Agmon-type decay of eigenfunctions for a class of Schrödinger operators with non-compact classically allowed region
aa r X i v : . [ m a t h . SP ] J a n AGMON-TYPE DECAY OF EIGENFUNCTIONS FOR A CLASSOF SCHR ¨ODINGER OPERATORS WITH NON-COMPACTCLASSICALLY ALLOWED REGION
CHRISTOPH A. MARX AND HENGRUI ZHU
Abstract.
An important result by Agmon implies that an eigenfunction ofa Schr¨odinger operator in R n with eigenvalue E below the bottom of the es-sential spectrum decays exponentially if the associated classically allowed re-gion t x P R n : V p x q ď E u is compact. We extend this result to a classof Schr¨odinger operators with eigenvalues, for which the classically allowed re-gion is not necessarily compactly supported: We show that integrability of thecharacteristic function of the classically allowed region with respect to an in-creasing weight function of bounded logarithmic derivative leads to L -decay ofthe eigenfunction with respect to the same weight. Here, the decay is measuredin the Agmon metric, which takes into account anisotropies of the potential.In particular, for a power law (or, respectively, exponential) weight, our mainresult implies that power law (or, respectively, exponential) decay of “the sizeof the classically allowed region” allows to conclude power law (or, respectively,exponential) decay, in the Agmon metric, of the eigenfunction. Introduction
In a series of lectures given in 1980 at the University of Virginia, later pub-lished as [1], S. Agmon proposed his now celebrated method for proving expo-nential decay of eigenfunctions for Schr¨odinger type operators H “ ´ ∆ ` V in R n for eigenvalues below the bottom of the essential spectrum. One of themain accomplishments of Agmon’s approach was to account for anisotropies ofthe potential function V captured through a pseudo-metric, now known as the Agmon metric ; see (3) below. While the spectral condition on the eigenvalue E can be expressed more generally as Agmon’s λ -condition (see (1.7) in [1]), it istypically formulated by assuming compactness of the classically allowed region t x P R n : V p x q ď E ` δ u , for some δ ą
0. In this situation, Persson’s character-ization of the essential spectrum [19] (see also [16, Theorem 14.11]) implies that E is strictly below the bottom of the essential spectrum (separated by at least δ ). We refer the reader to [16, chapter 3] or [18, chapter 8.5] for a pedagogicaltreatment of Agmon’s method for this basic set-up.In the 40 years since, Agmon’s result has triggered a vast body of literature thatdevelops further the subject of proving decay-estimates. An extensive discussionof both the history and a summary of the works up to the year 2000 can be found in the review article [15]. Of the developments in more recent years, wemention without aiming for a comprehensive list: applications to tight bindingtype models (see e.g. [7, 23]), decay estimates for magnetic Schr¨odinger operators(see e.g. [17, 20]), decay estimates in superconductivity (for a review, see [9]) andfor the Robin problem (see e.g. [13, 14]), Agmon estimates on quantum graphs(see e.g. [12, 2, 11]), and the landscape function approach (see e.g. [3, 22, 8]).The subject of this short paper is to examine anisotropic decay for the basicSchr¨odinger set-up in R n , without assuming compactness of the classically allowedregion. Specifically, we show that if the characteristic function of the classicallyallowed region associated with an eigenvalue E is merely integrable with respectto (the square of) an increasing weight function 1 ď φ P C pr , `8qq with boundedlogarithmic derivative , i.e. if for some 0 ă ǫ ă δ ą ż χ t V ď E ` δ u p x q φ pp ´ ǫ q ρ E p x qq d n x ă `8 , (1)then the associated eigenfunctions exhibit L -decay with respect to the sameweight. Here, the decay is measured in the Agmon distance ρ E p x q of x to theorigin (see (3)-(5) for the definition), which takes into account anisotropies ofthe potential. Our main result is formulated in Theorem 2.3, the precise set-upof which is summarized in hypotheses (H1) and (H2) of section 2. Particularlyrelevant examples for admissible weight functions φ in (1) are power functions φ p t q “ p ` t q r with r ą
0, and exponentials φ p t q “ e αt with α ą
0. In thiscontext we also point out that our integrability condition (1) still implies thatthe eigenvalue E is at a positive distance of at least δ below the bottom of theessential spectrum, see Proposition 3.1 in section 3.2.1.Morally, the integrability condition in (1) can be understood as allowing for aclassically allowed region which stretches out to , however in such a way that itsmeasure decays like 1 { φ , when measured in the Agmon metric; see also Remark2.2 for the situation in one dimension. In particular, for power law weights in(1), φ p t q “ p ` t q r with exponent r ą
0, our main result in Theorem 2.3 impliesthat power law decay of “the size of the classically allowed region” still allowsto conclude power law decay with same exponent of the wave function. Explicitexamples for potentials satisfying the condition (1) are constructed in section 3.2.2.We note that, since for compact classically allowed region, exponential decay ofeigenfunctions is known to be optimal [5], our condition of bounded logarithmicderivative for the weight function φ in (1) is in general necessary. Finally, wemention that while our result could be related the more general framework ofAgmon’s λ -condition, we believe that a direct proof is valuable in its own right;to our knowledge, this type of question has not been examined explicitly in theliterature.We structure the paper as follows: In section 2, we describe the precise set-up (see hypotheses (H1) - (H2), and (H3), respectively) for our main results,Theorem 2.3 and its extension, Theorem 2.4. The proofs of Theorems 2.3 and ECAY OF EIGENFUNCTIONS ON A BOUNDED BELOW POTENTIAL 3 E for which theintegrability condition in (1) is satisfied are still strictly below the bottom of theessential spectrum (see Proposition 3.1). In section 3.2.2, we construct explicitexamples of potentials for which the integrability condition in (1) holds. Finally, insection 4, we prove point-wise decay of eigenfunctions for more regular potentials(Theorem 4.2); this extends the well known result for an exponential weight goingback to Agmon [1] to the more general weight functions with bounded logarithmicderivative, considered in this article. Acknowledgements:
The authors would like to thank Peter D. Hislop for nu-merous valuable discussions while preparing this manuscript.2.
