Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval
SSPECTRAL GAP LOWER BOUND FOR THE ONE-DIMENSIONALFRACTIONAL SCHR ¨ODINGER OPERATOR IN THE INTERVAL
KAMIL KALETA
Abstract.
We prove the uniform lower bound for the difference λ − λ between first two eigen-values of the fractional Schr¨odinger operator ( − ∆) α/ + V , which is related to the Feynman-Kacsemigroup of the symmetric α -stable process killed upon leaving open interval ( a, b ) ∈ R with sym-metric differentiable single-well potential V in the interval ( a, b ), α ∈ (1 , λ − λ ≥ C α ( b − a ) − α is independent of the po-tential V . In general case of α ∈ (0 , λ ∗ − λ ,where λ ∗ denotes the smallest eigenvalue related to the antisymmetric eigenfunction ϕ ∗ . We discusssome properties of the corresponding ground state eigenfunction ϕ . In particular, we show thatit is symmetric and unimodal in the interval ( a, b ). One of our key argument used in proving thespectral gap lower bound is some integral inequality which is known to be a consequence of theGarsia-Rodemich-Rumsey-Lemma. Introduction and statement of results
The main purpose of this paper is to prove an uniform lower bound for the spectral gap of thefractional Schr¨odinger operator with symmetric differentiable single-well potential on a boundedinterval of the real line. Such an operator is related to the Feynman-Kac semigroup of the killedsymmetric α -stable process. To obtain this bound we study some basic properties of the first andsecond eigenfunction of this operator such as monotonicity and differentiability. Another main argu-ment used in proving our spectral gap lower bound is some integral inequality which has importantconsequences in the embedding theory of Sobolev spaces of fractional order. This inequality is aconsequence of the Garsia-Rodemich-Rumsey-Lemma (abbreviated as GRR-Lemma) [34]. The factthat this inequality follows from GRR-Lemma was observed by M. Kassmann in [39].Our work is motivated by the classical results of M. Ashbaugh and R. Benguria obtained in[3, 4], where the similar spectral problem was studied for the classical Schr¨odinger operator withthe symmetric single-well potential on the interval.Before we describe our results in details let us recall the basic definitions and facts. Let ( X t ) t ≥ be the symmetric α -stable process of order α ∈ (0 ,
2) in R . This process is a Markov process withstationary independent increments and the characteristic function of the form E [exp ( iξX t )] =exp( − t | ξ | α ), ξ ∈ R , t >
0. As usual, E x denotes the expected value of the process starting at x ∈ R .Let ( a, b ) ⊂ R , −∞ < a < b < ∞ , be an open interval and let τ ( a,b ) = inf { t ≥ X t / ∈ ( a, b ) } bethe first exit time of X t from ( a, b ).The Feynman-Kac semigroup ( T t ) t ≥ for the symmetric α -stable process X t killed upon leaving( a, b ) and for potential V ∈ V α (( a, b )) is defined as T t f ( x ) = E x (cid:20) exp (cid:18) − (cid:90) t V ( X s ) ds (cid:19) f ( X t ); τ ( a,b ) > t (cid:21) , f ∈ L (( a, b )) , t > , x ∈ ( a, b ) , (1)where V α (( a, b )) is a class of functions V : ( a, b ) → R specified by the following three conditions:(i) integrability: V extended to R by putting 0 outside ( a, b ) is in the Kato class K α for thesymmetric α -stable process X t . Formal definition of K α is given in Section 2.(ii) symmetry: V ( x ) = V ( b + a − x ) for x ∈ ( a, b ). Research supported by the Polish Ministry of Science and Higher Education grant no. N N201 527338. a r X i v : . [ m a t h . P R ] M a y KAMIL KALETA (iii) differentiability and monotonicity: V (cid:48) exists in ( a, b ) and V (cid:48) ( x ) ≤ x ∈ ( a, ( a + b ) / V are defined on the interval ( a, b ).However, very often, it will be useful to see the potential V as a function extended to whole real line R by putting 0 outside ( a, b ). Notice also that the assumption (i) is an integrability condition underwhich the above Feynman-Kac semigroup is well defined (see [14, 15]). Moreover, it immediatelyfollows from the assumptions (ii) and (iii) that V is a symmetric function, which is continuous,bounded from below, nonincreasing in ( a, ( a + b ) /
