On a scattering length for additive functionals and spectrum of fractional Laplacians with non-local perturbations
aa r X i v : . [ m a t h . P R ] O c t On a scattering length for additive functionals and spectrum offractional Laplacian with a non-local perturbation
Daehong Kim ∗ and Masakuni Matsuura Abstract
In this paper we study the scattering length for positive additive functionals of symmetricstable processes on R d . The additive functionals considered here are not necessarily continuous.We prove that the semi-classical limit of the scattering length equals the capacity of the supportof a certain measure potential, thus extend previous results for the case of positive continuousadditive functionals. We also give an equivalent criterion for the fractional Laplacian with ameasure valued non-local operator as a perturbation to have purely discrete spectrum in termsof the scattering length, by considering the connection between scattering length and the bottomof the spectrum of Schr¨odinger operator in our settings. Keywords
Additive functionals, Dirichlet forms, Discrete spectrum, Non-local perturbation,Schr¨odinger operators, Scattering lengths, Stable processes.
Mathematics Subject Classification (2010)
Primary 60J45, secondary , 60F17, 60J57, 35J10,60J55, 60J35
In [10, 11], Kac and Luttinger studied a connection between the scattering length Γ ( V ) of a positiveintegrable potential V and Brownian motion B t on R . They gave a probabilistic expression of Γ ( V ) as Γ ( V ) = lim t →∞ t Z R (cid:16) − E x h e − R t V ( B s )d s i(cid:17) d x, where E x denotes the expectation of B t started at x ∈ R . In addition, they proved that if V = K for a compact set K ⊂ R satisfying the so called Kac’s regularity (the Lebesgue penetration timeof K by B t is the same as the hitting time of K ), then lim p →∞ Γ ( p K ) = Cap( K ). Moreover,they conjectured that for any positive integrable function V with compact support satisfying theregularity as above lim p →∞ Γ ( pV ) = Cap(supp[ V ]) . (1.1) ∗ The first named author is partially supported by a Grant-in-Aid for Scientific Research (C) No. 17K05304 fromJapan Society for the Promotion of Science. − ∆ on R d . More precisely, fora positive integrable function V on R d , let U V be the capacitary potential of V defined by U V ( x ) = lim ε → ( ε + V − ∆) − V ( x ). Taylor defined the scattering length Γ ( V ) as Γ ( V ) = − Z R d ∆ U V ( x )d x and proved Kac and Luttinger’s formula ([24, Proposition 1.1]), which makes it natural that thescattering length is analogous to the capacity. Indeed, for a compact set K ⊂ R d with the Kac’sregularity, the capacitary potential U K of K is given by U K ( x ) = 1 − E x [ e − R ∞ V K ( B t )d t ] for V K = ∞ on K and 0 off K . Then − ∆ U K is equal to γ K the equilibrium measure on K . Hence Γ ( V K ) = − R R d ∆ U K ( x )d x = R R d γ K (d x ) = Cap( K ), where Cap denotes the Wiener capacity. Therefore, thephenomenon of (1.1) is expected naturally.The Kac-Luttinger’s conjecture (1.1) was confirmed by Taylor [24]. For more general symmetricMarkov processes, Takahashi [20] gave a new probabilistic representation of the scattering lengthof a continuous potential which makes the limit (1.1) quite transparent. For symmetric Markovprocesses again, Takeda [22] considered the behaviour of the scattering length of a positive smoothmeasure potential by using the random time change argument for Dirichlet forms and gave a simpleelegant proof of the analog of (1.1) without Kac’s regularity. The result in [22] was extended toa non-symmetric case by He [9]. For general right Markov processes, Fitzsimmons, He and Ying[7] extended Takahashi’s result by using the tool of Kutznetsov measure and proved the analog of(1.1) for a positive continuous additive functional.Some interesting applications for spectral properties of − ∆+ V have been studied with scatteringlengths. Taylor [24] gave a two-sided bound for the bottom of the spectrum of − ∆+ V on a boundedregion with the Neumann boundary condition via Γ ( V ). This result was extended by Siudeja [17]in the context of isotropic stable processes. Furthermore, Taylor [25] gave the following equivalentcriterion for discreteness of the spectrum of − ∆ + V in terms of Γ ( V ): for given c >
0, there exists r = r ( c ) ∈ (0 ,
1] and R : (0 , r ] → (0 , ∞ ) such that Γ ( r V r,ξ ) ≥ r c, for r ∈ (0 , r ] , | ξ | ≥ R ( r ) , (1.2)where V r,ξ is the function supported on the unit cube D , in R d defined by V r,ξ ( x ) = V ( rx + ξ ).In the case V = ∞ on R d \ Ω, the condition (1.2) becomes Cap( D r,ξ \ Ω) ≥ r c Cap( D r,ξ ) for r ∈ (0 , r ], | ξ | ≥ R ( r ), where D r,ξ is the cube in R d with side length r and center ξ . This is knownas one of the equivalent criteria for discreteness of the spectrum of − ∆ on L (Ω) with the Dirichletboundary condition on ∂ Ω ([15]).Scattering lengths cited so far were considered for positive continuous additive functionals. Inthe present paper, we first define the scattering length of a positive additive functional of the form A µt + X
2) in R d . Here A µt is the positive continuous additive functional of X with a positive smooth measure µ on R d as its Revuz measure and F is a symmetric positivebounded Borel function on R d × R d vanishing on the diagonal. Let F ( p ) be a non-local linearoperator defined by F ( p ) f ( x ) = C d,α Z R d (cid:0) − e − pF ( x,y ) (cid:1) f ( y ) | x − y | d + α d y, p ≥ f on R d , where C d,α := α α − π − d/ Γ( d + α )Γ(1 − α ) − . Put F f := F (1) f . We assume that F ( p ) ∈ L ( R d ) for any p ≥
1. Let U µ + F be the capacitary potential relativeto the additive functional (1.3) defined by(1.5) U µ + F ( x ) := 1 − E x h e − A µ ∞ − P t> F ( X t − ,X t ) i . In this paper, we shall define the scattering length Γ ( µ + F ) relative to (1.3) by Γ ( µ + F ) := Z R d (1 − U µ + F )( x ) µ (d x ) + Z R d F (1 − U µ + F )( x )d x. We will explain in Section 2 why the expression as above is natural for the definition of the scatteringlength relative to (1.3). We will also give another expression for the scattering length above, whichplays a crucial role throughout this paper (see Lemma 2.1(1)).Our first result of this paper is about the semi-classical limit of the scattering length. Wewill investigate the behaviour of the scattering length Γ ( pµ + pF ) when p → ∞ . More precisely,let τ t be the right continuous inverse of the positive continuous additive functional A µ + F t := A µt + R t F X s )d s defined by τ t := inf { s > | A µ + F s > t } . Denote by S µ + F the fine support of A µ + F t , the topological support of µ + F X , S µ + F = n x ∈ R d | P x ( τ = 0) = 1 o . Theorem 1.1.
