 2 dβ (2 γ + d ) and β = ( d + 2 γ )( d − / [2( d − γ )] . If | V | ∈ A ( R d ) then there exists a constant c γ,d,λ,µ > independent on V such that σ p ( − ∆ ∗ + V ) ⊂ (cid:26) z ∈ C : | z | γ ≤ c γ,d,λ,µ Q ( | V | ) γ + d k| V | β k β ( γ + d ) KS α ( R d ) (cid:27) . Corollary 1.3. If d ≥ and c ,d,λ,µ Q ( | V | ) d k| V | d − k dd − KS d − < , then − ∆ ∗ + V has no eigenvalues. Furthermore, for d = 3 the constant is explicitly given by c , ,λ,µ := (cid:16) c KS (1 + 6 C )min { µ, λ + 2 µ } (cid:17) , with c KS as in (7) and C > independent on V. In the three dimensional case, the third author with Krejˇciˇr´ık and Vega proved in [38, Thm. 1] that thespectrum of the three dimensional Schr¨odinger operator is stable under perturbations which satisfy the followingsubordination relation ∃ a < Z R d | V || u | dx ≤ a Z R d |∇ u | dx, ∀ u ∈ H ( R d ) . (12)5n other words, under this assumption they not only show that the point spectrum is empty, but that thewhole spectrum is absolutely continuous and equal to the spectrum of the unperturbed operator. Their resultrelies on the proof of a variational one-sided version of the conventional Birman-Schwinger principle extendedto possible eigenvalues embedded in the essential spectrum. This approach turned out to be very robust, it wasindeed adopted to investigate on the spectrum of other Hamiltonians than the Schr¨odinger operators: see [37]and [50] for an adaptation to non-self-adjoint Dirac and biharmonic operators, respectively. We refer the readerto the recent work [49] by Hansmann and Krejˇciˇr´ık for a systematic exposition of abstract Birman-Schwingerprinciples and their rigorous applications in spectral theory. To show the analogue result in the context ofperturbed Lam´e operator, we need to recall that there exist C > W ∈ A ( R ) the followingsharp bound on the weighted L operator norm of the Riesz transform R = ( R , R , . . . , R d ) is available, seeLemma 2.4: kR j k L ( W dx ) → L ( W dx ) ≤ c W := C Q ( W ) , for all j = 1 , . . . , d, (13)where Q ( W ) is the 0–homogeneous A constant of W defined in (18) below. Theorem 1.4.
Let d = 3 . Assume that V : R → C × , | V | ∈ A ( R ) and ∃ a < min { µ, λ + 2 µ } c V such that Z R | V || u | dx ≤ a Z R |∇ u | dx, ∀ u ∈ H ( R ) , (14) with c V = c | V | given by (13) . Then σ ( − ∆ ∗ + V ) = σ c ( − ∆ ∗ + V ) = [0 , ∞ ) . Remark . Thanks to (7), a necessary and sufficient condition for the Hardy-type inequality in (14) tohold is that V belongs to the Kerman-Saywer class KS ( R ); furthermore in this case, a = c KS k V k KS ( R ) .This entails that the subordination condition (14) holds if and only if V ∈ KS ( R ) and c KS k V k KS ( R ) < min { µ, λ + 2 µ } / (1 + 6 c V ) . We decided anyway to state Theorem 1.4 with the smallness condition (14) insteadof requiring smallness of the Kerman-Sawyer norm of V in line with what just pointed out, in order to keepwith the subordination relation (12) introduced in [38]. Remark . In order to define − ∆ ∗ + V as an m-sectorial operator it is sufficient to assume a < min { µ, λ + 2 µ } ,see Section 2. The stronger condition on a in (14) is needed to ensure the boundedness of the Birman-Schwingeroperator K z := | V | / ( − ∆ ∗ − z ) − V / with bound strictly less than one. We stress that the demand for thestronger smallness condition in (14) is connected to the elasticity framework of the Lam´e operator and theneed of the Helmholtz decomposition, refer to the proof of Lemma 4.1. On the other hand, for the Birman-Schwinger operator associated to the Laplacian K ∆ z := | V | / ( − ∆ − z ) − V / , the validity of (12) directly gives k K ∆ z k ≤ a < , with a the same constant as in (12), refer to [38, Lemma 1].In the following theorem we obtain the spectral stability stated in Theorem 1.4 in the case that V belongsto the Morrey-Campanato class L ,p ( R ) , < p ≤ /
2. Notice that L , / ( R ) = L / ( R ) is also covered. Theorem 1.5.
Let d = 3 . Assume V ∈ L ,p ( R ) , < p ≤ / and c F (1 + 6 c V )min { µ, λ + 2 µ } k V k L ,p ( R ) < , (15) with c F as in (7) and c V = c | V | given by (13) . Then σ ( − ∆ ∗ + V ) = σ c ( − ∆ ∗ + V ) = [0 , ∞ ) . Remark . In Theorem 1.5 the assumption that the potential V is in the Morrey-Campanato class allowsto drop the assumption that it belongs to A ( R ). Thanks to Lemma 2.6, if V is in the Morrey-Campanatoclass, Q ( V ) can be bounded by a constant independent on V ; in turn from (13) one has that c V in (15) isindependent on V too. Remark . Notice that the smallness condition (15) ensures k K z k < K z the Birman-Schwingeroperator ( cfr. (37) in Lemma 2.9). From (15) one has in particular that c F k V k L ,p ( R ) < min { µ, λ + 2 µ } , hence − ∆ ∗ + V is well defined as an m-sectorial operator due to (6) and (7) (see also Section 2).6ven though the case of V ∈ L / ( R ) is covered by the previous result ( p = 3 / L p framework of the Hardy-Littlewood-Sobolev inequality (see proof of (36) inLemma 2.9). More specifically, we can prove the following L p framed result. Theorem 1.6.