2) and nondecreasing in (( a + b ) / , b ). In theterminology of [3] potentials with a such monotonicity property are called symmetric single-well. Werefer to the potentials from the class V α (( a, b )) as the symmetric differentiable single-well potentials on the interval ( a, b ).The operators T t are symmetric and form a strongly continuous semigroup on L (( a, b )). Theinfinitesimal generator L of the semigroup ( T t ) t ≥ is defined formally by Lf = lim t ↓ T t f − ft for such f ∈ L (( a, b )) for which this limit exists in L (( a, b )). The set of all such functions is calledthe domain of L and is denoted by D ( L ). Similarly, we define Lf ( x ) = lim t ↓ ( T t f ( x ) − f ( x )) /t forany f ∈ C (( a, b )) and x ∈ ( a, b ) for which the limit exists. It is easy to show that if f ∈ C ∞ c (( a, b )),then Lf ( x ) is well defined and Lf ( x ) = − ( − ∆) α/ f ( x ) − V ( x ) f ( x ) , x ∈ ( a, b ) , where − ( − ∆) α/ is the fractional Laplacian of order α (see [16, p. 11-13]). Note that the operator( − ∆) α/ will not be used in any essential way in proving our eigenvalue gap estimates. We justwant to emphasize the connection between the operators L and ( − ∆) α/ . By this correspondence,we refer to the operator − L as the fractional Schr¨odinger operator with the potential V on interval( a, b ).In recent years Schr¨odinger operators based on non-local pseudodifferential operators have beenintensively studied. One of the most well known result is the so-called Hardy-Lieb-Thirring in-equality obtained in 2008 by R. Frank, E. Lieb and R. Seiringer [36], which is connected with theproblem of the stability of matter [47]. In the last 30 years many results concerning the fractionalSchr¨odinger operators and relativistic Schr¨odinger operators have been obtained [20, 35, 53, 22, 23,24, 14, 15, 43, 37, 38, 49]. These results are about functional integration, structure of spectrum,conditional gauge theorem, estimates of eigenfunctions and intrinsic ultracontractivity. Most ofthese results are obtained by using the probabilistic and potential theoretic methods.The fact that the interval ( a, b ) is bounded implies that for any t > T t maps L (( a, b )) into L ∞ (( a, b )). It follows from the general theory of semigroups that there exists anorthonormal basis of eigenfunctions { ϕ n } in L (( a, b )) and corresponding sequence of eigenvalues λ < λ ≤ λ ≤ ... → ∞ satisfying T t ϕ n = e − λ n t ϕ n ,Lϕ n = − λ n ϕ n , n ≥ . We may and do choose the basis { ϕ n } so that ϕ n is either symmetric (i.e., ϕ n ( x ) = ϕ n ( a + b − x )for x ∈ ( a, b )) or antisymmetric (i.e., ϕ n ( x ) = − ϕ n ( a + b − x ) for x ∈ ( a, b )). Moreover, eacheigenfunction ϕ n is continuous and bounded and all λ n have finite multiplicities. Additionally, λ is simple and the corresponding eigenfunction, the so-called ground state eigenfunction, canbe assumed to be strictly positive on ( a, b ). Our main concern in this paper is the difference λ − λ >
0, which is called the spectral gap. All above defined objects depend on the stability
PECTRAL GAP FOR FRACTIONAL SCHR ¨ODINGER OPERATORS 3 parameter α ∈ (0 , a, b ) and the potential V ∈ V α (( a, b )). However, for simplicity,we prefer to omit this dependence in our notation.Let us point out that ( − ∆) α/ ϕ n ( x ) is well defined for all x ∈ ( a, b ), n ≥