Assume that there exists a positive function ψ ( p ) satisfying ψ ( p ) ≤ p and ψ ( p ) → ∞ as p → ∞ , the non-local operator F ( p ) induced by F satisfies the following condition: for large p ≥ and a constant C > F ( p ) x ) ≥ Cψ ( p ) F x ) for x ∈ R d . (1.6) Then lim p →∞ Γ ( pµ + pF ) = Cap( S µ + F ) . Here Cap is the capacity relative to the Dirichlet form ( E , F ) of X . Theorem 1.1 can be regarded as a generalization of the result in [22] (in the framework ofsymmetric stable processes). In Section 2, we will prove Theorem 1.1 with the help of Lemmas 2.1and 2.3 and confirm the condition (1.6) with some concrete examples of F s.Sections 3 and 4 are devoted to the spectral theory of the Neumann fractional Laplacian with apositive potential V and a measure valued non-local operator d F defined by d F f ( x ) := F f ( x )d x as3erturbations, including an equivalent criterion for discreteness of the spectrum of the Schr¨odingeroperator L V + F = ( − ∆) α/ + V + d F . In Section 3, we give a two-side bound for the bottom of the spectrum of L V + F with the Neumannboundary condition on the unit cube D , in R d via the scattering length Γ ( V + F ). The proofs areanalogous to corresponding results in [17] with some additional modification due to F . It is wellknown that an operator H has discrete spectrum if its spectrum set σ ( H ) consists of eigenvalues offinite multiplicity (with the only accumulated point ∞ ). We will abbreviate this with the notation σ ( H ) = σ d ( H ). We also give an equivalent criterion for σ ( L V + F ) = σ d ( L V + F ) in terms of thescattering length for V and F , by using the results obtained in the previous section.Let F r,ξ be the function supported on D , × D , defined by F r,ξ ( x, y ) := F ( rx + ξ, ry + ξ ).The second result of this paper is the following. Theorem 1.2.
The following conditions are equivalent. (i)
For given c > , there exists r := r ( c ) ∈ (0 , and R := R ( c ) > such that Γ ( r α V r,ξ + F r,ξ ) ≥ r α c for | ξ | ≥ R. (ii) For given c > , there exists r := r ( c ) ∈ (0 , and R : (0 , r ] → (0 , ∞ ) such that Γ ( r α V r,ξ + F r,ξ ) ≥ r α c for | ξ | ≥ R ( r ) , r ∈ (0 , r ] . (iii) σ ( L V + F ) = σ d ( L V + F ) . In Section 4, we give the proof of Theorem 1.2. The sufficient condition for discreteness of thespectrum in terms of the scattering length (((i)) = ⇒ ((iii))) is proved by Proposition 4.1. Thenecessary condition (((ii)) = ⇒ ((i))) follows from Proposition 3.1 in combination with Lemmas 4.2and 4.3. Finally we also discuss an easier-to-handle sufficient condition for σ ( L V + F ) = σ d ( L V + F )based on the concept of thin at infinity studied in [14, 23] in our settings.Throughout this paper, we use c, C, c ′ , C ′ , c i , C i ( i = 1 , , . . . ) as positive constants which maybe different at different occurrences. For notational convenience, we let a ∧ b := min { a, b } for any a, b ∈ R . Let X = ( X t , P x ) be the symmetric α -stable process in R d with 0 < α < d ≥
1, that is X is airreducible and conservative L´evy process whose characteristic function is given by exp( − t | ξ | α ) ( ξ ∈ R d ). For simplicity, we assume that d > α , the transience of X . Note that X admits a strictlypositive joint continuous transition density function p t ( x, y ) on (0 , ∞ ) × R d × R d satisfying thefollowing two-sided bound: for some c > d and α , c − (cid:18) t − d/α ∧ t | x − y | d + α (cid:19) ≤ p t ( x, y ) ≤ c (cid:18) t − d/α ∧ t | x − y | d + α (cid:19) , x, y ∈ R d , t > − ∆) α/ be the fractional Laplacian on R d , the generator of X . The Dirichlet form( E , F ) on L ( R d ) associated with X (or ( − ∆) α/ ) is given by E ( f, g ) = C d,α Z R d Z R d ( f ( x ) − f ( y ))( g ( x ) − g ( y )) | x − y | d + α d x d y F = (cid:26) f ∈ L ( R d ) : Z R d Z R d ( f ( x ) − f ( y )) | x − y | d + α d x d y < ∞ (cid:27) , where C d,α := α α − π − d/ Γ( d + α )Γ(1 − α ) − . It is known that X has a L´evy system ( N ( x, d y ) , d t )where N ( x, d y ) = C d,α | x − y | − ( d + α ) d y , that is, E x X s ≤ t φ ( X s − , X s ) = E x (cid:20)Z t Z R d C d,α φ ( X s , y ) | X s − y | d + α d y d s (cid:21) for any non-negative Borel function φ on R d × R d vanishing on the diagonal and any x ∈ R d (cf.[5]).Let µ be a positive smooth measure on R d and denote by A µt a positive continuous additivefunctional (PCAF in abbreviation) of X in the Revuz correspondence to µ : for any boundedmeasurable function f on R d ( B b ( R d ) in notation), Z R d f ( x ) µ (d x ) = ↑ lim t → t Z R d E x (cid:20)Z t f ( X s )d A µs (cid:21) d x. It is known that the family of positive smooth measures and the family of equivalence classes of theset of PCAFs are in one to one correspondence under Revuz correspondence ([8, Theorem 5.1.4].Let F ( x, y ) be a symmetric positive bounded Borel function on R d × R d vanishing on the diagonal.Then P
0, we define the β -order resolvent kernel r β ( x, y ) = R ∞ e − βt p t ( x, y )d t , x, y ∈ R d .Since X is transient and β r β ( x, y ) is decreasing, one can define the 0-order resolvent kernel r ( x, y ) := lim β → r β ( x, y ) < ∞ for x, y ∈ R d with x = y . It is known that r ( x, y ) is nothing butthe Riesz kernel, r ( x, y ) = C d,α | x − y | α − d . For a non-negative Borel measure ν , we write Rν ( x ) := C d,α Z R d | x − y | α − d ν (d y ) = E x (cid:20)Z ∞ d A νt (cid:21) := E x [ A ν ∞ ]and Rf ( x ) := Rν ( x ) when ν (d x ) = f ( x )d x for any f ∈ B b ( R d ).We say that a non-negative Borel measure ν on R d (resp. a non-negative symmetric Borelfunction φ on R d × R d vanishing on the diagonal) is Green-bounded ( ν ∈ S D ( X ) (resp. φ ∈ J D ( X )),in notations) if sup x ∈ R d E x [ A ν ∞ ] < ∞ , resp . sup x ∈ R d E x "X t> φ ( X t − , X t ) < ∞ ! . F ∈ J D ( X ).Let F be a non-local linear operator defined as in (1.4). We assume that µ is finite, µ ( R d ) < ∞ and F ∈ L ( R d ). Now we shall define the scattering length Γ ( µ + F ) relative to the additivefunctional (1.3). In analogy with classical potential theory, it seems to be natural to define Γ ( µ + F )by the total mass of ( − ∆) α/ U µ + F , Γ ( µ + F ) = Z R d ( − ∆) α/ U µ + F ( x )d x. We note that the capacitary potential U µ + F defined in (1.5) satisfies the following formal equation( − ∆) α/ U µ + F = (1 − U µ + F ) µ + F − d F U µ + F . (2.2)Indeed, let e X = ( X t , e P x ) be the transformed process of X by the pure jump Girsanov transformdefined by Y Ft := exp − X F and sup x ∈ R d E x "Z ∞ Z R d (cid:0) e − F ( X s ,y ) − (cid:1) | X s − y | d + α d y d t ≤ sup x ∈ R d E x (cid:20)Z ∞ Z R d F ( X s , y ) | X s − y | d + α d y d t (cid:21) = sup x ∈ R d E x "X t> F ( X t − , X t ) < ∞ . From this fact, we see that the transformed process e X is to be a transient and conservative sym-metric stable-like process on R d in the sense of [6]. Let ( − e ∆) α/ be the generator of e X . Then( − e ∆) α/ is formally given by ( − e ∆) α/ = ( − ∆) α/ + d F − F . (2.4)It is known that a PCAF of X can be regarded as a PCAF of e X ([18, Lemma 2.2]). Thus we seefrom [12, Lemma 4.9] and [21, Lemma 3.2] that U µ + F ( x ) = 1 − E x h e − A µ ∞ − P t> F ( X t − ,X t ) i = 1 − e E x h e − A µ ∞ − R ∞ F X t )d t i = e E x (cid:20)Z ∞ e − A µt − R t F X s )d s (d A µt + F X t )d t ) (cid:21) . (2.5) 6he equation (2.5) implies that U µ + F satisfies the following formal equation (cid:16) µ + F − e ∆) α/ (cid:17) U µ + F = µ + F . (2.6)Hence we have (2.2) by applying (2.4) to (2.6). We note that the relation (2.2) is rigorouslyestablished for µ and F whenever µ and F L ( R d ).Now, let us define the scattering length Γ ( µ + F ) relative to (1.3) by the total mass of ( − ∆) α/ U µ + F ,that is,(2.7) Γ ( µ + F ) := Z R d (1 − U µ + F )( x ) µ (d x ) + Z R d F (1 − U µ + F )( x )d x. The following expression and monotonicity of the scattering length (2.7) play a crucial rolethroughout this paper.