Set-up and main result
Let H “ ´ ∆ ` V be a closed operator acting on L p R n q with σ p H q Ď R , where V is real-valued, continuous, and bounded below. While certain aspects of thispaper may hold true for more general situations, for simplicity and concreteness,we consider the following set-up: Hypothesis . (a) Let V be real-valued continuous and bounded below, ´ 8 ă m V : “ inf x P R n V p x q , (2)and suppose that, as a multiplication operator with natural self-adjointdomain D p V q : “ t f P L p R n q : p V f q P L u , V is relatively p´ ∆ q -boundedwith relative bound strictly less than one; here, as usual, we define p´ ∆ q as a self-adjoint operator on H p R n ). In this case, H “ ´ ∆ ` V is aself-adjoint operator on the domain H p R n q Ď D p V q .(b) Let ψ be an eigenfunction of H with associated eigenvalue E P R whichalso satisfies that ψ P L p R n q . Remark . In view of hypothesis (H1) part (b), we note that for n ď
3, theSobolev Embedding Theorem embeds H p R n q into the continuous functions van-ishing at infinity, so that the condition ψ P L p R n q holds trivially for any eigen-function ψ of H . Moreover, in any dimension n P N , an eigenfunction ψ of H also satisfies ψ P L p R n q if the potential V is sufficiently regular (with regularitydepending on n ); indeed, V P C k p R n q with bounded derivatives, implies that anyeigenvector ψ of H automatically satisfies ψ P H k ` p R n q and thus is continuousand vanishing at infinity if k ` ą n , see e.g. [15, Proposition 1.2]. The latterwill also play a role in the point-wise bounds discussed in section 4.We recall that for a continuous potential V and E P R , the Agmon metric isdefined by ρ E p x, y q : “ inf γ P P x,y ż p V p γ p t qq ´ E q { ` | γ p t q| d t , (3) C. A. MARX AND H. ZHU where P x,y : “ t γ : r , s Ñ R n | γ p q “ x, γ p q “ y, and γ P AC r , su , and p V ´ E q ` : “ max t V ´ E, u . In particular, the distance in the Agmon metric iszero precisely if two points can be connected by a path which remains completelyinside the classically allowed region t V ď E u . We note that in one dimension, thedefinition in (3) reduces to the WKB factor: ρ E p x, y q “ sgn p y ´ x q ż yx p V p t q ´ E q { ` d t , for x, y P R . (4)We will write ρ E p x q : “ ρ E p x, q , (5)for the Agmon distance of x P R n to the origin.Finally, we recall the key property of the Agmon metric which provides a re-lation to the classically allowed region: x ÞÑ ρ E p x q is locally Lipschitz on R n , inparticular, for a.e. in x P R n , ρ E p x q is differentiable with | ∇ ρ E p x q| ď p V p x q ´ E q ` , (6)see, e.g., Proposition 3.3 in [16]. Hypothesis . Let E P R be an eigenvalue satisfying hypothesis (H1). Forsome function 1 ď φ P C pr , `8qq with propertieslim t Ñ`8 φ p t q “ 8 and (7)0 ă sup ˇˇˇˇ φ φ ˇˇˇˇ “ : M φ ă 8 , (8)suppose there exist ǫ, δ ą , ´ M ´ φ ( ă ǫ ă χ t V ď E ` δ u φ pp ´ ǫ q ρ E q P L p R n q . (10)Here, ρ E p x q is the distance to the origin in the Agmon metric, associated with theeigenvalue E , defined in (4)-(5).We will refer to a function 1 ď φ P C pr , `8qq which satisfies the conditions(7)-(8) as an admissible weight function . Note that condition (8) is equivalentto assuming that φ has bounded logarithmic derivative . In view of our mainresult, particularly relevant examples for admissible weight functions are powerfunctions φ p t q “ p ` t q r with r ą
0, and exponentials φ p t q “ e αt with α ą φ in (8) is in general necessary. Condition (10) in hypothesis(H2) thus requires that the “ δ -enlarged” classically allowed region t V ď E ` δ u be φ -integrable in the Agmon metric. ECAY OF EIGENFUNCTIONS ON A BOUNDED BELOW POTENTIAL 5
Remark . To further interpret the meaning of the integrability condition in(10), observe that in one dimension ( n “ V implies that thesub-level set t V ď E ` δ u is a countable union of closed intervals, i.e. t V ď E ` δ u “ ď j P Z r a j , b j s , (11)with r a j , b j s mutually disjoint except possibly for endpoints. Here, without lossof generality, we take r a j , b j s Ď r , `8q for j ě r a j , b j s Ď p´8 , s for j ă ρ E p x q increases on r , `8q and decreases on p´8 , s (not necessarilystrictly). Thus, since φ is increasing, the integrability condition (10) requires thatthe length of each of the intervals in the decomposition (11) be summable withrespect to φ pp ´ ǫ q ρ E q , i.e. ÿ j ě φ pp ´ ǫ q ρ E p a j qq | b j ´ a j | ď } χ t V ď E ` δ uXr , `8q φ pp ´ ǫ q ρ E q } ď ÿ j ě φ pp ´ ǫ q ρ E p b j qq | b j ´ a j | , (12)and similarly for } χ t V ď E ` δ uXp´8 , sq φ pp ´ ǫ q ρ E q } , where j ă a j and b j in (12) are interchanged.Our result shows that φ -integrability of the classically allowed region in the senseof (10) in hypothesis (H2) implies L -decay of the eigenfunction with respect tothe same weight, specifically: Theorem 2.3.