Let d = 3 . Assume V ∈ L / ( R ) . If / (1 + 6 cot ( π/ π / min { µ, λ + 2 µ } k V k L / ( R ) < , (16) then σ ( − ∆ ∗ + V ) = σ c ( − ∆ ∗ + V ) = [0 , ∞ ) . Remark . Observe that the smallness condition (16) ensures k K z k < cfr. (36) in Lemma 2.9). Furthermorenotice that it follows from H¨older inequality and Sobolev embedding that Z R | V || u | dx ≤ k V k L / ( R ) k u k L ( R ) ≤ / π / k V k L / ( R ) Z R |∇ u | dx. (17)From (16) one has in particular that 2 / / (3 π / ) k V k L / ( R ) < min { µ, λ + 2 µ } , hence − ∆ ∗ + V is well definedas an m-sectorial operator. Remark . Observe that both Corollaries 1.1–1.3 and Theorems 1.4–1.6 are providing with sufficient smallness-type conditions on the perturbation V whichensure stability (in an appropriate sense) of the spectrum of the free Lam´e operator. Notice that if on one handCorollaries 1.1–1.3 seem more general as they are stated for any dimension d ≥ d = 3 only), on the other hand Theorems 1.4–1.6 give a more complete description of the spectrum ofthe perturbed Hamiltonian ensuring the full stability σ ( − ∆ ∗ + V ) = [0 , ∞ ) = σ ( − ∆ ∗ ) instead of stability of thesole point spectrum, i.e. σ p ( − ∆ ∗ + V ) = ∅ = σ p ( − ∆ ∗ ) , as in Corollaries 1.1–1.3. Nevertheless, as far as the case d = 3 is considered and if we focus on the point spectrum only, then Corollaries 1.1–1.3 and Theorems 1.4–1.6equal one another, in the sense that they provide the same smallness conditions on the potentials to guaranteethe stated stability.The rest of the paper is organized as follows: in Section 2 we collect some preliminary results related tothe Lam´e operator which will be used later in the paper. The proofs of Theorems 1.1–1.3 are provided inSection 3. Section 4 is devoted to the proof of the spectral stability valid in the three dimensional setting,namely Theorems 1.4–1.6. Notations • For 1 ≤ p < ∞ and u = ( u , . . . , u d ) ∈ C d we denote | u | p := ( P dj =1 | u j | p ) /p . Also, we will drop thesubscript for p = 2, writing | u | := | u | . • For u = ( u , . . . , u d ) ∈ L p ( R d ) d , we denote k u k L p ( R d ) d := k| u ( · ) | p k L p ( R d ) • Let V : R d → C d × d ; we define | V ( x ) | p := sup u ∈ C d | V ( x ) u | p | u | p , for a.a. x ∈ R d , and k V k L p ( R d ) d × d := k| V ( · ) | p k L p ( R d ) . With a slight abuse of notation we consistently write k V k L p ( R d ) toindicate k V k L p ( R d ) d × d . • As customarily, the notation h· , ·i pp ′ is used to denote the duality pairing L p × L p ′ → C , with 1 /p +1 /p ′ = 1 . • Given a measurable function w , L p ( wdx ) stands for the w -weighted L p space on R d with measure w ( x ) dx. Given a measurable non-negative function w, we say that w belongs to the A p ( R d ) Muckenhoupt class ofweights, for 1 < p < ∞ if the following quantity Q p ( w ) := sup Q | Q | Z Q w ( x ) dx ! | Q | Z Q w ( x ) − p − dx ! p − (18)is finite. Here the supremum is taken over any cube Q in R d . • Any given u ∈ L ( R d ) d can be decomposed as u = u S + u P , where u S is a divergence-free vector field and u P is a gradient, see Lemma 2.1. • The Riesz transform R = ( R , R , . . . , R d ) is defined through the Fourier transform by d R j f ( ξ ) = − i ξ j | ξ | b f ( ξ ) , j = 1 , , . . . , d, for all f ∈ L ( R d ) . • Let d = 3 and let z ∈ C \ [0 , ∞ ) . We denote by G ζ ( x, y ) the integral kernel associated to the resolvent ofthe Laplacian ( − ∆ − z ) − . Its explicit expression is given G ζ ( x, y ) := 14 π e −√− ζ | x − y | | x − y | . (19)Here and in the sequel we choose the principal branch of the square root. • We use the notations C ( • ) , C • or c ( • ) , c • to emphasize the dependence of C or c on • . If not stated, theseconstants are in principle not explicit. Acknowledgment
L.C. thanks O. Ibrogimov for manifesting an interest in the contents of the paper which strongly motivated thisproject. The authors are also grateful to D. Krejˇciˇr´ık and R. L. Frank for useful correspondences.B.C. is supported by Fondo Sociale Europeo – Programma Operativo Nazionale Ricerca e Innovazione 2014-2020, progetto PON: progetto AIM1892920-attivit`a 2, linea 2.1. The research of L.C. is supported by theDeutsche Forschungsgemeinschaft (DFG) through CRC 1173.
In this section we collect some preliminary results on the Lam´e operator, referring to [21] and references thereinfor more details. We start recalling the Helmholtz decomposition of vector fields in L ( R d ) d . Lemma 2.1 ([21, Theorem 2.1, Lemma 2.2]) . Any vector field u ∈ L ( R d ) d can be uniquely split into itsdivergence-free (transversal) part u S and its gradient-type (longitudinal) part u P . More precisely, u can be de-composed as follows u = u S + u P , with u P = ∇ ϕ and u S = u − u P , where the scalar potential ϕ satisfies ∆ ϕ = div u ensuring that div u S = 0 . Furthermore the decomposition is L orthogonal, that is k u k L ( R d ) = k u S k L ( R d ) + k u P k L ( R d ) . Moreover ( π S u ) j = u S ,j = u j + d X k =1 R j R k u k , and ( π P u ) j = u P ,j = − d X k =1 R j R k u k , j = 1 , , . . . , d, (20) being R = ( R , . . . , R d ) is the Riesz transform. Lemma 2.2 ([7, Lemma 2.2], [21, Lemma 2.6]) . Let d ≥ and let f be a regular vector field sufficiently rapidlydecaying at infinity. Then − ∆ ∗ acts on f = f S + f P as − ∆ ∗ f = − µ ∆ f S − ( λ + 2 µ )∆ f P , (21) where f S is a divergence free vector field and f P a gradient. Lemma 2.3 ([7, Lemma 2.2], [21, Lemma 2.7]) . Let z ∈ C \ [0 , ∞ ) and g ∈ L ( R d ) d . Then the identity ( − ∆ ∗ − z ) − g = 1 µ (cid:0) − ∆ − zµ (cid:1) − g S + 1 λ + 2 µ (cid:0) − ∆ − zλ +2 µ (cid:1) − g P (22) holds true, where g = g S + g P is the Helmholtz decomposition of g. Thanks to the representation (21) of − ∆ ∗ in terms of Laplace operators, the quadratic form h associatedwith − ∆ ∗ has the following expression: h [ u ] := µ Z R d |∇ u S | dx + ( λ + 2 µ ) Z R d |∇ u P | dx, D ( h ) := H ( R d ) d . (23)Let V : R d → C d × d be a measurable matrix-valued function such that ∃ a < min { µ, λ + 2 µ } such that Z R d | V || u | dx ≤ a Z R d |∇ u | dx, ∀ u ∈ H ( R d ) , thus the quadratic form v [ u ] := Z R d V u · u dx, D ( v ) := n u ∈ L ( R d ) d : Z R d | V || u | dx < ∞ o (24)is relatively bounded with respect to h with relative bound less than one. As a consequence, the sum h V := h + v is a closed form with D ( h V ) = H ( R d ) d which gives rise to an m-sectorial operator in L ( R d ) d via therepresentation theorem ( cf . [53, Thm. VI.2.1]).In the following lemma we gather some boundedness results for the Riesz transform. Lemma 2.4.
Let < p < ∞ and p ′ such that /p + 1 /p ′ = 1 and let w ∈ A p ( R d ) . Then, for any j = 1 , , . . . , d, the following bounds on the operator norms of the Riesz transform R j hold true: kR j k L p ( R d ) → L p ( R d ) = c p := cot (cid:16) π { p, p ′ } (cid:17) , (25) kR j k L p ( wdx ) → L p ( wdx ) ≤ c w := C Q p ( w ) max { ,p ′ /p } , (26) for some C > independent on w and for Q p ( w ) defined as in (18) .Proof. For the proof of (25) refer to [3] (see also [13]); inequality (26) is proved in [70] (see also [19]).Thanks to Lemma 2.1 and Lemma 2.4, one shows almost-orthogonality of the S and P components in theHelmholtz decomposition. Lemma 2.5.
Let g = g S + g P be the Helmholtz decomposition of g. For < p < ∞ the following estimates holdtrue: k g S k L p ( R d ) d + k g P k L p ( R d ) d ≤ (1 + 2 dc p ) k g k L p ( R d ) d , (27) k g S k L ( wdx ) d + k g P k L ( wdx ) d ≤ (1 + 2 dc w ) k g k L ( wdx ) d , (28) with c p , c w > defined in Lemma 2.4. roof. We prove only (27), the proof of (28) is similar. Let g ∈ L p ( R d ) d : from (20) one has k g S k L p ( R d ) d + k g P k L p ( R d ) d ≤ k g k L p ( R d ) d + 2 k P dk =1 RR k g k k L p ( R d ) d . Using (25) one gets k P dk =1 RR k g k k L p ( R d ) d ≤ P dk =1 ( P dj =1 kR j R k g k k pL p ( R d ) ) p ≤ d p c p P dk =1 k g k k L p ( R d ) ≤ dc p k g k L p ( R d ) d , where in the last inequality we have used the H¨older inequality for discrete measure. Gathering the two previousbounds gives (27).The following result shows the good behavior of the Morrey-Campanato space in relation with the Muchen-houpt class of weights. Lemma 2.6 ([18, Lemma 1]) . Let < α < d , < p ≤ d/α and let V ∈ L α,p ( R d ) , V ≥ . If p ∈ (1 , p ) , then W = ( M V p ) /p ∈ A ( R d ) ∩ L α,p ( R d ) , where M denotes the usual Hardy-Littlewood maximal operator, and V ( x ) ≤ W ( x ) for almost all x ∈ R d . Moreover, there exists a constant
C > independent on V, such the A constant for W is less than C and k W k L α,p ( R d ) ≤ C k V k L α,p ( R d ) . (29)In the next lemma we list uniform estimates for the operator norm of the resolvent ( − ∆ ∗ − z ) − , z ∈ C \ [0 , ∞ ) . Lemma 2.7.