Lemma 2.1. (1)
The scattering length (2.7) also can be rewritten as Γ ( µ + F ) = Z R d E x h e − A µ ∞ − P t> F ( X t − ,X t ) i ( µ (d x ) + F x )d x ) . (2.8)(2) Let ν be a positive finite smooth measure on R d and G be a symmetric positive bounded Borelfunction on R d × R d vanishing on the diagonal set satisfying G ∈ L ( R d ) , where G is the non-localoperator defined as in (1.4) for G . If µ ≤ ν and F ≤ G , then Γ ( µ + F ) ≤ Γ ( ν + G ) .Proof. (1): The expression (2.8) is a consequence of the symmetry of F . Indeed, Z R d F (1 − U µ + F )( x )d x = C d,α Z R d Z R d (cid:0) − e − F ( x,y ) (cid:1) E y [ e − A µ ∞ − P t> F ( X t − ,X t ) ] | x − y | d + α d y d x = Z R d E x h e − A µ ∞ − P t> F ( X t − ,X t ) i F x )d x. (2): The proof is a mimic of the proof of [17, Proposition 3.2] in our settings. It is clear that U µ + F ≤ U ν + G under the assumption. By virtue of [12, Lemma 4.9] and [21, Lemma 3.2], U µ + F ( x ) = 1 − E x h e − A µ ∞ − P t> F ( X t − ,X t ) i = 1 − e E x h e − A µ ∞ − R ∞ F X t )d t i = e E x (cid:20)Z ∞ e E X t h e − A µ ∞ − R ∞ F X s )d s i (d A µt + F X t )d t ) (cid:21) = e E x (cid:20)Z ∞ E X t h e − A µ ∞ − P s> F ( X s − ,X s ) i (d A µt + F X t )d t ) (cid:21) := e R (cid:16) E · h e − A µ ∞ − P t> F ( X t − ,X t ) i ( µ + F (cid:17) ( x ) , (2.9)where e R denotes the 0-order resolvent operator of e X with the resolvent kernel e r ( x, y ). Let K ⊂ R d be a Kac’s regular set in the sense that the Lebesgue penetration time of K by the transformed7rocess e X is the same as the hitting time of K . In this case, the capacity potential e U K of K isgiven by e U K ( x ) = 1 − e E x h e − R ∞ V K ( X t )d t i for V K = ∞ on K and 0 off K . Moreover, analogously to (2.9), we see that e U K ( x ) = e Rγ K ( x ) forthe equilibrium measure γ K on K (cf. [19]).First, we suppose that the topological supports supp[ µ + F
1] and supp[ ν + G
1] are boundedsuch that supp[ µ + F ∪ supp[ ν + G ⊂ K . In view of (2.9) and the fact that e U K = e Rγ K = 1 onsupp[ µ + F ∪ supp[ ν + G Γ ( µ + F ) = Z supp[ µ + F e Rγ K ( x ) E x h e − A µ ∞ − P t> F ( X t − ,X t ) i ( µ (d x ) + F x )d x )= Z supp[ µ + F Z R d e r ( x, y ) E x h e − A µ ∞ − P t> F ( X t − ,X t ) i ( µ (d x ) + F x )d x ) γ K (d y )= Z supp[ µ + F e R (cid:16) E · h e − A µ ∞ − P t> F ( X t − ,X t ) i ( µ + F (cid:17) ( y ) γ K (d y )= Z supp[ µ + F U µ + F ( y ) γ K (d y ) ≤ Z supp[ ν + G U ν + G ( y ) γ K (d y ) = Γ ( ν + G ) . (2.10)Now we prove the monotonicity (2.10) without assumptions on the supports. Let ν n (resp. G n )be a non-decreasing sequence of finite positive smooth measures on R d (resp. a non-decreasingsequence of symmetric positive bounded Borel functions on R d × R d vanishing on the diagonalsatisfying G n ∈ L ( R d )) such that supp[ ν n + G n
1] is bounded, supp[ ν n + G n ⊂ K for any n ≥ ν n + G n ր ν + G as n → ∞ . Set µ n := ν n ∧ µ and F n := G n ∧ F . Then we have Z R d E x h e − A µn ∞ − P t> F n ( X t − ,X t ) i ( µ n (d x ) + F n x )d x )= Γ ( µ n + F n ) ≤ Γ ( ν n + G n ) = Z R d E x h e − A νn ∞ − P t> G n ( X t − ,X t ) i ( ν n (d x ) + G n x )d x ) . Both integrands are bounded above by R R d ( ν (d x ) + G x )d x ) < ∞ , hence we have the assertionby the dominated convergence theorem. Remark 2.2.
The scattering length (2.7) is also expressed as Γ ( µ + F ) = lim t →∞ t Z R d (cid:16) − E x h e − A µt − P | A µs + R s F X u )d u > t } . Let denote by S µ + F the fine support of A µt + R t F X s )d s , S µ + F = n x ∈ R d (cid:12)(cid:12)(cid:12) P x ( τ = 0) = 1 o , and by Cap the capacity relative to the Dirichlet form ( E , F ) (see [5], [8]). For p ≥
1, set F ( p ) x ) := C d,α Z R d (cid:0) − e − pF ( x,y ) (cid:1) | x − y | d + α d y. Clearly, F x )(= F (1) x )) ≤ F ( p ) x ) for p ≥