Let ψ P L p R n q be an eigenfunction of H satisfying hypothesis(H1). Suppose that for an associated eigenvalue E and a weight function φ as inhypothesis (H2), the integrability condition (10) holds for some ă ǫ ă as in(9) and δ ą . Then, there exists a constant ă c ǫ,δ ă 8 , such that ż φ pp ´ ǫ q ρ E p x qq | ψ p x q| d n x ď c ǫ,δ . We note that the positive lower bound on the value of ǫ in (9) is a result ofisolating the contribution of the “ δ -enlarged classically allowed region,” t V ă E ` δ u ; see also (28) in the proof below. It drops to zero if M φ ď
1, i.e. if φ ď φ for all t P r , `8q . By Gr¨onwall’s inequality, the latter is equivalent to φ p x q ď e t ,for all t P r , . For the power law weights φ p t q “ p ` t q r with r ą ǫ ą ´ r . (14)This positive lower bound on ǫ can however be removed by replacing hypothesis(H2) with the following: C. A. MARX AND H. ZHU
Hypothesis . (a) Let 1 ď φ P C pr , `8qq satisfy (7) and (8), as well asthat lim x Ñ8 φ p x q φ p x q “ . (15)(b) For V and E P R as in (H1), assume thatlim x Ñ8 ρ E p x q “ 8 . (16)Suppose that there exist 0 ă ǫ ă δ ą ǫ in (9): Theorem 2.4.
Suppose ψ P L p R n q is an eigenfunction of H in the sense ofhypothesis (H1) and that for an associated eigenvalue E the condition (16) inhypothesis (H3) holds. Given a weight function φ , satisfying the hypothesis (H3)item (a), one has: for each ă ǫ ă and δ ą for which the integrabilitycondition (10) holds, there exists a constant ă ˜ c ǫ,δ ă 8 such that ż φ pp ´ ǫ q ρ E p x qq | ψ p x q| d n x ď ˜ c ǫ,δ . L decay Proof of the main results.
Following Agmon, the key to proving Theo-rem 2.3 is to perform a gauge transform, which we will adapt to the weight φ .Specifically, given ǫ ą α ą
0, we consider the operator H f α : “ φ p f α q Hφ p f α q ´ , (17)where f α : “ p ´ ǫ q ρ E ` α p ´ ǫ q ρ E ; (18)To avoid confusion, we mention that φ p f α q ´ denotes φ p f α q and not the inversefunction of φ p f α q . Since we will eventually take the limit α Ñ ` , we also notethat f α p x q Õ p ´ ǫ q ρ E p x q “ : f p x q , (19) monotonically for each x P R n .Finally, given an eigenfunction ψ as in hypothesis (H1), we associate with thetransformed operator (17) the transformed wave function:Φ α : “ φ p f α q ψ . (20)We then prove the following key lemma: ECAY OF EIGENFUNCTIONS ON A BOUNDED BELOW POTENTIAL 7
Lemma 3.1.
Given max , ´ M ´ φ ( ă ǫ ă and δ ą as in hypothesis (H2).For each α ą , one then has that Φ α P H p R n q X D p V q and || Φ α || ď η ǫ δ " ℜ x Φ α , p H f α ´ E q Φ α y ` ż | Φ α | p V ´ E q ´ * ` ż t V ď E ` δ u | Φ α | , (21) where ă η ǫ : “ ´ M φ p ´ ǫ q ă . (22)Here, as common, we denote p V ´ E q ´ : “ | V ´ E | ´ p V ´ E q ` . Proof.