Let z ∈ C \ [0 , ∞ ) . Then the following estimates for the resolvent ( − ∆ ∗ − z ) − hold true.i) Let < p ≤ / if d = 2 , d/ ( d + 2) ≤ p ≤ d + 1) / ( d + 3) if d ≥ and let p ′ such that /p + 1 /p ′ = 1 . Then there exists a universal constant c p,d,λ,µ > such that k ( − ∆ ∗ − z ) − k L p ( R d ) → L p ′ ( R d ) ≤ c p,d,λ,µ | z | − d +22 + dp . (30) ii) Let / < α < if d = 2 , d/ ( d + 1) < α ≤ if d ≥ and let ( d − / α − < p ≤ d/α. Then thereexists a universal constant c α,p,d,λ,µ > such that for any non-negative function V in L α,p ( R d ) k ( − ∆ ∗ − z ) − k L ( V − dx ) → L ( V dx ) ≤ c α,p,d,λ,µ | z | − α k V k L α,p ( R d ) . (31) iii) Let / ≤ α < if d = 2 , d − ≤ α < d if d ≥ and let β = (2 α − d + 1) / . Then there exists a universalconstant c α,d,λ,µ > such that for any non-negative function V such that | V | β ∈ KS α ( R d ) k ( − ∆ ∗ − z ) − k L ( V − dx ) → L ( V dx ) ≤ c α,d,λ,µ Q ( V ) | z | − α − d +12 α − d +1 k| V | β k β KS α ( R d ) . (32) Proof.
For a proof refer to [21, Thm. 2.3] (see also [7, Thm. 1.1]). These estimates are consequence of thecorresponding bounds for the resolvent of the free Schr¨odinger operator ( − ∆ − z ) − ( cfr. [21, Thm. 2.2] and [7,Thm. 3.8] for the collection of statements and references therein for the explicit proof), after using the explicitrepresentation (22) and the boundedness properties of the Riesz transform 2.4.From Lemma 2.7, the following estimates descend for the Birman-Schwinger operator associated to the Lam´eoperator. Lemma 2.8.
Let z ∈ C \ [0 , ∞ ) , < γ ≤ / if d = 2 and ≤ γ ≤ / if d ≥ . Then the following estimatefor the L − L operator norm of the Birman-Schwinger operator K z := | V | / ( − ∆ ∗ − z ) − V / hold true: k K z k L ( R d ) → L ( R d ) ≤ c γ,d,λ,µ | z | − γ γ + d k V k L γ + d ( R d ) . (33)10 or p and α as in Theorem 1.2 one has k K z k L ( R d ) → L ( R d ) ≤ c γ,p,d,λ,µ | z | − γ γ + d k V k L α,p ( R d ) . (34) If, in addition, V ∈ A ( R d ) , then for α and β as in Theorem 1.3 one has k K z k L ( R d ) → L ( R d ) ≤ c γ,d,λ,µ Q ( | V | ) | z | − γ γ + d k| V | β k β KS α ( R d ) . (35) Proof.
The three estimates (33), (34) and (35) follow straightforwardly from the validity of (30), (31) and (32),respectively. An explicit proof is obtained in [21] as a byproduct of the proofs of Theorem 1.2, Theorem 1.3and Theorem 1.4 there.For later purposes, we rewrite the statement of Lemma 2.8 in the case that d = 3 and γ = 0, since inthis situation we are able to provide more explicit information on the bound of the operator norm of theBirman-Schwinger operator. Lemma 2.9.
Let d = 3 and let z ∈ C \ [0 , ∞ ) . Assume < p ≤ / . Then the following estimates for theBirman-Schwinger operator K z := | V | / ( − ∆ ∗ − z ) − V / hold true: k K z k L ( R ) → L ( R ) ≤ / (1 + 6 cot ( π/ π / min { µ, λ + 2 µ } k V k L / ( R ) , (36) k K z k L ( R ) → L ( R ) ≤ c F (1 + 6 C )min { µ, λ + 2 µ } k V k L ,p ( R ) , (37) with c F as in (7) and C > . If, in addition, V ∈ A ( R ) , then k K z k L ( R ) → L ( R ) ≤ c KS (1 + 6 c V )min { µ, λ + 2 µ } k V k KS ( R ) , (38) with c KS as in (7) and c V := C Q ( | V | ) , for C > .Proof. Bounds (36)-(38) are simply bounds (33)-(35) in the specific framework considered here. To get theexplicit values of the constants in in (36)-(38), we provide a direct proof which relies on the explicit expressionof the integral kernel G z ( x, y ) of ( − ∆ − z ) − in d = 3 . To bound the operator norm of K z we estimate the inner product h f, K z g i , for any f, g ∈ L ( R ) . Therelation (22) gives |h f, K z g i| ≤ µ |h f, | V | / ( − ∆ − zµ ) G S i| + 1 λ + 2 µ |h f, | V | / ( − ∆ − zλ +2 µ ) G P i| , (39)where we set G = G S + G P := V / g. First we estimate |h f, | V | / ( − ∆ − zµ ) G S i| . Given the explicit expression (19)for the integral kernel of ( − ∆ − ζ ) − , ζ ∈ C \ [0 , ∞ ) , one has that G ζ ( x, y ) is bounded in absolute value by theGreen function G ( x, y ) := (4 π | x − y | ) − , i.e. , |G ζ ( x, y ) | ≤ G ( x, y ) . Hence |h f, | V | / ( − ∆ − zµ ) G S i| ≤ h| f | , | V | / | ( − ∆ − zλ +2 µ ) G S |i = Z Z R × R | f | ( x ) | V ( x ) | / |G z/µ ( x, y ) || G S ( y ) | dx dy ≤ π Z Z R × R | f ( x ) | | V ( x ) | / | G S ( y ) || x − y | dx dy ≤ / π / k| f | | V | / k L / ( R ) k| G S |k L / ( R ) , where in the last inequality we used the sharp Hardy-Littlewood-Sobolev inequality (see [66], [67, Thm. 4.3]).Analogous computations for |h f, | V | / ( − ∆ − zµ ) G P i| give |h f, | V | / ( − ∆ − zµ ) G P i| ≤ / π / k| f | | V | / k L / ( R ) k| G P |k L / ( R ) . |h f, K z g i| ≤ { µ, λ + 2 µ } / π / k f | V | / k L / ( R ) (cid:0) k G S k L / ( R ) + k G P k L / ( R ) (cid:1) . Using the orthogonality property in Lemma 2.5, thanks to the H¨older inequality one has |h f, K z g i| ≤ { µ, λ + 2 µ } / π / (1 + 6 cot ( π/ k f | V | / k L / ( R ) k V / g k L / ( R ) ≤ { µ, λ + 2 µ } / π / (1 + 6 cot ( π/ k V k L / ( R ) × k f k L ( R ) k g k L ( R ) . Taking the supremum over all f, g ∈ L ( R ) with norm equal to one, gives the bound in (36).Now we are in position to prove (37) and (38). As a starting point we observe that under the assumptionsof the lemma the Hardy type inequality (6) with (7) holds true.It is known that estimates of type (6) are equivalent to weighted boundedness properties of the 1- fractionalintegral operator I = H − / where H := − ∆ (see [38, Lemma 1]). Then (6) is equivalent to k| V | / H − / k L → L ≤ √ a (40)and, by taking the adjoint, one also has k H − / | V | / k L → L ≤ √ a, (41)where a is as in (7).Let us prove (37) first, that is let us assume that V ∈ L ,p ( R ). By Lemma 2.6, there exists W ∈ A ( R ) ∩L ,p ( R ) such that V ( x ) ≤ W ( x ) for almost all x ∈ R . Using this fact and the pointwise bound |G ζ ( x, y ) | ≤G ( x, y ) , z ∈ C \ (0 , ∞ ) , x, y ∈ R , we have |h f, | V | / ( − ∆ − zµ ) G S i|≤ h| f | , | W | / H − | G S |i≤ k f k L ( R ) k| W | / H − / k L → L k H − / | W | / k L → L k G S k L ( | W | − dx ) ≤ a k f k L ( R ) k G S k L ( | W | − dx ) , (42)where in the last inequality we have used bounds (40) and (41). Similar computations for the term in (39)involving the P component, namely |h f, | V | / ( − ∆ − zλ +2 µ ) G P i| give |h f, | V | / ( − ∆ − zλ +2 µ ) G P i| ≤ a k f k L ( R ) k G P k L ( | W | − dx ) . (43)Plugging (42) and (43) in (39) one has |h f, K z g i| ≤ a min { µ, λ + 2 µ } k f k L ( R ) (cid:0) k G S k L ( | W | − dx ) + k G S k L ( | W | − dx ) (cid:1) . Using the orthogonality property (28) stated in Lemma 2.5, recalling that | W | − ≤ | V | − almost everywhereand using that G := V / g we get |h f, K z g i| ≤ a C Q ( | W | ) min { µ, λ + 2 µ } k f k L ( R ) k g k L ( R ) . (44)From Lemma 2.6 we know that Q ( | W | ) is less than a constant independent on W , so estimate (37) followsfrom (44) using the bound (29) in a = c F k W k L ,p ( R ) .The proof of (38) descends from (44) with W = V and a = c KS k V k KS ( R ) . The next lemma represents another relevant consequence of the boundedness of the Birman-Schwingeroperator. We are grateful to R.L. Frank for showing us the argument.12 emma 2.10.