1. For notational convenience, we let A F t := R t F X s )d s . Lemma 2.3.
For any ε > p →∞ Γ (cid:0) pF + p ε µ + p ε F (cid:1) = Cap( S µ + F ) . In particular, lim sup p →∞ Γ ( pµ + pF ) ≤ Cap( S µ + F ) .Proof. The last assertion easily follows from the first one with the monotonicity of the scatteringlength. Put k = 1 / (1 + ε ). From the expression (2.8), one can easily see that Γ (cid:16) p k F + pµ + p F (cid:17) = Z R d E x h e − p k P t> F ( X t − ,X t ) − pA µ ∞ − pA F ∞ i (cid:16) pµ (d x ) + F ( p k ) x )d x + p F x )d x (cid:17) . Since F ( q ) ≤ q F q ≥ Γ (cid:16) p k F + pµ + p F (cid:17) ≤ Z R d E x h e − pA µ ∞ − pA F ∞ i (cid:16)(cid:16) p k − (cid:17) pµ (d x ) + (cid:16) p k F p F (cid:17) ( x )d x (cid:17) = (cid:16) p k − (cid:17) Γ ( pµ + p F . Therefore we have by the monotonicity of the scattering length that Γ ( pµ + p F ≤ Γ (cid:16) p k F + pµ + p F (cid:17) ≤ (cid:16) p k − (cid:17) Γ ( pµ + p F . By applying (2.11), the scattering lengths of both sides of the above converge to Cap( S µ + F ) as p → ∞ , which implies the first assertion. 9ow we prove the first main result of this paper. Proof of Theorem 1.1 : Let ψ ( p ) be the function which appeared in the condition (1.6). By themonotonicity of the scattering length, we have for some C > Γ (cid:18) ψ ( p ) n µ + Cψ ( p ) n F (cid:19) ≤ Γ (cid:18) pF + ψ ( p ) n µ + Cψ ( p ) n F (cid:19) ≤ Γ (cid:0) pF + p ε µ + p ε F (cid:1) for any n ≥ ε >
0. Then, by Lemma 2.3 and applying (2.11) again, one can get thatlim p →∞ Γ (cid:18) pF + ψ ( p ) n µ + Cψ ( p ) n F (cid:19) = Cap( S µ + F ) . From this and the condition (1.6), we see thatlim inf p →∞ Γ ( pµ + pF ) ≥ lim inf p →∞ Γ (cid:18) ψ ( p ) n µ + pF (cid:19) = lim inf p →∞ Z R d E x h e − p P t> F ( X t − ,X t ) − ψ ( p ) n A µ ∞ i (cid:18) ψ ( p ) n µ (d x ) + F ( p ) x )d x (cid:19) = nn + 1 lim inf p →∞ Z R d E x h e − p P t> F ( X t − ,X t ) − ψ ( p ) n A µ ∞ i · (cid:18) n + 1 n ψ ( p ) n µ (d x ) + F ( p ) x )d x + 1 n F ( p ) x )d x (cid:19) ≥ nn + 1 lim inf p →∞ Z R d E x h e − p P t> F ( X t − ,X t ) − ψ ( p ) n A µ ∞ − Cψ ( p ) n A F ∞ i · (cid:18) F ( p ) x )d x + ψ ( p ) n µ (d x ) + Cψ ( p ) n F x )d x (cid:19) = nn + 1 lim p →∞ Γ (cid:18) pF + ψ ( p ) n µ + Cψ ( p ) n F (cid:19) = nn + 1 Cap( S µ + F ) . Letting n → ∞ , we have Cap( S µ + F ) ≤ lim inf p →∞ Γ ( pµ + pF ) . (2.12)The proof will be finished by the last assertion of Lemma 2.3 and (2.12).Now, we consider some concrete examples of F s satisfying the condition (1.6). Denote by B ( a, b )the open ball in R d with center a and radius b . Example 2.4.
Let F be the function on R d × R d such that for R, R ′ > F ( x, y ) = 12 φ ( | x − y | ) (cid:0) B ( x,R ′ ) ( y ) B (0 ,R ) ( x ) + B ( y,R ′ ) ( x ) B (0 ,R ) ( y )+ B ( y,R ′ ) ( x ) B ( x,R ′ ) ( y ) B (0 ,R + R ′ ) \ B (0 ,R ) ( x ) B (0 ,R + R ′ ) \ B (0 ,R ) ( y ) (cid:1) , where φ is a non-negative strictly increasing smooth function on [0 , ∞ ) satisfying φ (0) = 0 , φ ( t ) = o ( t α ) ( t → and φ − ( p − t ) α ≤ ψ ( p ) φ − ( t ) α , t ≥ , large p ≥ or a positive function ψ ( p ) such that ψ ( p ) ≤ p and ψ ( p ) → ∞ as p → ∞ . Then the condition (1.6) holds for this F . First, let us take x ∈ B (0 , R ) . In this case, F is given by F ( x, y ) = φ ( | x − y | ) y ∈ B ( x, R ′ ) ∩ B (0 , R ) φ ( | x − y | ) y ∈ B ( x, R ′ ) ∩ B (0 , R ) c . Then we have F x ) = C d,α Z R d − e − F ( x,y ) | x − y | d + α d y = C d,α Z B ( x,R ′ ) − e − F ( x,y ) | x − y | d + α d y = C d,α (Z B ( x,R ′ ) ∩ B (0 ,R ) − e − φ ( | x − y | ) | x − y | d + α d y + Z B ( x,R ′ ) ∩ B (0 ,R ) c − e − φ ( | x − y | ) | x − y | d + α d y ) ≤ C d,α Z B ( x,R ′ ) − e − φ ( | x − y | ) | x − y | d + α d y. (2.14) By using integration by parts, the right-hand side of (2.14) is equal to C ′ d,α Z R ′ − e − φ ( r ) r α d r = C ′ d,α ( e − φ ( R ′ ) − α ( R ′ ) α + 1 α Z R ′ r − α φ ′ ( r ) e − φ ( r ) d r ) = C ′ d,α ( e − φ ( R ′ ) − α ( R ′ ) α + 1 α Z φ ( R ′ )0 φ − ( t ) α e − t d t ) , (2.15) where C ′ d,α is a positive constant depending on d and α . On the other hand, by a similar calculationabove with the inequality − e − a − b ≤ (1 − e − a ) + (1 − e − b ) for any a, b ≥ and the condition (2.13) ,we see F ( p ) x ) = C d,α (Z B ( x,R ′ ) ∩ B (0 ,R ) − e − pφ ( | x − y | ) | x − y | d + α d y + Z B ( x,R ′ ) ∩ B (0 ,R ) c − e − p φ ( | x − y | ) | x − y | d + α d y ) ≥ C d,α Z B ( x,R ′ ) − e − p φ ( | x − y | ) | x − y | d + α d y ≥ C d,α Z B ( x,R ′ ) − e − pφ ( | x − y | ) | x − y | d + α d y = C ′ d,α ( e − pφ ( R ′ ) − α ( R ′ ) α + 1 α Z pφ ( R ′ )0 φ − ( p − t ) α e − t d t ) ≥ ψ ( p )2 C ′ d,α ( e − φ ( R ′ ) − α ( R ′ ) α + 1 α Z φ ( R ′ )0 φ − ( t ) α e − t d t ) (2.16) for large p ≥ . Hence we can confirm by (2.14) , (2.15) and (2.16) that F ( p ) x ) ≥ ψ ( p ) F x ) , x ∈ B (0 , R ) , large p ≥ . (2.17) Next, take x ∈ B (0 , R + R ′ ) \ B (0 , R ) . In this case, F is given by F ( x, y ) = φ ( | x − y | ) y ∈ B ( x, R ′ ) ∩ B (0 , R ) φ ( | x − y | ) y ∈ B ( x, R ′ ) ∩ B (0 , R ) c . hen, by the same calculations as (2.14) , (2.15) and (2.16) F x ) ≤ C d,α Z B ( x,R ′ ) − e − φ ( | x − y | ) | x − y | d + α d y = C ′ d,α ( e − φ ( R ′ ) − α ( R ′ ) α + 1 α Z φ ( R ′ )0 φ − ( t ) α e − t d t ) and F ( p ) x ) ≥ C d,α Z B ( x,R ′ ) − e − pφ ( | x − y | ) | x − y | d + α d y ≥ ψ ( p )2 C ′ d,α ( e − φ ( R ′ ) − α ( R ′ ) α + 1 α Z φ ( R ′ )0 φ − ( t ) α e − t d t ) = ψ ( p )2 F x ) for large p ≥ . Therefore, we can confirm (2.17) for x ∈ B (0 , R + R ′ ) \ B (0 , R ) . For x ∈ B (0 , R + R ′ ) c , (2.17) is trivial because F ( p ) x ) = 0 for any p ≥ . Hence we have (2.17) for any x ∈ R d . Moreover, for x ∈ B (0 , R + R ′ ) F ( p ) x ) = C d,α Z B (0 ,R + R ′ ) ∩ B ( x, − e − pF ( x,y ) | x − y | d + α d y + Z B (0 ,R + R ′ ) ∩ B ( x, c − e − pF ( x,y ) | x − y | d + α d y ! ≤ C d,α Z B ( x, − e − p φ ( | x − y | ) | x − y | d + α d y + C d,α Z B (0 ,R + R ′ ) ∩ B ( x, c (cid:16) − e − p k φ k ∞ (cid:17) d y ≤ C ′ d,α Z − e − p φ ( r ) r α +1 d r + C d,α (cid:12)(cid:12)(cid:12) B (0 , R + R ′ ) (cid:12)(cid:12)(cid:12) . Since φ ( t ) = o ( t α ) ( t → , it follows that F ( p ) is bounded on B (0 , R + R ′ ) and is zero on B (0 , R + R ′ ) c for any p ≥ . This shows that F ( p ) ∈ L ( R d ) for any p ≥ . By a similar way as inthe proof of [4, Proposition 7.10(3)], one can also prove that F ( p ) ∈ L ℓ ( R d )( ℓ ≥ for any p ≥ .We omit the details.There are many functions satisfying the condition (2.13) . For instance, they can be given by φ ( t ) = t β , φ ( t ) = t β / (1 + t ) β , φ ( t ) := φ (1) ( t ) = log(1 + t β ) and its iterated function φ ( n ) ( t ) = φ ( φ ( n − ( t )) ( n ≥ ) for β > α . For these functions, it holds that F ∈ J D ( X ) (cf. [2, Example2.1]) and we can take the function ψ ( p ) which appeared in (2.13) as ψ ( p ) = p α/β . Hence, the scattering length Γ ( pµ + pF ) induced by the functions φ above converges to Cap( S µ + F ) as p → ∞ , in view of Theorem 1.1. The behaviour of scattering lengths for small potentials is of independent of interest. Let usconsider the case that µ (d x ) = V ( x )d x with V being a positive L ( R d )-function. Proposition 2.5.
Suppose that V and F have bounded supports in R d . Then lim ε → ε Γ ( εV + εF ) = Z R d (cid:16) V + b F (cid:17) ( x )d x, where b F x ) := C d,α R R d F ( x, y ) | x − y | − d − α d y . roof. It is clear from (2.8) that Γ ( V + F ) ≤ R R d ( V ( x )d x + F x )d x ). For convenience, assumethat V and F B ⊂ R d . By (2.9), we see that U V + F ( x ) := e R (cid:16) E · h e − R ∞ V ( X t )d t − P t> F ( X t − ,X t ) i ( V + F (cid:17) ( x ) . (2.18)It is known in [18, Corollary 2.8] that the resolvent kernel e r ( x, y ) of e R satisfies C − | x − y | d − α ≤ e r ( x, y ) ≤ C | x − y | d − α , x, y ∈ R d (2.19)for some C >
0. Then Z B U V + F ( x )d x = Z B e R (cid:16) E · h e − R ∞ V ( X t )d t − P t> F ( X t − ,X t ) i ( V + F (cid:17) ( x )d x ≤ C Z B Z R d | x − y | d − α E y h e − R ∞ V ( X t )d t − P t> F ( X t − ,X t ) i ( V + F y )d y d x ≤ C ( B ) Γ ( V + F ) ≤ C ( B ) Z R d ( V + F x )d x, (2.20)where C ( B ) := C sup y ∈ R d R B | x − y | α − d d x . From this and the fact that F ( ε ) → ε →
0, we see Z B U εV + εF ( x )d x −→ , as ε → , which implies that U εV + εF → B . The same are true for U εV + εF V and ε − U εV + εF F ( ε ) ε − F ( ε ) → b F ε →
0. Now, by the definition of the scattering length and the dominatedconvergence theorem, (cid:12)(cid:12)(cid:12)(cid:12)Z R d (cid:16) V + ε − F ( ε ) (cid:17) ( x )d x − ε − Γ ( εV + εF ) (cid:12)(cid:12)(cid:12)(cid:12) = Z B U εV + εF ( x ) (cid:16) V + ε − F ( ε ) (cid:17) ( x )d x −→ , as ε → . The proof is complete.
In the rest of the sections, we are going to consider the case that µ is an absolutely continuousmeasure with respect to the Lebesgue measure having V ≥ µ (d x ) = V ( x )d x .In this section, we give a two-sided bound for the bottom of the spectrum of the fractionalNeumann Laplacian on the unit cube D := D , , the cube of side length 1 centered at 0, in R d withlocal and non-local perturbations. We assume that V and F D . 13et ( − ∆) α/ N be the fractional Laplacian on L ( D ) with the Neumann boundary condition, asthe generator of the Dirichlet form on L ( D ) defined by E ref ( f, g ) = C d,α Z D Z D ( f ( x ) − f ( y ))( g ( x ) − g ( y )) | x − y | d + α d x d y F ref a = (cid:26) f ∈ L ( D ) : Z D Z D ( f ( x ) − f ( y )) | x − y | d + α d x d y < ∞ (cid:27) . It is known that ( E ref , F ref a ) is the active reflected Dirichlet form of ( E D , F D ), the Dirichlet formon L ( D ) associated with the part process X D of X on D . The stochastic process Y = ( Y t , P x )associated with ( E ref , F ref a ) (or ( − ∆) α/ N ) is then the reflected stable process on D (cf. [1]).Define the bottom of the spectrum of the formal Schr¨odinger operator L NV + F := ( − ∆) α/ N + V + d F on L ( D ) by λ N ( V + F ) := inf ϕ ∈F ref a E ref ( ϕ, ϕ ) + H V + FD ( ϕ, ϕ ) R D ϕ ( x ) d x , (3.1)where H V + FD ( ϕ, ϕ ) = Z D ϕ ( x ) V ( x )d x + Z D Z D ϕ ( x ) ϕ ( y ) (cid:0) − e − F ( x,y ) (cid:1) | x − y | d + α d x d y. (3.2)The next two propositions are originally due to [17] when F ≡ λ N ( V + F ). Proposition 3.1.