Fix α ą
0. First, we verify that Φ α P H p R n q . Observe that Φ α P L since ψ P L and φ p f α q is bounded by0 ď f α ď min " f , α * . (23)We also need to show that ∇ Φ α P L , where ∇ Φ α “ φ p f α q ψ ∇ f α ` φ p f α q ∇ ψ . (24)Using (6), for a.e. x P R n , | ∇ f α | is estimated by | ∇ f α | “ ˇˇˇˇ ∇ f p ` αf q ˇˇˇˇ ď | ∇ f | ď p ´ ǫ qp V ´ E q ` . (25)Therefore, the bound in (23) and the hypothesis in (H1) that ψ P D p V q yield ż | φ p f α q ψ | | ∇ f α | ď p ´ ǫ q sup ď t ď { α | φ p t q| ż | ψ | p V ´ E q ` “ p ´ ǫ q sup ď t ď { α | φ p t q| "ż t V ´ E ě u | ψ | p V ´ E q ` ` ż t ď V ´ E ă u | ψ | p V ´ E q ` * ď sup ď t ď { α | φ p t q| || ψ p V ´ E q|| ` || ψ || ( ă 8 . (26)To see that also the second term in (24) is in L , observe that by (23), φ p f α q is bounded, and that the hypotheses in (H1) imply that ψ P H p R n q Ď D p V q .In summary, we conclude that Φ α P H p R n q ; moreover, Φ α P D p V q because ψ P D p V q and φ p f α q P L .Now, we compute the gauge transform: H f α “ ´ φ p f α q ∆ φ p f α q ´ ` V “ ´ “ φ p f α q ∇ φ p f α q ´ ‰ ¨ “ φ p f α q ∇ φ p f α q ´ ‰ ` V , where φ p f α q ∇ φ p f α q ´ “ ∇ ´ φ p f α q φ p f α q ∇ f α “ : ∇ ´ ω α . Whence, we have H f α “ ´p ∇ ´ ω α q ` V “ ´ ∆ ` r ∇ , ω α s ` ´ | ω α | ` V , (27) C. A. MARX AND H. ZHU where r ∇ , ω α s ` is the anti-commutator of ∇ and ω α . Then, since Φ α P H p R n q X D p V q , we estimate, using (27) and (25): ℜ x Φ α , p H f α ´ E q Φ α y ě x Φ α , p V ´ | ω α | ´ E q Φ α yě x Φ α , p V ´ ˇˇˇˇ φ p f α q φ p f α q ˇˇˇˇ p ´ ǫ qp V ´ E q ` ´ E q Φ α yě x Φ α , p V ´ M φ p ´ ǫ qp V ´ E q ` ´ E q Φ α y . (28)Here, we used that r ∇ , ω α s ` is antisymmetric, so its real part vanishes. Thus,substituting for M φ p ´ ǫ q “ ´ η ǫ , and taking δ ą ℜ x Φ α , p H f α ´ E q Φ α y ě x Φ α , p´p V ´ E q ´ ` η ǫ p V ´ E q ` q Φ α y“ η ǫ "ż t V ą E ` δ u | Φ α | p V ´ E q ` ` ż t E ă V ď E ` δ u | Φ α | p V ´ E q ` * ´ ż | Φ α | p V ´ E q ´ ě η ǫ δ ż t V ą E ` δ u | Φ α | ´ ż | Φ α | p V ´ E q ´ “ η ǫ δ " || Φ α || ´ ż t V ď E ` δ u | Φ α | * ´ ż | Φ α | p V ´ E q ´ . (29)Finally, to show the finiteness of the last two terms on the right hand side of(29), we write S E,V,ǫ,δ : “ || χ t V ď E ` δ u φ pp ´ ǫ q ρ E q|| ă 8 , (30)whose existence is guaranteed by (10) in (H2). Then, we estimate, using ourhypotheses in (H1) that V is bounded below and that ψ P L : ż | Φ α | p V ´ E q ´ ď ż t V ď E u | Φ α | p E ´ V q “ ż χ t V ď E u | ψ | φ p f α q p E ´ V qď ż χ t V ď E ` δ u | ψ | φ p f q p E ´ m V qď p E ´ m V q|| ψ || S E,V,ǫ,δ “ : C p E, m V , S E,V,ǫ,δ q ă 8 . (31)Similarly, we have ż t V ď E ` δ u | Φ α | ď || ψ || S E,V,ǫ,δ “ : C p E, m V , S E,V,ǫ,δ q ă 8 . (32)For later purposes, note that both the bounds in (31) and (32) are uniform in α ą
0. In summary, combining (29) with (31) - (32), we thus arrive at the claimin (21). (cid:3)
Now, we are ready to prove the main result in Theorem 2.3:
ECAY OF EIGENFUNCTIONS ON A BOUNDED BELOW POTENTIAL 9
Proof of Theorem 2.3.