Let γ, p and α as in Lemma 2.8. If V ∈ L γ + d/ ( R d ) , V ∈ L α,p ( R d ) , or V ∈ KS α ( R d ) , then themultiplication by | V | / is a bounded operator from H ( R d ) d to L ( R d ) d . Proof.
Minor modifications of the argument in Lemma 2.8 ensure that the operator | V | / ( − ∆ ∗ − z ) − | V | / with z ∈ ( −∞ ,
0] is a bounded operator in L , more precisely k| V | / ( − ∆ ∗ − z ) − | V | / k ≤ C ( z, V ) k V k , where k V k denotes k V k = k V k L γ + d ( R d ) , k V k = k V k L α,p ( R d ) or k V k = k V k KS α ( R d ) and C ( z, V ) is a constantthat may depend on | z | and V ( cfr. (33)–(35)). Since z ∈ ( −∞ , − ∆ ∗ − z ) is a positive operator. Wewrite | V | / ( − ∆ ∗ − z ) − | V | / = AA ∗ , with A := | V | ( − ∆ ∗ − z ) − / . Using that k AA ∗ k = k A ∗ k = k A k = k| V | / ( − ∆ ∗ − z ) − / k , one has Z R d | V || u | = k| V | / ( − ∆ ∗ − z ) − / ( − ∆ ∗ − z ) / u k L ( R d ) ≤ C ( z, V ) k V kk ( − ∆ ∗ − z ) / u k L ( R d ) = C ( z, V ) k V kh u, ( − ∆ ∗ − z ) u i = C ( z, V ) k V k (cid:0) h [ u ] − z k u k L ( R d ) (cid:1) , (45)where h denotes the quadratic form associated to the Lam´e operator − ∆ ∗ defined in (23). Using the explicitexpression (23) for the quadratic form h one can rewrite (45) as Z R d | V || u | ≤ C ( z, V, λ, µ ) k V k (cid:0) k∇ u k L ( R d ) − z k u k L ( R d ) (cid:1) , (46)where C ( z, V, λ, µ ) = C ( z, V ) max { µ, λ + 2 µ } and C ( z, V ) is as in (45). Hence, | V | / u ∈ L ( R d ) d whenever u ∈ H ( R d ) d . Remark . Assuming an Hardy-type condition like Z R d | V || u | dx ≤ C ( V ) Z R d |∇ u | dx, ∀ u ∈ C ∞ ( R d ) (47)would also serve the purpose of ensuring boundedness of | V | / as an operator from H ( R d ) d to L ( R d ) d . Asa matter of fact, it is known that (47) holds true if V ∈ L d/ ( R d ) , that is V ∈ L γ + d/ ( R d ) and γ = 0 (as aconsequence of H¨older inequality and Sobolev embedding), if V ∈ L α,p ( R d ) with α = 2 (which, in turn, gives γ = 0 , recall α := 2 d/ (2 γ + d )) and 1 < p ≤ d , fact that was discovered by Fefferman in [40] (see also Chiarenza-Frasca [18]) and if V ∈ KS α ( R d ) , with α = 2 (notice that as α = 2 dβ/ (2 γ + d ) and β = ( d +2 γ )( d − / [2( d − γ )] ,α = 2 gives γ = d (3 − d ) / . Since γ ≥ d = 3 and so γ = 0 . ). Thus estimate (46) generalizes (47),which corresponds to γ = 0 and after letting z go to zero (notice that if γ = 0 the constant C ( z, V, λ, µ ) in (46)is no more dependent on z, see (33)–(35)). In this section we provide the proofs of Theorems 1.1–1.3 valid in dimension d ≥ . We give two different proofsof Theorem 1.1: the proof in Section 3.1 is strongly sensitive of the L p framework, while the proof in Section 3.2is more robust and it is adapted to prove also Theorem 1.2 and Theorem 1.3. The strategy of the proof of Theorem 1.1 follows the one of [48, Thm. 3.2] with the modifications necessary totreat the Lam´e operator.For notation convenience we define p such that p/ (2 − p ) = γ + d/ . Thus the assumptions on γ give1 < p ≤ / d = 2 and 2 d/ ( d + 2) ≤ p ≤ d + 1) / ( d + 3) if d ≥ . V ∈ L p − p ( R d ) is a bounded operator from L p ′ ( R d ) d to L p ( R d ) d with 1 /p + 1 /p ′ = 1 . Let z ∈ C be an eigenvalue of − ∆ ∗ + V in L ( R d ) d with eigenfunction u. Since − ∆ ∗ + V is defined via m -sectorial forms, we know a-priori that an eigenfunction satisfies u ∈ H ( R d ) d . Inparticular, by Sobolev embedding, u ∈ L r ( R d ) d , for 2 ≤ r ≤ d/ ( d −
2) and so u ∈ L p ′ ( R d ) d . We start considering the easiest situation, i.e. , when z ∈ C \ [0 , ∞ ) . In this case the resolvent operator( − ∆ ∗ − z ) − ∈ B ( L p ( R d ) d ; L p ′ ( R d ) d ) and from Lemma 2.7 one has k ( − ∆ ∗ − z ) − k L p ( R d ) d → L p ′ ( R d ) d ≤ N ( z ) , N ( z ) = c p,d,λ,µ | z | − d +22 + dp . (48)Using that ( − ∆ ∗ + V ) u = zu one can write u = ( − ∆ ∗ − z ) − ( − ∆ ∗ − z ) u = − ( − ∆ ∗ − z ) − V u. (49)From the previous expression and the resolvent estimate (48), one has k u k L p ′ ( R d ) d = k ( − ∆ ∗ − z ) − V u k L p ′ ( R d ) d ≤ N ( z ) k V u k L p ( R d ) d ≤ N ( z ) k V k L p − p ( R d ) k u k L p ′ ( R d ) d . Using that N ( z ) = c p,d,λ,µ | z | − d +22 + dp , we have1 ≤ c p,d,λ,µ | z | − d +22 + dp k V k L p − p ( R d ) , (50)which gives the thesis once we replace p/ (2 − p ) = γ + d/ . It is clear that z = 0 belongs to the right hand side of (11), then it remain to consider the case z ∈ (0 , ∞ ) . In this situation, since z belongs to the spectrum of the free Lam´e operator − ∆ ∗ , the expression (49) no longermakes sense. On the other hand, taking ε > , the operator ( − ∆ ∗ − z − iε ) − is well defined and bounded from L p ( R d ) d to L p ′ ( R d ) d . Thus, for u such that ( − ∆ ∗ + V ) u = zu, one considers an approximating eigenfunction u ε defined as u ε = ( − ∆ ∗ − z − iε ) − ( − ∆ ∗ − z ) u = − ( − ∆ ∗ − z − iε ) − V u.