There exists a constant C ( D ) > such that λ N ( V + F ) ≤ C ( D ) Γ ( V + F ) provided Γ ( V + F ) is small.Proof. Note that the infimum in (3.1) may be taken over L ( D ). Put ϕ := 1 − U V + F . Clearly ϕ ∈ L ( D ). Let V n := B (0 ,n ) ( V ∧ n ) and F n ( · , y ) := B (0 ,n ) ( F ( · , y ) ∧ n ) for y ∈ R d , where B (0 , n )is the open ball in R d with center 0 and radius n . Since V n and F n L ( R d ), the relation(2.2) is rigorously established for V n and F n . Therefore we have by Fatou’s lemma that λ N ( V + F ) Z D ϕ ( x ) d x ≤ C d,α Z R d Z R d lim inf n →∞ ( U V n + F n ( x ) − U V n + F n ( y )) | x − y | d + α d x d y + Z R d lim inf n →∞ (1 − U V n + F n ) ( x ) V n ( x )d x + Z R d lim inf n →∞ (1 − U V n + F n )( x ) F n (1 − U V n + F n )( x )d x ≤ C d,α n →∞ Z R d Z R d ( U V n + F n ( x ) − U V n + F n ( y )) | x − y | d + α d x d y
14 lim inf n →∞ Z R d (1 − U V n + F n ) ( x ) V n ( x )d x + lim inf n →∞ Z R d (1 − U V n + F n )( x ) F n (1 − U V n + F n )( x )d x = lim inf n →∞ (cid:26)Z R d U V n + F n ( x )( − ∆) α/ U V n + F n ( x )d x + Γ ( V n + F n ) − Z R d U V n + F n ( x )(1 − U V n + F n )( x ) V n ( x )d x − Z R d U V n + F n ( x ) F n (1 − U V n + F n )( x )d x (cid:27) . = lim inf n →∞ Γ ( V n + F n ) . The sequences V n and F n converge to V and F , respectively. Thus by the monotonicities of thecapacitary potential and the scattering length we see that U V n + F n ր U V + F and Γ ( V n + F n ) ր Γ ( V + F ) as n → ∞ , respectively. Hence we have λ N ( V + F ) Z D (1 − U V + F ) ( x )d x ≤ lim inf n →∞ Γ ( V n + F n ) = Γ ( V + F ) . (3.3)From (2.20), it holds that Z D (1 − U V + F ) ( x )d x ≥ − Z D U V + F ( x )d x ≥ − C ( D ) Γ ( V + F ) , Applying this to (3.3), we obtain that λ N ( V + F )(1 − C ( D ) Γ ( V + F )) ≤ Γ ( V + F ) . Now the assertion holds if we make Γ ( V + F ) so small that Γ ( V + F ) ≤ / (4 C ( D )). Remark 3.2.
The result of Proposition 3.1 is valid for any bounded domain D . Next, we turn to a lower bound for λ N ( V + F ). To do this, we need some facts on subordinatedprocesses. Let B = ( B t , P x ) be a Brownian motion in R d running twice the usual speed. Let Z = ( Z t , P x ) be a reflected Brownian motion on D , that is, Z is the process generated by theLaplacian with the Neumann boundary condition in D . We will derive symmetric stable processesfrom B and Z by using a subordination technique. Let S t be a positive α/ B and Z . Then the symmetric α -stable process X is nothing but the subordinatedprocess of B by S t , that is, X t = B S t . Let W = ( W t , P x ) be the subordinated process of Z by S t .It is known that W is a stable-like process studied in [6], but it is, in general, different from Y thereflected stable process on D associated with ( E ref , F ref a ) (or ( − ∆) α/ N ) (cf. [1]).Let denote by −W the generator of W and λ W ( V + F ) the bottom of the spectrum of theSchr¨odinger operator −W + V + d F . It is easy to check that for some c > c > c λ W ( V + F ) ≤ λ N ( V + F ) ≤ c λ W ( V + F ) . (3.4)Moreover, we can prove the following relation by using a simlilar method as in [17, Lemma 4.1]:for any t > E x h e − R t V ( W s )d s − P such that C ( D ) Γ ( V + F ) ≤ λ N ( V + F ) . Proof.
In view of (3.4), it is enough to show that there exists a constant c ( D ) > c ( D ) Γ ( V + F ) ≤ λ W ( V + F ). To do this, we prove that there exist T > c ( D ) > x ∈ D (cid:0) − U TV + F ( x ) (cid:1) ≤ e − c ( D ) Γ ( V + F ) . (3.6)Then, we see from (3.5) that (cid:13)(cid:13)(cid:13) e − T ( V + d F −W ) (cid:13)(cid:13)(cid:13) , ≤ sup x ∈ D E x h e − R T V ( W s )d s − P s ≤ T F ( W s − ,W s ) i ≤ sup x ∈ D (cid:0) − U TV + F (cid:1) ≤ e − c ( D ) Γ ( V + F ) , which implies c ( D ) Γ ( V + F ) ≤ λ W ( V + F ). Here k · k , means the operator norm from L ( D ) to L ( D ). Now we shall prove (3.6). By the Markov property, for any t, s > U t + sV + F ( x ) − U tV + F ( x )= E x h e − R t V ( X u )d u − P F ( X t − ,X t ) i ( V + F (cid:17) ( X t ) i = Z R d p t ( x, y ) e R (cid:16) E · h e − R ∞ V ( X t )d t − P t> F ( X t − ,X t ) i ( V + F (cid:17) ( y )d y ≤ C Z R d p t ( x, y ) Z R d E z [ e − R ∞ V ( X t )d t − P t> F ( X t − ,X t ) ]( V + F z ) | y − z | d − α d z d y ≤ C sup x ∈ D,z ∈ R d Z R d p t ( x, y ) | y − z | d − α d y ! Γ ( V + F ) . (3.7)In view of (2.1), since Z R d p t ( x, y ) | y − z | d − α d y ≤ C Z R d Z ∞ p t ( x, y ) p τ ( y, z )d τ d y = C Z ∞ t p τ ( x, z )d τ ≤ C ′ Z ∞ t τ − d/α d τ = C ′ t − d/α +1 −→ , as t → ∞ , t → ∞ , locally uniformly in x . Thus we can take largeenough T > U V + F − U TV + F ≤ U V + F /
2. On the other hand, we have by (2.18) and (2.19)again U V + F ( x ) = e R (cid:16) E · h e − R ∞ V ( X t )d t − P t> F ( X t − ,X t ) i ( V + F (cid:17) ( x ) ≥ C − Z D E y [ e − R ∞ V ( X t )d t − P t> F ( X t − ,X t ) ]( V + F y ) | x − y | d − α d y ≥ C − (diam D ) α − d Γ ( V + F ) , x ∈ D. Hence we obtain1 − U TV + F ( x ) ≤ − U V + F ( x ) ≤ − c ( D ) Γ ( V + F ) ≤ e − c ( D ) Γ ( V + F ) uniformly in x ∈ D . The proof is complete. In this section, we give an equivalent characterization for discreteness of the spectrum of the formalSchr¨odinger operator L V + F := ( − ∆) α/ + V + d F in terms of the scattering length, by using the results obtained in the previous section.Let D r,ξ be the d -dimensional cube of the form D r,ξ = n x ∈ R d : | ξ j − x j | ≤ r , j = 1 , , · · · , d o , ξ ∈ R d . In the sequel, we use the notation ( − ∆) α/ N,r,ξ for the Neumann fractional Laplacian on L ( D r,ξ ), toemphasize the dependence of its side length r and center ξ . Similarly to (3.1), we write λ N,r,ξ ( V + F )for the bottom of the spectrum of ( − ∆) α/ N,r,ξ + V + d F . Let V r,ξ be a function supported on D , given by V r,ξ ( x ) = V ( rx + ξ ) and F r,ξ a non-negative function supported on D , × D , given by F r,ξ ( x, y ) := F ( rx + ξ, ry + ξ ). Then, by the definition, we can easily check that λ N,r,ξ ( V + F ) = r − α λ N, , ( r α V r,ξ + F r,ξ ) . First, we give the sufficient condition for the discreteness of the spectrum of L V + F in terms ofthe scattering length relative to V and F restricted to cubes. By σ ess ( H ) we mean the essentialspectrum set of a operator H . Let F r,ξ be the non-local linear operator defined in (1.4) for F r,ξ . Proposition 4.1.