First, from Lemma 3.1 and the bounds in (31) - (32), weknow that || Φ α || ď η ǫ δ " ℜ x Φ α , p H f α ´ E q Φ α y ` ż | Φ α | p V ´ E q ´ * ` ż t V ď E ` δ u | Φ α | ď η ǫ δ ℜ x Φ α , p H f α ´ E q Φ α y ` C p E, m V , S E,V,ǫ,δ q η ǫ δ ` C p E, m V , S E,V,ǫ,δ q . (33)Observe that x Φ α , p H f α ´ E q Φ α y “ x φ p f α q ψ, p H ´ E q ψ y “ , (34)hence, by (33) and (34), we have: || Φ α || ď C p E, m V , S E,V,ǫ,δ q η ǫ δ ` C p E, m V , S E,V,ǫ,δ q “ : c ǫ,δ ă 8 . (35)Since c ǫ,δ is independent of α ą
0, we can take the limit α Ñ ` , and thus arriveat the claim by Fatou’s Lemma, ż φ pp ´ ǫ q ρ E p x qq | ψ p x q| ď lim inf α Ñ ` || Φ α || ď c ǫ,δ ă 8 . (cid:3) We turn to the proof of Theorem 2.4 which removes the positive lower bound on ǫ in (9) by replacing hypothesis (H2) with (H3). As mentioned earlier in section2, in dimension one (16) is redundant: Remark . Note that (16) is always satisfied in dimension one ( n “ |t V ď E ` δ u X R ˘ | ă || χ t V ď E ` δ u || ă || χ t V ď E ` δ u φ pp ´ ǫ q ρ E q|| ă 8 , (36)whence lim x Ñ˘8 |t V ą E ` δ u X r , x s| “ |t V ą E ` δ u X R ˘ | “ 8 . (37)Hence, using the special form of the Agmon metric in R given in (4), we havelim inf x Ñ˘8 ρ E p x q “ lim inf x Ñ˘8 ż x p V ´ E q { ` d t ě lim inf x Ñ˘8 ż t V ą E ` δ uXr ,x s p V ´ E q { ` d t ě ? δ |t V ą E ` δ u X R ˘ | “ 8 (38)Equation (16) may however no longer hold in higher dimensions, R n with n ě x Ñ8 ρ E p x q ă lim sup x Ñ8 ρ E p x q . (39) The additional hypotheses formulated in (H3) will allow us to remove the posi-tive lower bound on ǫ in (9) by modifying Φ α in Lemma 3.1 with a smooth cut-off χ R : Specifically, consider Φ α,R : “ χ R Φ α “ χ R φ p f α q ψ , (40)where χ R P C p R n q satisfies 0 ď χ R ď χ R | B R p q “ χ R | R n z B R ` p q “
1, andthat ∇ χ R is compactly supported with || ∇ χ R || ď
1. Here, taking advantage of(15), we choose R ą x ą R | φ p x q{ φ p x q| ď α,R is still in H p R n q X D p V q , as both χ R and ∇ χ R are bounded,and that ∇ χ R is compactly supported. Then, the inner product in (28) withΦ α replaced by Φ α,R is well defined in the sense of quadratic forms and can bere-estimated: ℜ x Φ α,R , p H f α ´ E q Φ α,R y “ x Φ α,R , p V ´ | ω α | ´ E q Φ α,R yě x Φ α,R , p V ´ ˇˇˇˇ φ p f α q φ p f α q ˇˇˇˇ p ´ ǫ qp V ´ E q ` ´ E q Φ α,R yě x φ p f α q ψ, p V ´ p ´ ǫ qp V ´ E q ` ´ E q φ p f α q ψ y , (42)where we have used that χ R | φ { φ | ď
1. Hence, we no longer need to adjust ǫ tocause the multiplicative factor of p V ´ E q ` in (28) be less than 1, i.e. η ǫ in (29)is now replaced by ǫ , which, in turn, replaces the conclusion of Lemma 3.1 with: || Φ α,R || ď ǫδ " ℜ x Φ α,R , p H f α ´ E q Φ α,R y ` ż | Φ α,R | p V ´ E q ´ * ` ż t V ď E ` δ u | Φ α,R | , (43)However, this comes at the cost that (34) is no longer true, i.e. ℜ x Φ α,R , p H f α ´ E q Φ α,R y ‰ Lemma 3.3.
Given the setup outlined above, with Φ α,R defined in (40), we have ℜ x Φ α,R , p H f α ´ E q Φ α,R y “ x ξ α φ p f α q ψ, ψ y , (45) where ξ α : “ | ∇ χ R | ` p ∇ χ R ¨ ∇ f α q χ R φ p f α q φ p f α q . (46)Here, as mentioned earlier, Φ α,R P H p R n q X D p V q , implies that the innerproduct in (45) is well-defined in the sense of quadratic forms. ECAY OF EIGENFUNCTIONS ON A BOUNDED BELOW POTENTIAL 11
Proof.
Since, p H ´ E q ψ “
0, we obtain for the left side of (45): x Φ α,R , p H f α ´ E q Φ α,R y “ x χ R φ p f α q ψ, φ p f α qp H ´ E q χ R ψ y“ x χ R φ p f α q ψ, p´ ∆ χ R ´ ∇ χ R ¨ ∇ q ψ y (47)Using integration by parts, we compute x χ R φ p f α q ψ, ´ ∇ χ R ¨ ∇ ψ y “ xr p ∆ χ R q χ R φ p f α q ` | ∇ χ R | φ p f α q ` p ∇ χ R ¨ ∇ f α q χ R φ p f α q φ p f α qs ψ, ψ y ` x χ R φ p f α q ∇ χ R ¨ ∇ ψ, ψ y , (48)which, since the last term in (48) is antisymmetric, yields ℜ x´ p ∇ χ R q χ R φ p f α q ψ, ∇ ψ y“ xrp ∆ χ R q χ R φ p f α q ` | ∇ χ R | φ p f α q ` p ∇ χ R ¨ ∇ f α q χ R φ p f α q φ p f α qs ψ, ψ y Combining this with (47), we thus arrive at the claim: ℜ x Φ α , p H f α ´ E q Φ α y “ xr| ∇ χ R | φ p f α q ` p ∇ χ R ¨ ∇ f α q χ R φ p f α q φ p f α qs ψ, ψ y . (cid:3) Lemma 3.3 allows to prove Theorem 2.4, which, by requiring the additionalhypotheses in (H3) on the both the weight (15) and the behavior of the Agmonmetric at infinity (16), removes the positive lower bound on ǫ imposed in (9): Proof of Theorem 2.4.