Since V ∈ B ( L p ′ ( R d ) d ; L p ( R d ) d ) and ( − ∆ ∗ − z − iε ) − ∈ B ( L p ( R d ) d ; L p ′ ( R d ) d ) , we infer that u ε ∈ L p ′ ( R d ) d and k u ε k L p ′ ( R d ) d = k ( − ∆ ∗ − z − iε ) − V u k L p ′ ( R d ) d ≤ N ( z + iε ) k V u k L p ( R d ) d ≤ N ( z + iε ) k V k L p − p ( R d ) k u k L p ′ ( R d ) d . (51)From its explicit expression, one sees that N ( z + iε ) converges to N ( z ) as ε → , thus the sequence u ε isuniformly bounded in L p ′ ( R d ) d and therefore converges (up to subsequences) weakly in L p ′ ( R d ) d . Now we wantto show that u ε converges strongly in L ( R d ) d to u as ε approaches zero. Due to the L orthogonality of the S and P component of the Helmholtz decomposition it is enough to check that ( u ε ) S converges to u S , theconvergence of ( u ε ) P to u P follows similarly. Using the expressions (21) and (22) and applying Planchereltheorem one has k ( u ε ) S − u S k L ( R d ) d = (cid:13)(cid:13)(cid:2)(cid:0) − ∆ − z + iεµ (cid:1) − ( − ∆ − zµ ) − I (cid:3) u S (cid:13)(cid:13) L ( R d ) d = (cid:13)(cid:13)(cid:2)(cid:0) | ξ | − z + iεµ (cid:1) − ( | ξ | − zµ ) − (cid:3)b u S (cid:13)(cid:13) L ( R d ) d , then the conclusion follows from dominated convergence theorem.To show that u ε converges weakly to u in L p ′ , it is enough to prove that h u ε , ϕ i p ′ p converges to h u, ϕ i p ′ p forall ϕ ∈ L ∩ L p , that is immediate from Cauchy-Schwarz inequality and the strong convergence of u ε to u in L . Finally, using the weak lower semi-continuity of the norm and the preliminary estimate (51), one has k u k L p ′ ( R d ) d ≤ lim inf ε → k u ε k L p ′ ( R d ) d ≤ lim inf ε → N ( z + iε ) k V k L p − p ( R d ) k u k L p ′ ( R d ) d = N ( z ) k V k L p − p ( R d ) k u k L p ′ ( R d ) d . From this, as above, one concludes that the bound (50) holds, which gives the thesis.14 emark . In the proof of the previous result two ingredients have been used in a crucial way: the uni-form resolvent estimate (30) from Lemma 2.7, which holds true for spectral parameters outside the spectrum σ ( − ∆ ∗ ) = [0 , ∞ ), and the continuity of N ( z ) up to (0 , ∞ ) , that is N ( z + iε ) → N ( z ) as ε goes to zero and z ∈ (0 , ∞ ) , which allows us to cover also the case of possible embedded eigenvalues z ∈ (0 , ∞ ) . Recently, Kwon, Lee and Seo [62], adapting recent sharp resolvent estimates obtained for the Laplacian bytwo of the three authors in [61], were able to prove analogous estimates for the Lam´e operator, which improvethe one stated in Lemma 2.7. More precisely, they proved the following result:
Theorem 3.1 ([62, Theorem 1.3]) . Let d ≥ , z ∈ C \ [0 , ∞ ) , < p ≤ q < ∞ . If ( p , q ) ∈ R ∪ e R ∪ e R ∪ e R ′ (see [62, Def. 1.1]), then one has k ( − ∆ ∗ − z ) − k L p ( R d ) → L q ( R d ) ≤ N ( z ) , where N ( z ) = c p,q,d,λ,µ | z | − d (cid:0) p − q (cid:1) dist( z/ | z | , [0 , ∞ )) − γ p,q , with γ p,q := max { , − d +12 ( p − q ) , d +12 − dp , dq − d − } . As a consequence, the following result on location of discrete eigenvalues is also proven.
Corollary 3.1 ([62, Corollary 1.4]) . Let d ≥ , < p ≤ q < ∞ , ( p , q ) ∈ R ∪ e R ∪ e R ∪ e R ′ (see [62, Def. 1.1]).Then any eigenvalue z ∈ C \ [0 , ∞ ) of − ∆ ∗ + V acting on L q ( R d ) d satisfies | z | − d ( p − q ) dist( z/ | z | , [0 , ∞ )) γ p,q ≤ c p,q,d,λ,µ k V k L pqq − p ( R d ) . For the explicit expressions of the ranges R , e R , e R and e R ′ of allowed indexes p, q we refer the reader tothe original paper [62] (Definition 1.1 and Figure 1 and 2 there); we give a few comments on their result here.The region R is represented by p, q such that2 d + 1 ≤ p − q ≤ d , p > d + 12 d , q < d − d , (52)and p − q = d if d = 2 . In particular, the duality line p + p ′ = 1 restricted to dd +2 ≤ p ≤ d +1) d +3 ofestimate (30) is contained in the range R . Notice that if p, q satisfies (52), then γ p,q in Theorem 3.1 is zero,that is, within R , N ( z ) depends only on | z | , whereas outside R it also depends on the distance from thespectrum σ ( − ∆ ∗ ) = [0 , ∞ ) . Thus, in light of Remark 3.1 above, outside R , since N ( z ) becomes singular as z approaches the positive real axis, Corollary 3.1 cannot be improved to cover also possible embedded eigenvalues z ∈ (0 , ∞ ) . Finally, notice that the range R allows for a larger collection of indexes p, q than just the self-dualcase p, p ′ . On the other hand, as long as one is interested in finding bounds on the location of eigenvalues interms of norms of the potential, considering the whole range R (instead of just the duality line 1 /p + 1 /p ′ = 1)does no provide with more information. Indeed, in this context, it is the local integrability/asymptotic behaviorof the potential V that matters, or better for which class of potential such a bound holds true, in other words onetakes into account not the pair (1 /p, /q ) but rather the difference 1 /r := 1 /p − /q (notice that pq/ ( q − p ) = r and V ∈ L r ). In this section we use an adaptation of the Birman-Schwinger principle in the spirit of the proof providedabove. Nonetheless, the proof presented here allows to treat at a time the L p framework as well as the Morrey-Campanato and the Kerman-Sawyer setting.Let z ∈ C be an eigenvalue of − ∆ ∗ + V in L ( R d ) d with corresponding eigenfunction u ∈ H ( R d ) d . We firstconsider the easiest case of eigenvalues outside the spectrum of − ∆ ∗ , namely z ∈ C \ [0 , ∞ ) . In this situation thestandard Birman-Schwinger principle applies: if z ∈ C \ [0 , ∞ ) is an eigenvalue of the perturbed Lam´e operator − ∆ ∗ + V with corresponding eigenfunction u ∈ H ( R d ) d , then − K z := | V | / ( − ∆ ∗ − z ) − V / with eigenvector φ := | V | / u ∈ L ( R d ) d (see, for example, Thm. III.1215nd Thm. III.14 in [74]). Notice that φ := | V | / u ∈ L ( R d ) d for any u ∈ H ( R d ) d by Lemma 2.10. Since φ = −| V | / ( − ∆ ∗ − z ) − V / φ, using the bounds in Lemma 2.8 we get k φ k L ( R d ) d ≤ k| V | / ( − ∆ ∗ − z ) − V / k L → L k φ k L ( R d ) d ≤ c γ,d,λ,µ | z | − γ γ + d k V kk φ k L ( R d ) d , where k V k denotes k V k L γ + d ( R d ) , k V k L α,p ( R d ) or Q ( | V | ) k| V | β k β KS α ( R d ) depending on which operator estimatefrom Lemma 2.8 we used to bound the norm of the Birman-Schwinger operator, namely (33), (34) or (35),respectively. This gives the proof of Theorems 1.1–1.3 for z ∈ C \ [0 , ∞ ) . Now, let z ∈ [0 , ∞ ) . Observe that for any ε > z ∈ [0 , ∞ ) the operator ( − ∆ ∗ − z − iε ) − is well defined.The approximating eigenfunction u ε := ( − ∆ ∗ − z − iε ) − ( − ∆ ∗ − z ) u, satisfies the corresponding problem( − ∆ ∗ − z − iε ) u ε = − V u.
Defining the auxiliary functions φ := | V | / u and φ ε := | V | / u ε one easily gets the following identity φ ε = −| V | / ( − ∆ ∗ − z − iε ) − V / φ. Passing to the norms and using Lemma 2.8 we get k φ ε k L ( R d ) d ≤ k| V | / ( − ∆ ∗ − z − iε ) − V / k L → L k φ k L ( R d ) d ≤ c γ,d,λ,µ ( | z | + ε ) − γ γ + d k V kk φ k L ( R d ) d , where, as above, k V k denotes k V k L γ + d ( R d ) , k V k L α,p ( R d ) or Q ( | V | ) k| V | β k β KS α ( R d ) . Thus the theses of Theo-rem 1.1, Theorem 1.2 and Theorem 1.3 follow letting ε go to zero as soon as one proves that φ ε converges to φ in L ( R d ) d . Notice first that using the dominated convergence theorem in Fourier space, one easily checks as inSubsection 3.1 that u ε := ( − ∆ ∗ − z − iε ) − ( − ∆ ∗ − z ) u converges to u in H ( R d ) d . Then the convergence of φ ε to φ in L ( R d ) d follows as a consequence the boundedness of | V | / as an operator from H ( R d ) d to L ( R d ) d ,see Lemma 2.10. This section is devoted to the proof of Theorems 1.4–1.6 which show that the spectrum of the perturbed Lam´eoperator in d = 3 remains stable under suitable small perturbations ( cfr. (14), (15) and (16)).We first prove Theorem 1.4: Theorem 1.5 and Theorem 1.6 are obtained with minor modifications of theargument. The proof of Theorem 1.4 follows the strategy developed in [38] to prove the analogous result forthree dimensional Schr¨odinger operators and it will be obtained as a consequence of some preliminary resultswhich are contained in the following subsections. The final proof of Theorems 1.4, and then of Theorem 1.5and Theorem 1.6, can be found in Section 4.5.In the following lemma we show that under the assumption (14) the Birman-Schwinger operator K z isbounded with bound strictly less than one, using the explicit formula (19) for the Green function G z ( x, y ) of − ∆ − z . Lemma 4.1.
Let d = 3 and assume (14) . Then there exists a positive constant a < such that k K z k L ( R ) → L ( R ) ≤ a , for all z ∈ C \ (0 , ∞ ) . (53) Proof.
To bound the operator norm of K z we estimate the inner product h f, K z g i , for any f, g ∈ L ( R ) . Usingthe same strategy of the proof of Lemma 2.9 one has |h f, K z g i| ≤ a c V min { µ, λ + 2 µ } k f k L ( R ) k g k L ( R ) . Thanks to (14), we get the thesis for a := a (1 + 6 c V ) / min { µ, λ + 2 µ } < .1 Absence of eigenvalues As a starting point we observe that under the assumption (14), Corollary 1.3 ensure the absence of the pointspectrum, more precisely we have the following result.
Proposition 4.1 (Absence of eigenvalues) . Let d = 3 and assume (14) . Then σ p ( − ∆ ∗ + V ) = ∅ . Proof.
Taking into account Remark 1.3 the proposition is an easy consequence of Corollary 1.3.The next step we accomplish is to show the absence of the continuous spectrum outside [0 , ∞ ) . [0 , ∞ ) We need the following lemma which is valid for any dimension d ≥ . Lemma 4.2.
Let d ≥ and assume (14) . If k ( − ∆ ∗ + V ) u n − zu n k L ( R d ) d → as n → ∞ with some z ∈ C \ R and { u n } n ∈ N ⊂ H ( R d ) d such that k u n k L ( R d ) d = 1 for all n ∈ N , then φ n := | V | / u n obeys lim n →∞ h φ n , K z φ n ik φ n k L ( R d ) d = − . Proof.
Given z ∈ C \ R and using the explicit representation of the resolvent given in (22) one has h ϕ n , K z φ n i = h ϕ n , | V | / ( − ∆ ∗ − z ) − V / φ n i = h ϕ n , | V | / ( − ∆ ∗ − z ) − V u n i = 1 µ h ϕ n , | V | / ( − ∆ − zµ ) − ( V u n ) S i + 1 λ + 2 µ h ϕ n , | V | / ( − ∆ − zλ +2 µ ) − ( V u n ) P i = I + II. (54)We consider only I as II can be treated similarly.Defining F n = F n, S + F n, P := V u n we have I := 1 µ h ϕ n , | V | / ( − ∆ − zµ ) − F n, S i = 1 µ Z Z R × R ϕ n ( x ) | V | / ( x ) G zµ ( x, y ) F n, S ( y ) dx dy = 1 µ Z R η n,µ ( y ) F n, S ( y ) dy, (55)where η n,µ := Z R G zµ ( x, · ) | V | / ( x ) ϕ n ( x ) dx = ( − ∆ − zµ ) − | V | / ϕ n , (56)where the second equality holds due to the symmetry G ζ ( x, y ) = G ζ ( y, x ) . The analogous computations for II give II = 1 λ + 2 µ Z R η n,λ +2 µ ( y ) F n, P ( y ) dy, (57)where η n,λ +2 µ is defined analogously to η n,µ in (56).Using (55) and (57) in (54) gives h φ n , K z φ n i = 1 µ Z R d η n,µ ( y ) F n, S ( y ) dy + 1 λ + 2 µ Z R d η n,λ +2 µ ( y ) F n, P ( y ) dy. (58)Notice that η n,µ , η n,λ +2 µ ∈ H ( R d ) d . Indeed, writing H := − ∆ , we have η n,µ = ( H − z/µ ) − H / H − / | V | / φ n , (59)17ince φ n ∈ L ( R d ) d by (14), H − / | V | / is bounded due to (40) and ( H − z/µ ) − H / maps L ( R d ) d to H ( R d ) d , one has k η n,µ k L ( R d ) d ≤ C z/µ √ a k φ n k L ( R d ) d , where C z/µ := sup ξ ∈ [0 , ∞ ) (cid:12)(cid:12)(cid:12)(cid:12) ξξ − z/µ (cid:12)(cid:12)(cid:12)(cid:12) . (60)Due to the L -orthogonality of the S and P component of the Helmholtz decomposition and using that theprojection into the S and P components commutes with the Laplacian ( cfr. (20)), from k ( − ∆ ∗ + V ) u n − zu n k L ( R d ) d → , we get k ( − ∆ − z/µ )( u n ) S + µ F n, S k L ( R d ) d → , and k ( − ∆ − z/ ( λ + 2 µ ))( u n ) P + λ +2 µ F n, P k L ( R d ) d → . (61)Let us define the following quantities R µ := h∇ η n,µ , ∇ ( u n ) S i − zµ h η n,µ , ( u n ) S i , R λ +2 µ := h∇ η n,λ +2 µ , ∇ ( u n ) P i − zλ + 2 µ h η n,λ +2 µ , ( u n ) P i . Thanks to (59), we have R µ = h∇ ( u n ) S , ∇ η n,µ i − zµ h ( u n ) S , η n,µ i = h H / ( u n ) S , H / ( H − z/µ ) − H / H − / | V | / φ n i − zµ h ( u n ) S , η n,µ i = h H / ( u n ) S , H − / | V | / φ n i + zµ h H / ( u n ) S , ( H − z/µ ) − H − / | V | / φ n i − zµ h ( u n ) S , η n,µ i = h H / ( u n ) S , H − / | V | / φ n i = h ( H − / | V | / ) ∗ H / ( u n ) S , φ n i = h| V | / ( u n ) S , φ n i . Similar computations for R λ +2 µ give R λ +2 µ = h| V | / ( u n ) P , φ n i . Adding and subtracting the quantities R µ and R λ +2 µ to (58) and noticing that R µ + R λ +2 µ = h| V | / ( u n ) S , φ n i + h| V | / ( u n ) P , φ n i = h| V | / ( u n ) , φ n i = k φ n k L ( R d ) d , one has h φ n , K z φ n i = R µ + 1 µ Z R d η n,µ ( y ) F n, S ( y ) dy + R λ +2 µ + 1 λ + 2 µ Z R d η n,λ +2 µ ( y ) F n, P ( y ) dy − k φ n k L ( R d ) d (62)Since k ( − ∆ ∗ + V ) u n − zu n k L ( R d ) d = sup ϕ ∈ L ( R d ) d ϕ =0 h ϕ, ( − ∆ ∗ + V ) u n − zu n ik ϕ k L ( R d ) d ≥ | µ k∇ u n, S k L ( R d ) d + ( λ + 2 µ ) k∇ u n, P k L ( R d ) d + v [ u n ] − z | , where the inequality is obtained choosing ϕ = u n , and the left-hand side vanishes as n goes to infinity, we have ℑ v [ u n ] tends to ℑ z = 0 as n goes to infinity. In particular, lim inf n →∞ k φ n k L ( R d ) d > . From (62) one has h φ n , K z φ n ik φ n k L ( R d ) d = 1 k φ n k L ( R d ) d h R µ + 1 µ Z R d η n,µ ( y ) F n, S ( y ) dy i + 1 k φ n k L ( R d ) d h R λ +2 µ + 1 λ + 2 µ Z R d η n,λ +2 µ ( y ) F n, P ( y ) dy i − I + II − . I and II tend to zero as n goes to infinity. Using the explicit expressions for R µ and R λ +2 µ and estimate (60), one has | I | = |h η n,µ , ( − ∆ − z/µ )( u n ) S + µ F n, S i|k φ n k L ( R d ) d ≤ k η n,µ k L ( R d ) d k ( − ∆ − z/µ )( u n ) S + µ F n, S k L ( R d ) d k φ n k L ( R d ) d ≤ C z/µ √ a k ( − ∆ − z/µ )( u n ) S + µ F n, S k L ( R d ) d k φ n k L ( R d ) d . Since lim inf n →∞ k φ n k L ( R d ) d > n goesto infinity. Analogous computations show that also II vanishes as n → ∞ . This yieldslim n →∞ h φ n , K z φ n ik φ n k L ( R d ) d = − , ∞ ) . Proposition 4.2.
Let d = 3 and assume (14) . Then σ c ( − ∆ ∗ + V ) ⊂ [0 , ∞ ) . Proof.
Consider ℜ h V [ u ] , where h V [ u ] is the quadratic form associated with − ∆ ∗ + V (see (23),(24)). One has ℜ h V [ u ] = µ Z R |∇ u S | dx + ( λ + 2 µ ) Z R |∇ u P | dx + ℜ Z R V u · u dx. By assumption (14), ℜ h V [ u ] ≥ (min { µ, λ + 2 µ } − a ) k∇ u k ≥ u ∈ H ( R ) . Since − ∆ ∗ + V is m-sectorial, then its spectrum is contained in the right complex half-plane ( cf . [53, Thm. V.3.2]). Now, assumeby contradiction that there exists z ∈ C with ℜ z ≥ ℑ z = 0 such that z ∈ σ c ( − ∆ ∗ + V ) . Then z belongsto the kind of essential spectrum which is characterized by the existence of a singular sequence of − ∆ ∗ + V corresponding to z ( cf. [33, Thm. IX.1.3]): there exists { u n } n ∈ N ⊂ H ( R ) such that k u n k L ( R ) = 1 for all n ∈ N , k ( − ∆ ∗ + V − z ) u n k L ( R ) → n → ∞ and { u n } n ∈ N is weakly converging to zero. By Lemma 4.2and (53), one has a > k K z k ≥ (cid:12)(cid:12)(cid:12) lim n →∞ h u n , K z u n ik u n k L ( R ) (cid:12)(cid:12)(cid:12) = 1 , which is a contradiction as a < . [0 , ∞ ) in the spectrum Now we show that the semi axis [0 , ∞ ) lies in the spectrum. In order to do that we shall use the followingcriterion. Lemma 4.3 ([38, Lemma 4]) . Let H be an m-sectorial accretive operator in a complex Hilbert space H which isassociated with a densely defined, closed, sectorial) sesquilinear form h. Given z ∈ C , assume that there existsa sequence { φ n } n ∈ N ⊂ D ( h ) such that k φ n k H = 1 for all n ∈ N and sup ψ ∈D ( h ) ψ =0 | h ( φ n , ψ ) − z ( φ n , ψ ) |k ψ k D ( h ) −−−−→ n →∞ , (63) where k ψ k D ( h ) := p ℜ h [ ψ ] + k ψ k . Then z ∈ σ ( H ) . In the following we construct an appropriate sequence { φ n } n ∈ N to apply Lemma 4.3 to H = − ∆ ∗ + V and z ∈ [0 , ∞ ) , showing then that [0 , ∞ ) ⊂ σ ( − ∆ ∗ + V ) . In [38] the authors proved the analogous result for theSchr¨odinger operator taking as { φ n } n ∈ N the standard singular sequence for the Laplacian. In order to adaptthat construction to this setting, we perform a suitable diagonalization argument operated on the symbol of theLam´e operator. 19 emma 4.4. Let d ≥ . For any z ∈ (0 , ∞ ) there exists a classical solution u ∈ C ∞ ( R d ) d to − ∆ ∗ u − zu = 0 (64) such that | u ( x ) | = 1 for all x ∈ R and its derivatives are bounded.Proof. For simplicity of notation, we give a proof in the case that d = 3. The general case d ≥ − ∆ ∗ , u is solution to (64) if and only if its Fourier trasform b u := F u satisfies L ( ξ ) b u ( ξ ) − z b u ( ξ ) = 0 , for a.a. ξ ∈ R , where L ( ξ ) = µ | ξ | b u ( ξ ) + ( λ + µ ) ξξ t b u ( ξ ) = µ | ξ | + ( λ + µ ) ξ ( λ + µ ) ξ ξ ( λ + µ ) ξ ξ ( λ + µ ) ξ ξ µ | ξ | + ( λ + µ ) ξ ( λ + µ ) ξ ξ ( λ + µ ) ξ ξ ( λ + µ ) ξ ξ µ | ξ | + ( λ + µ ) ξ . For a.e. ξ ∈ R we have that P − ( ξ ) L ( ξ ) P ( ξ ) = D ( ξ ) := µ | ξ | µ | ξ |
00 0 ( λ + 2 µ ) | ξ | , with P ( ξ ) = − ξ − ξ ξ ξ ξ ξ ξ . Determining a solution u of (64) is equivalent to find a vector field b v = ( b v , b v , b v ) := P − b u such that D ( ξ ) b v ( ξ ) − z b v ( ξ ) = 0 . Using the inverse Fourier transform, one is reduced to determine v = ( v , v , v ) a solution to theHelmholtz-type system − ∆ v − zµ v = 0 , − ∆ v − zµ v = 0 , − ∆ v − zλ +2 µ v = 0 , (65)and a solution u to (64) is given by u = F − P F v = i − ∂ v − ∂ v + ∂ v ∂ v + ∂ v ∂ v + ∂ v . For k := (0 , z/µ, v ( x ) := ( µe ik · x /z, ,
0) is clearly solution to (65). So, the function u ( x ) =( e ik · x , ,
0) is solution to (64), | u ( x ) | = 1 for almost all x ∈ R and all its derivatives are bounded.With this result at hand, we are in position to prove the following theorem guaranteeing that the semi-axis[0 , ∞ ) belongs to the spectrum of − ∆ ∗ + V. Proposition 4.3.