Let C ( D , ) be the positive constant as in Proposition 3.3. Suppose that forgiven c > there exists r := r ( c ) ∈ (0 , and R := R ( c ) > such that C ( D , ) Γ ( r α V r,ξ + F r,ξ ) ≥ r α c for | ξ | ≥ R. (4.1) Then σ ( L V + F ) = σ d ( L V + F ) . roof. Let us denote by λ ess ( V + F ) the bottom of the set σ ess ( L V + F ). To end the proof, we willshow that λ ess ( V + F ) = ∞ under (4.1). It follows from (4.1) and Proposition 3.3 that there existsa constant C ( D , ) > r − α λ N, , ( r α V r,ξ + F r,ξ ) ≥ r − α C ( D , ) Γ ( r α V r,ξ + F r,ξ ) ≥ c for | ξ | ≥ R and which yields that σ (cid:16) r − α (cid:16) ( − ∆) α/ N, , + r α V r,ξ + d F r,ξ (cid:17)(cid:17) ⊂ [ c, ∞ ) for | ξ | ≥ R. (4.2)Note that the operators ( − ∆) α/ N,r,ξ + V + d F and r − α (cid:16) ( − ∆) α/ N, , + r α V r,ξ + d F r,ξ (cid:17) are unitarilyequivalent. Therefore, (4.2) is equivalent to σ (cid:16) ( − ∆) α/ N,r,ξ + V + d F (cid:17) ⊂ [ c, ∞ ) for | ξ | ≥ R. (4.3)By a standard argument involving Rellich’s theorem, (4.3) implies that σ ess ( L V + F ) ⊂ [ c, ∞ ), thatis, λ ess ( V + F ) ≥ c . Since c is arbitrary, we have the assertion by letting c → ∞ .Now we turn to the necessary condition. Let C ∞ ( R d ) be the set of all C ∞ functions withcompact support on R d . Set M := n f ∈ C ∞ ( R d ) : E ( f, f ) + H V + F ( f, f ) ≤ o , where H V + F ( f, f ) is the bilinear form defined as in (3.2) with D replaced by R d . In a similar wayof [13, Lemma 2.2], we see that σ ( L V + F ) = σ d ( L V + F ) if and only if M is precompact in L ( R d ).For the Schr¨odinger operator ( − ∆) α/ D,r,ξ + V + d F , where ( − ∆) α/ D,r,ξ is the fractional Laplacianon L ( D r,ξ ) with the Dirichlet boundary condition, we denote by λ D,r,ξ ( V + F ) the bottom of itsspectrum. Lemma 4.2. If σ ( L V + F ) = σ d ( L V + F ) , then for each r ∈ (0 , , λ D,r,ξ ( V + F ) → ∞ as | ξ | → ∞ .Proof. The proof is similar to that of [13, Lemma 2.3 and Proposition 2.4]. We address here theproof for reader’s convenience. For a fixed x ∈ R d and R >
0, the compact embedding theoremsays that M B = { f | B ( x ,R ) ∈ L ( B ( x , R )) : f ∈ M } is precompact in L ( B ( x , R )). From thisfact, we see that the precompactness of M is equivalent to the precompactness of M B with thecondition: for any ε > R := R ( ε ) > Z B ( x ,R ) c f ( x ) d x ≤ ε, for any f ∈ M. (4.4)Now, choose a sufficiently small ε > D r,ξ ⊂ B ( x , R ) c where R = R ( ε ) corre-sponds to ε according to (4.4). Then for any f ∈ C ∞ ( D r,ξ ) with E D r,ξ ( f, f ) + H V + FD r,ξ ( f, f ) ≤ R D r,ξ f ( x ) d x ≤ ε . Therefore E D r,ξ ( f, f ) + H V + FD r,ξ ( f, f ) R D r,ξ f ( x ) d x ≥ ε . This implies that λ D,r,ξ ( V + F ) → ∞ as | ξ | → ∞ .18 emma 4.3. If for each r ∈ (0 , λ D,r,ξ ( V + F ) → ∞ as | ξ | → ∞ , then so does λ N,r,ξ ( V + F ) .Proof. Let λ := λ D, , ( r α V r,ξ + F r,ξ ). For a fixed a ∈ (0 , τ := λ − a . Since the transitiondensity p t ( x, y ) of ( − ∆) α/ satisfies (2.1) for x, y ∈ D , and t >
0, we have for some
C > (cid:13)(cid:13)(cid:13)(cid:13) e − τ (cid:16) r α V r,ξ + d F r,ξ +( − ∆) α/ D, , (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) , ≤ Cτ − d/α , (cid:13)(cid:13)(cid:13)(cid:13) e − τ (cid:16) r α V r,ξ + d F r,ξ +( − ∆) α/ D, , (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) , ∞ ≤ Cτ − d/α , while the definition of λ gives (cid:13)(cid:13)(cid:13)(cid:13) e − τ (cid:16) r α V r,ξ + d F r,ξ +( − ∆) α/ D, , (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) , ≤ e − τλ . Here k · k p,q means the operator norm from L p ( D , ) to L q ( D , ) for 1 ≤ p, q ≤ ∞ . Then (cid:13)(cid:13)(cid:13)(cid:13) e − τ (cid:16) r α V r,ξ + d F r,ξ +( − ∆) α/ D, , (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) ∞ = (cid:13)(cid:13)(cid:13)(cid:13) e − τ (cid:16) r α V r,ξ + d F r,ξ +( − ∆) α/ D, , (cid:17) e − τ (cid:16) r α V r,ξ + d F r,ξ +( − ∆) α/ D, , (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ (cid:13)(cid:13)(cid:13)(cid:13) e − τ (cid:16) r α V r,ξ + d F r,ξ +( − ∆) α/ D, , (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) , ∞ (cid:13)(cid:13)(cid:13)(cid:13) e − τ (cid:16) r α V r,ξ + d F r,ξ +( − ∆) α/ D, , (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) , ≤ Cτ − d/α e − τλ = Cλ ad/α e − λ − a , that is, the transition density p D, r α V r,ξ + F r,ξ t ( x, y ) of ( − ∆) α/ D, , + r α V r,ξ + d F r,ξ satisfies0 ≤ p D, r α V r,ξ + F r,ξ τ ( x, y ) ≤ Cλ ad/α e − λ − a , for x, y ∈ D , . By p N, r α V r,ξ + F r,ξ t ( x, y ), we denote the transition density of ( − ∆) α/ N, , + r α V r,ξ + d F r,ξ . Put q r α V r,ξ + F r,ξ t ( x, y ) := p N, r α V r,ξ + F r,ξ t ( x, y ) − p D, r α V r,ξ + F r,ξ t ( x, y ) , for t ∈ (0 , τ ] , x, y ∈ D , . Then q r α V r,ξ + F r,ξ t ( x, y ) satisfies the following fractional heat equation: ( (cid:16) ∂ t − ( − ∆) α/ − r α V r,ξ − d F r,ξ (cid:17) q r α V r,ξ + F r,ξ t ( · , y ) = 0 on (0 , ∞ ) × D , q r α V r,ξ + F r,ξ ( x, y ) = 0 , q r α V r,ξ + F r,ξ t ( x, y ) = p N, r α V r,ξ + F r,ξ t ( x, y ) on x ∈ ∂D , . (4.5)Therefore 0 ≤ q r α V r,ξ + F r,ξ t ( x, y ) ≤ p N, , t ( x, y ) . x ∈ ∂D , , (4.6)where p N, , t ( x, y ) is the transition density of ( − ∆) α/ N, , . Set D ( τ )1 , := n y ∈ D , : dist( y, ∂D , ) ≥ τ / ( d + α +1) o . p N, , t ( x, y ) ≤ Ct | x − y | d + α ≤ Cτ − d + αd + α +1 , x ∈ ∂D , , y ∈ D ( τ )1 , , t ∈ (0 , τ ] . (4.7)So applying the maximum principle for the fractional Laplacian to (4.5), together with (4.6) and(4.7), gives q r α V r,ξ + F r,ξ t ( x, y ) ≤ Cτ − d + αd + α +1 , x ∈ D , , y ∈ D ( τ )1 , , t ∈ (0 , τ ] , and hence, for sufficiently large λ , we have0 ≤ p N, r α V r,ξ + F r,ξ τ ( x, y ) = q r α V r,ξ + F r,ξ τ ( x, y ) + p D, r α V r,ξ + F r,ξ τ ( x, y ) ≤ Cτ − d + αd + α +1 + Cλ ad/α e − λ − a = λ − a ( − d + αd + α +1 ) + Cλ ad/α e − λ − a ≤ C ′ λ ad/α e − λ − a , x ∈ D , , y ∈ D ( τ )1 , . (4.8)By using the semigroup property of e − t ( r α V r,ξ + d F r,ξ +( − ∆) α/ N, , ) , the estimate (4.8) can be extendedfor any t ∈ [3 τ, ∞ ),0 ≤ p N, r α V r,ξ + F r,ξ t ( x, y ) ≤ C ′ λ ad/α e − λ − a , x ∈ D , , y ∈ D ( τ )1 , , t ∈ [3 τ, ∞ ) . (4.9)In particular, if λ is large enough that 3 τ = 3 λ − a <
1, then the estimate (4.9) holds for p N, r α V r,ξ + F r,ξ ( x, y ).On the other hand, since p N, r α V r,ξ + F r,ξ ( x, y ) ≤ p N, , ( x, y ) ≤ C, x ∈ D , , y ∈ D , \ D ( τ )1 , , it follows that Z D , p N, r α V r,ξ + F r,ξ ( x, y )d y = Z D ( τ )1 , p N, r α V r,ξ + F r,ξ ( x, y )d y + Z D , \ D ( τ )1 , p N, r α V r,ξ + F r,ξ ( x, y )d y ≤ Cλ ad/α e − λ − a + Cλ − a/ ( d + α +1) . We also have a similar bound for R D , p N, r α V r,ξ + F r,ξ ( x, y )d x by the symmetry. Hence we can deducethat (cid:13)(cid:13)(cid:13)(cid:13) e − (cid:16) r α V r,ξ + d F r,ξ +( − ∆) α/ N, , (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) , ≤ Cλ ad/α e − λ − a + Cλ − a/ ( d + α +1) which implies the assertion.We are now ready to prove the necessary condition for the discreteness of the spectrum of L V + F in terms of the scattering length. 20 roposition 4.4. Suppose that σ ( L V + F ) = σ d ( L V + F ) . Then for given c > , there exists r := r ( c ) ∈ (0 , and R : (0 , r ] → (0 , ∞ ) such that Γ ( r α V r,ξ + F r,ξ ) ≥ r α c for | ξ | ≥ R ( r ) , r ∈ (0 , r ] . (4.10) Proof.
We argue by contradiction. For given c >
0, choose r := r ( c ) so small that Proposition3.1 can be applied, so that Γ ( r α V r,ξ + F r,ξ ) ≤ r α c implies λ N, , ( r α V r,ξ + F r,ξ ) ≤ C ( D , ) Γ ( r α V r,ξ + F r,ξ ) . As a consequence, if Γ ( r α V r,ξ + F r,ξ ) ≤ r α c for r ∈ (0 , r ], then λ N,r,ξ ( V + F ) = r − α λ N, , ( r α V r,ξ + F r,ξ ) ≤ r − α C ( D , ) Γ ( r α V r,ξ + F r,ξ ) ≤ cC ( D , ) . (4.11)However, we cannot have the bound (4.11) for large enough | ξ | in view of Lemma 4.2 and Lemma4.3. The proof is complete.Now, we can finish the proof of Theorem 1.2. Proof of Theorem 1.2 : It is clear that (4.10) implies (4.1). Hence we can conclude fromProposition 4.1 and Proposition 4.4 that the assertions in Theorem 1.2 are equivalent.
Remark 4.5.
It is well known that the compactness of a bounded semigroup of linear operators isequivalent to the discreteness of the spectrum of the corresponding generator. Therefore we see thatthe assertions ((i)) and ((ii)) in Theorem 1.2 are equivalent to the compactness of the followingnon-local Feynman-Kac semigroup p V + Ft on L ( R d ) p V + Ft f ( x ) = E x h e − R t V ( X s )d s − P
Let V ∈ S D ( X ) (resp., or F ∈ J D ( X ) ). Assume that for any c > there exists r = r ( c ) ∈ (0 , and lim | ξ |→∞ |{ V ≤ c } ∩ D r,ξ | = 0 (cid:18) resp., or lim | ξ |→∞ |{ F ≤ c } ∩ D r,ξ | = 0 (cid:19) . (4.12) Then σ ( L V + F ) = σ d ( L V + F ) .Proof. First we prove the assertion for V . In view of the assumption (4.12), it follows that for any c > r = r ( c ) ∈ (0 ,
1] and R = R ( c ) > | ξ | ≥ R |{ V > c } ∩ D r,ξ | ≥ | D r,ξ | = 12 | D r, | . (4.13) 21ence, we have by (4.13) Γ ( r α V r,ξ + F r,ξ ) ≥ Γ ( r α V r,ξ )= r α Z D , E x h e − r α R ∞ V r,ξ ( X s )d s i V r,ξ ( x )d x = r α − Z D r,ξ E r − ( x − ξ ) h e − r α R ∞ V r,ξ ( X s )d s i V ( x )d x ≥ r α − Z D r,ξ e − r α sup y ∈ D , E y [ R ∞ V r,ξ ( X s )d s ] V ( x )d x ≥ r α − Z { V >c }∩ D r,ξ e − r α sup y ∈ Dr,ξ E y [ R ∞ V ( X s )d s ] V ( x )d x ≥ r α − e − r α ℓ Z { V >c }∩ D r,ξ V ( x )d x ≥ r α − e − r α ℓ c | D r, | which implies Theorem 1.2((i)). The proof of the assertion for F can be deduced in a similar way,by using the fact that F r,ξ r − ( x − ξ )) = r d + α − F x ) on D r,ξ .The condition (4.12) means that the sublevel set { V ≤ c } (resp. { F ≤ c } ) is to be thin atinfinity. More results for compactness of Schr¨odinger operators based on the concept of thin atinfinity can be found in [14, 23] (see also [16, 26] as special cases). Acknowledgement.
The authors would like to thank the referee for valuable comments andsuggestions. The authors also thank to Professor Kazuhiro Kuwae for his helpful comments.
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