First, notice that since χ R ď
1, the upper bounds on theright most sides of (31)-(32) remain unaffected when replacing Φ α with Φ α,R .Thus, combining (43) with the bounds in (31)-(32), we have || Φ α,R || ď ǫδ ℜ x Φ α,R , p H f α ´ E q Φ α,R y ` C ǫδ ` C . (49)We recall that the constants C and C in (31)-(32) are independent of α ą || Φ α,R || ď ǫδ x ξ α φ p f α q ψ, ψ y ` C ǫδ ` C ď || ψ || ǫδ sup x P supp | ∇ χ R | ξ α φ p f α q ( ` C ǫδ ` C “ || ψ || ǫδ sup x P supp | ∇ χ R | "„ | ∇ χ R | ` | ∇ χ R || ∇ f α | χ R φ p f α q φ p f α q φ p f α q * ` C ǫδ ` C ď || ψ || ǫδ sup x P supp | ∇ χ R | r| ∇ χ R | ` | ∇ χ R || ∇ f | φ p f q ( ` C ǫδ ` C “ : a ǫ,δ ă 8 , where, in the last step, we used (41) and that | ∇ f α | ď | ∇ f | , see (25). Again,since the constant a ǫ,δ is α -independent, Fatou’s Lemma thus yields || Φ R || : “ lim α Ñ ` || Φ α,R || ď a ǫ,δ ă 8 . (50) In summary, using (50) and (32), we can bound the integral in the claim: ż φ pp ´ ǫ q ρ E p x qq | ψ p x q| d n x “ "ż t V ą E ` δ u ` ż t V ď E ` δ u * φ pp ´ ǫ q ρ E p x qq | ψ p x q| d n x ď ż t V ą E ` δ u φ pp ´ ǫ q ρ E p x qq | ψ p x q| d n x ` C “ ż t V ą E ` δ u p ´ χ R q φ pp ´ ǫ q ρ E p x qq | ψ p x q| d n x ` || Φ R || ` C ď } ψ } sup x P B R ` p q φ pp ´ ǫ q ρ E p x qq ( ` a ǫ,δ ` C ,where the first term in the last step is finite because it is the supremum of acontinuous function over a compact domain. (cid:3) Concluding remarks.
We conclude this section with two remarks aboutabout the scope of Theorems 2.3 and 2.4: The first relates δ in the integrabilitycondition (10) to the distance of the eigenvalue under consideration to the essentialspectrum, the second provides examples of potentials which satisfy the hypothesesof Theorems 2.3-2.4.3.2.1. Integrability condition and the essential spectrum.
The classical results forexponential decay of the wave function, both isotropic [10, 4, 6, 21] and anisotropic[1], consider eigenvalues E below the bottom of the essential spectrum σ ess of theSchr¨odinger operator. Indeed, Agmon decay in its most basic form, assumescompactness of the enlarged classically allowed region t V ď E ` δ u , for some δ ą
0, which in turn implies that E ď inf σ ess ´ δ (e.g. use the well-knownPersson characterization for the essential spectrum [19] (see also [16, Theorem14.11, chapter 14.4]). The following shows that δ in the integrability condition(10) of Theorems 2.3-2.4 serves a similar purpose, specifically: Proposition 3.1.
Given a Schr¨odinger operator H “ ´ ∆ ` V where (i) V is real-valued and bounded below with V ě inf x P R n V “ : m V ą ´8 and (ii) V is relatively p´ ∆ q -bounded with relative bound strictly less than one, i.e. lim λ Ñ`8 || V p´ ∆ ` λ q ´ || ă , (51) in particular, H is self-adjoint on the domain H p R n q .Suppose that for some E P R , there exists δ ą such that the set A E ; δ : “ t x P R n : V ď E ` δ u (52) has finite Lebesgue measure. Then, one has σ ess p H q Ď r E ` δ, `8q . (53) ECAY OF EIGENFUNCTIONS ON A BOUNDED BELOW POTENTIAL 13
In view of Theorems 2.3-2.4, observe that, since χ t V ď E ` δ u ď χ t V ď E ` δ u φ pp ´ ǫ q ρ E q , (54)Proposition 3.1 interprets δ in the integrability condition (10) as a lower boundon the distance of the eigenvalue E to the bottom of the essential spectrum of H .The idea of the proof of Proposition 3.1 is to show that modifying the potentialappropriately on the finite measure set A E ; δ does not alter the essential spectrum.To keep the paper self-contained, we include the brief argument as follows: Proof.
Consider the perturbation W : “ χ t V ď E ` δ u ¨ p E ` δ ´ V q , (55)so that 0 ď W ď E ` δ ´ m V , which by (52), yields W P L X L p R n q with || W || ď p E ` δ ´ m V q| A E ,δ | . (56)Since W P L p R n q , we obtain that W is relatively compact with respect to p´ ∆ q (use e.g. Theorem 14.9 in [16]). Using (51), this shows that W is also relativelycompact with respect to H . In particular, we conclude that σ ess p H q “ σ ess p H ` W q . (57)By construction, one has V ` W ě E ` δ , whence (57) yields σ ess p H q “ σ ess p H ` W q Ď σ p H ` W q Ď r E ` δ, `8q , (58)proving the claim. (cid:3) Example for potentials satisfying the integrability condition.