Let d ≥ and assume (14) . Then [0 , ∞ ) ⊂ σ ( − ∆ ∗ + V ) . Proof.
Let z ∈ (0 , ∞ ) . We construct the sequence { φ n } n ∈ N from Lemma 4.3 applied to H = − ∆ ∗ + V and z ,making use of Lemma 4.4. Let u be as in Lemma 4.4: we set φ n ( x ) = ϕ n ( x ) u ( x ) , where ϕ n ( x ) := n − d/ ϕ ( x/n )for all n ≥ ϕ ∈ C ∞ ( R d ) , k ϕ k L ( R d ) = 1 . Clearly k ϕ n k L ( R d ) = k ϕ k L ( R d ) = 1 , k∇ ϕ n k L ( R d ) = n − k∇ ϕ k L ( R d ) , k ∂ j ∂ k ϕ n k L ( R d ) = n − k ∂ j ∂ k ϕ k L ( R d ) , (66)for any j, k = 1 , , . . . , d. Notice that as u is chosen such that | u ( x ) | = 1 , then k φ n k L ( R d ) d = 1 and clearly φ n ∈ D ( h ) = D ( h ) = H ( R d ) d for all n ∈ N . Moreover, using that u satisfies (64) and that u and its derivativesare bounded, one has (we hide the summation over repeated symbols) k − ∆ ∗ φ n − zφ n k L ( R d ) d = k − µ ∆ ϕ n u − µ ( ∇ ϕ n · ∇ ) u − ( λ + µ ) ∇ ϕ n div u − ( λ + µ ) ∂ j ∇ ϕ n u j − ( λ + µ ) ∂ j ϕ n ∇ u j k L ( R d ) d ≤ µ k ∆ ϕ n k L ( R d ) d k u k L ∞ ( R d ) d + 2 µ k ∂ j ϕ n k L ( R d ) d k ∂ j u k L ∞ ( R d ) d + ( λ + µ ) k∇ ϕ n k L ( R d ) d k div u k L ∞ ( R d ) d − ( λ + µ ) k ∂ j ∇ ϕ n k L ( R d ) d k u j k L ∞ ( R d ) d + ( λ + µ ) k ∂ j ϕ n k L ( R d ) d k∇ u j k L ∞ ( R d ) d . (67)20rom (66) it follows that the right hand side of (67) goes to zero as n tends to infinity.Using the Hardy-type subordination (14) one has | v [ φ n ] | = (cid:12)(cid:12)(cid:12) Z R d V φ n · φ n (cid:12)(cid:12)(cid:12) ≤ k| V | / ϕ n k L ( R d ) d ≤ a k∇ ϕ n k L ( R d ) d , (68)again using (66) it follows that the right hand side of (68) goes to zero as n tends to infinity. The numeratorin (63) can be estimated as follows | h ( φ n , ψ ) − z ( φ n , ψ ) | = | ( − ∆ ∗ φ n − zφ n , ψ ) + v ( φ n , ψ ) |≤ k − ∆ ∗ φ n − zφ n k L ( R d ) d k ψ k L ( R d ) d + p | v [ φ n ] | p | v [ ψ ] |≤ k − ∆ ∗ φ n − zφ n k L ( R d ) d k ψ k L ( R d ) d + p | v [ φ n ] |√ a k∇ ψ k L ( R d ) d ≤ (cid:0) k − ∆ ∗ φ n − zφ n k L ( R d ) d + p | v [ φ n ] |√ a (cid:1) k ψ k D ( h ) , where k · k D ( h ) is the usual H ( R d ) d norm. As for the denominator in (63), using again (14), it follows k ψ k D ( h ) = µ k∇ ψ S k L ( R d ) d + ( λ + 2 µ ) k∇ ψ P k L ( R d ) d + ℜ v [ ψ ] + k ψ k L ( R d ) d ≥ (min { µ, λ + 2 µ } − a ) k∇ ψ k L ( R d ) d + k ψ k L ( R d ) d ≥ min { , (min { µ, λ + 2 µ } − a ) }k ψ k D ( h ) . Using the previous estimates one hassup ψ ∈D ( h ) ψ =0 | h ( φ n , ψ ) − z ( φ n , ψ ) |k ψ k D ( h ) ≤ k − ∆ ∗ φ n − zφ n k + p | v [ φ n ] |√ a p min { , (min { µ, λ + 2 µ } − a ) } . Since the right hand side tends to zero due to (67) and (68), the sequence φ n satisfies the hypotheses ofLemma 4.3, thus (0 , ∞ ) ⊂ σ ( − ∆ ∗ + V ) . Since the spectrum is closed, we get the thesis.
In order to conclude the claimed stability, it is left to show that the residual spectrum of − ∆ ∗ + V is empty.This is the object of the next theorem. Proposition 4.4.
Let d ≥ . Then σ r ( − ∆ ∗ + V ) = ∅ . Proof.
Let define H V := − ∆ ∗ + V. It is easy to see that H ∗ V = H V t , where V t denotes the conjugate transposeof the matrix V. Let denote with J the complex-conjugation transposition operator defined by J ( Au ) = A t u, for any squarematrix A ∈ C d × d and any vector u ∈ C d : notice that J ( u ) = J ( I C d u ) = u, with I C d the d × d identity matrix,in other words, given any vector u = I C d u then J acts as the usual complex-conjugation operator. J as definedabove is a conjugation operator in the sense of [33, Sec. III.5]. One easily checks that H ∗ V = JH V J, i.e. , H V is J -self-adjoint, and thus it has no residual spectrum ( cfr. [10]) . Proof of Theorem 1.4.
The proof of Theorem 1.4 follows from Proposition 4.1, Proposition 4.2, Proposition 4.3and Proposition 4.4.Now we turn to the proof of Theorem 1.5. We stress that the validity of Propositions 4.1–4.4 (from whichthe stability of the spectrum of − ∆ ∗ + V follows) requires only two ingredients: first, one needs, D ( v ) ⊂ D ( h )( cfr . (23) and (24)) and secondly k K z k ≤ a < . As soon as we consider class of potentials such that these tworequests are satisfied, then one gets spectral stability of the perturbed Lam´e operators with such perturbationsas a consequence of Propositions 4.1–4.4. This allows us to prove Theorem 1.5 and Theorem 1.6.21 roof of Theorem 1.5.
Thanks to the Hardy-type inequality (6) with (7), if V ∈ L ,p ( R ) , < p ≤ / D ( v ) ⊂ D ( h ) . Moreover, from (37) and hypothesis (15) one has k K z k ≤ a < . In light of the remark above,this concludes the proof.
Proof of Theorem 1.6.
Thanks to (17), if V ∈ L / ( R ) then D ( v ) ⊂ D ( h ) . Moreover, from (36) and assump-tion (16) one has k K z k ≤ a < . This concludes the proof.
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