Our secondremark constructs explicit examples in dimension one ( n “
1) for potentials whichsatisfy the hypotheses of Theorems 2.3-2.4, specifically the integrability conditionin (10). In particular, this shows that these results apply to a nonempty classof Schr¨odinger operators. In the following construction, the dimension ( n “ R , see (73) - (74)below.In one dimension, take a potential V P C p R q with V ď
0, lim x Ñ8 V p x q “ σ disc p´ ∆ ` V q X p´8 , q ‰ H ;in particular, then σ ess p´ ∆ ` V q “ r , `8q . We thus have a negative eigenvalue E of p´ ∆ ` V q , ´8 ă m V : “ inf V ă E ă
0, with associated eigenfunction ψ , || ψ || “ φ P C pr , `8qq for which the conditions (7)and (8) hold, take a continuous V P L X L p R q , V ď
0, such that V : “ V ` V satisfies m V ď V ď t x P R : V p x q “ m V u is unbounded , (59)and that ż χ t V ď E u p x q φ p| m V | { | x |q d x ă 8 . (60) Note that since m V ă E ă x Ñ8 V p x q “
0, the set t V ď E u isbounded, whence such V can always be found independent of how V decays tozero. Morally, the perturbation V is used to add negative spikes to the originalpotential V reaching down to m V on the set t x P R : V ą E u , so that V ` V satisfies the condition in (60). Remark . To construct V more explicitly, fix 0 ă R ă `8 such that t x P R : V p x q ď E u Ď r´ R , R s . (61)Take p c j q j P N to be an increasing sequence in the set t x P p , `8q : V p x q ą E u such that lim j Ñ8 c j “ `8 . For j P N , let 0 ă l j ă ř j P N l j ă 8 , so thatthe collection of intervals r c j ´ l j , c j ` l j s are mutually disjoint and that r c j ´ l j , c j ` l j s Ď t x P p , `8q : V p x q ą E u . (62)Now, obtain V P C p R q by modifying V continuously on each of the mutuallydisjoint intervals r c j ´ l j , c j ` l j s so that, for j P N , one has V p x q “ m V , for x P r c j ´ l j , c j ` l j s , (63) m V ď V p x q ď x P r c j ´ l j , c j ` l j s , (64)while not changing V otherwise, i.e. V p x q “ V p x q , for x P R z j P N r c j ´ l j , c j ` l j s + . (65)Set V : “ V ´ V P L p R q . Since m V ď V ď
0, the properties of V in (63)-(65)imply } V } ď m V ÿ j P N l j ă `8 , (66)whence V P L X L p R q as claimed. By construction, we also obtain ď j P N r c j ´ l j , c j ` l j s Ď t x P R : V p x q “ m V u , (67)verifying (59). Finally, since t V ď E u Ď ď j P N r c j ´ l j , c j ` l j s Y r´ R , R s , (68) ECAY OF EIGENFUNCTIONS ON A BOUNDED BELOW POTENTIAL 15 we conclude, in analogy to (12) in Remark 2.2, that ż χ t V ď E u p x q φ p| m V | { | x |q d x ď R ¨ φ ` | m V | { R ˘ ` ÿ j P N l j ¨ φ ˆ | m V | { ˆ c j ` ˙˙ , (69)where we used that φ is increasing and that 0 ă l j ă
1, for all j P N .In summary, choosing the sequence p l j q to decay fast enough (adapted to therate of increase of φ ) such that the right hand side of (69) is finite, we obtain V (and thus implicitly V “ V ´ V ) such that (60) holds. Here, we note thatsince by hypothesis φ has bounded logarithmic derivative (8), i.e. by Gr¨onwall’sinequality φ p t q ď φ p q e M φ t for t P r , `8q , (69) is always satisfied if l j ď j e ´ M φ | m V | { p c j ` q ; (70)of course, a slower decay of p l j q is possible depending on the rate of increase of φ .Since V P L p R q , it is relatively compact to p´ ∆ ` V q . Hence, we know that σ ess p´ ∆ ` V ` V q “ σ ess p´ ∆ ` V q “ r , . (71)Moreover, by construction we have m V ă x ψ , Hψ y “ x ψ , p´ ∆ ` V q ψ y ` x ψ , V ψ y loooomoooon ă ă E , (72)thus the min-max principle guarantees the existence of an eigenvalue E of H with m V ă E ď E of H and an associated eigenfunction ψ , || ψ || “
1. Since V is bounded, the operator H “ ´ ∆ ` V is self-adjoint on the domain D p H q “ H p R q Ď D p V q , where D p V q is defined in (H1). In particular, by the SobolevEmbedding Theorem, the hypothesis ψ P L p R q is trivially satisfied.Now, we can bound the Agmon distance to the origin by the following: ρ E p x q “ sgn p x q ż x p V p t q ´ E q { ` d t ď | m V | { | x | , x P R , (73)which, together with (60), implies that for every 0 ă ǫ ă || χ t V ` V ď E u φ pp ´ ǫ q ρ E q|| ď || χ t V ` V ď E u φ pp ´ ǫ q ρ E q|| ď || χ t V ` V ď E u φ p| m V | { | x |q|| ă 8 . (74)Hence, the eigenfunction ψ associated with E satisfies the hypotheses of Theorem2.3 for every 0 ă δ ă E ´ E . Notice that by (59), since m V ă E , the set t x P R : V p x q “ m V u Ď t x P R : V p x q ď E u (75)is unbounded, in particular, the classically allowed region for the potential V andthe eigenvalue E of H is not compact. Finally, since by Remark 3.2, (16) in (H3) always holds for dimension n “ φ which also satisfy the condition in (15).4. Point-wise decay
In this last section we show that the L decay in Theorem 2.3 and 2.4 impliespoint-wise decay if the potential is sufficiently regular . The case where the weightfunction φ is exponential is well known [1, chapter 5], see also [16, chapter 3.5].Here, we will modify these arguments appropriately to obtain point-wise boundsfor general admissible weights 1 ď φ P C pr , `8qq with bounded logarithmicderivative. The latter in particular accounts for the power law weights φ p t q “p ` t q r with r ą
0. In the following we will consider potentials V P C kb p R n q , for k P N , where, as usual, C kb p R n q denotes the C k -functions whose partial derivatives,up to order k , are all bounded.Since, for all s ě p´ ∆ ` q ´ : H s p R n q Ñ H s ` p R n q , regularity of thepotential V P C kb p R n q implies that any eigenfunction ψ P L p R q of H “ ´ ∆ ` V automatically satisfies ψ P H k ` p R n q . In particular, taking k P N sufficiently largeto ensure k ` ą n or equivalently k ą p n ´ q{
2, one can then prove local boundsfor the L -norm of ψ by its L -norm, specifically: Lemma 4.1 ( cf. Theorem 5.1 in [1] or Lemma 3.9 in [16] ) . Let V P C kb p R n q for k P N with k ą p n ´ q{ . Suppose Hψ “ Eψ for ψ P L p R n q and E P R .Then, ψ P C b p R n q vanishing at infinity and there exists C E,V depending on E and sup x P R n | V p α q p x q| , | α | “ , ..., k , such that for all x P R n , one has max x P B { p x q | ψ p x q| ď C E,V || ψ || L p B p x qq . (76)Here, we mention that the condition k ` ą n ensures that p ψ P L p R n q , inparticular ψ P C b p R n q vanishing at infinity, so that the boundedness conditionon ψ in (H1) part (a) holds true for any such eigenfunction, irrespective of thedimension n ; see also Remark 2.1. Relying on Lemma 4.1, we can then claim: Theorem 4.2 (Point-wise Bound) . Let H “ ´ ∆ ` V be as in (H1) and assumethat moreover V P C kb p R n q for k P N with k ą p n ´ q{ . Let ψ P L p R n q be aneigenfunction of H with associated eigenvalue E P R such that, for some weightfunction ď φ P C pr , `8qq satisfying (8) and ă ǫ ă , one has φ pp ´ ǫ q ρ E q ψ P L p R n q .Then, there exists a constant ă C ǫ ă 8 such that | ψ p x q| ď C ǫ φ pp ´ ǫ q ρ E p x qq ´ , @ x P R n . (77) Proof of Theorem 4.2.
Fix an arbitrary x P R n . Then, by Lemma 4.1, thereexists C E,M,V such that (76) holds, whence since φ ě
1, we have uniformly over
ECAY OF EIGENFUNCTIONS ON A BOUNDED BELOW POTENTIAL 17 all x P B { p x q : || ψφ pp ´ ǫ q ρ E q|| ; B { p x q ď C E,V ˜ sup x P B p x q φ pp ´ ǫ q ρ E q ¸ || ψ || L p B p x qq ď C E,V ˜ sup x,y P B p x q φ pp ´ ǫ q ρ E p x qq φ pp ´ ǫ q ρ E p y qq ¸ ¨ || ψφ pp ´ ǫ q ρ E q|| L p B p x qq (78)By hypothesis in (8), φ has bounded logarithmic derivative, thus | log p φ p s qq ´ log p φ p t qq| ď M φ ¨ | s ´ t | , @ s, t P r , . (79)In particular, for all x, y P B p x q , we conclude ˇˇˇˇ log ˆ φ pp ´ ǫ q ρ E p x qq φ pp ´ ǫ q ρ E p y qq ˙ˇˇˇˇ ď M φ ¨ p ´ ǫ q| ρ E p x q ´ ρ E p y q| (80) ď M φ ¨ p ´ ǫ q ρ E p x, y q (81) ď M φ p ´ ǫ q c , (82)where c “ p max V ´ E q { ` using that V was assumed to be bounded. Observethat the upper bound in (82) is uniform in x P R n .In summary, we obtainsup x,y P B p x q ˇˇˇˇ φ pp ´ ǫ q ρ E p x qq φ pp ´ ǫ q ρ E p y qq ˇˇˇˇ ď e M φ p ´ ǫ q c , (83)which, combined with (78), yields || ψφ pp ´ ǫ q ρ E q|| ; B { p x q ď C E,V e M φ p ´ ǫ q c ¨ || ψφ pp ´ ǫ q ρ E q|| L p B p x qq ď C E,V e M φ p ´ ǫ q c ¨ || ψφ pp ´ ǫ q ρ E q|| L p R n q . (84)Since x P R n was arbitrary and the right-most side of (84) is independent of x ,we obtain the claim in (77). (cid:3) References [1] S. Agmon,
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Department of Mathematics, Oberlin College, Oberlin, Ohio 44074, USA
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