1, and we have − Lϕ n ( x ) = ( − ∆) α/ ϕ n ( x ) + V ( x ) ϕ n ( x ) = λ n ϕ n ( x ) , x ∈ ( a, b ) . By the fact that each ϕ n may be extended to whole R by putting 0 outside ( a, b ), the spectralproblem discussed in this paper can be seen as the eigenproblem for the fractional Schr¨odingeroperator on the interval ( a, b ) with Dirichlet exterior conditions (that is, outside the interval ( a, b )).Mentioned above spectral problem has been widely studied for classical Schr¨odinger operators − ∆ + V acting on L ( D ) with Dirichlet boundary conditions, where D is a bounded domain in R d , d ≥
1. Motivated by problems in mathematical physics concerning the behaviour of free Boson gases,M. van den Berg [11] made the following conjecture. If D ⊂ R d is convex such that diam( D ) < ∞ and V is nonnegative convex potential in D , then λ V ,D − λ V ,D > π diam( D ) , (2)where λ V ,D and λ V ,D are the first and the second eigenvalue of − ∆ + V acting on L ( D ) withDirichlet boundary conditions. This problem has been widely studied by many authors [50, 52,28, 10, 51, 48, 1]. In particular, the strict inequality (2) was obtained in 2010 by B. Andrews andJ. Clutterbuck [1]. Let us point out that the this conjecture for intervals on the real line and forarbitrary nonnegative convex potentials was proved earlier by R. Lavine [46].The classical result which is the most related to our one was obtained by M. Ashbaugh and R.Benguria [3, 4]. They studied this problem in one dimension when a domain D is just a boundedinterval and showed the inequality (2) for the different class of symmetric single-well potentials V that are integrable in D ⊂ R . This class includes the symmetric convex potentials, as well as avariety of nonconvex (but symmetric) potentials.The problem of eigenvalue estimates and the spectral gap lower bound has also been studiedfor fractional Laplacian − ( − ∆) α/ (i.e. V ≡
0) on bounded domains of R d with Dirichlet exteriorconditions [26, 29, 6, 7, 8, 32, 44, 30]. In one dimension (when D is an interval) eigenvalue gapsestimates follow from results in [5] ( α = 1) and [25] ( α > α = 1), [45] ( α ∈ (0 , L is a rather standard fact. In particular, this formula gives a variationalrepresentation for the spectral gap λ − λ and will be a starting point of our proofs. In fact, itis a fractional extension of the classical variational formula for the eigenvalue gaps of the classicalSchr¨odinger operators which can be found for example in [51]. For the version of this formulafor the fractional Laplacian (i.e. V ≡
0) we refer to [32]. Denote by L (( a, b ) , ϕ ) the space ofsquare-integrable functions on the interval ( a, b ) with measure ϕ ( x ) dx . Proposition 1.
Assume that α ∈ (0 , . Let V ∈ V α (( a, b )) , −∞ < a < b < ∞ . Then for every n ≥ we have λ n − λ = inf f ∈F n A − α (cid:90) ba (cid:90) ba ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy, (3) where F n = (cid:26) f ∈ L (( a, b ) , ϕ ) : (cid:90) ba f ( x ) ϕ ( x ) dx = 1 , (cid:90) ba f ( x ) ϕ ( x ) ϕ i ( x ) dx = 0 , ≤ i ≤ n − (cid:27) KAMIL KALETA and A γ = Γ((1 − γ ) / γ √ π | Γ( γ/ | . (4) Moreover, the infimum in (3) is achieved for f = ϕ n /ϕ . Proposition 1 is a consequence of the standard variational formula for eigenvalues and a special caseof [21, Theorems 2.6 and 2.8]. Its proof is given at the end of Section 2.Our first main results are the following theorems concerning the properties of eigenfunctions.The first one is about the monotonicity of ϕ . We extend some earlier ideas from [7, 9] and [10],and prove that the ground state eigenfunction is unimodal in the interval ( a, b ). This property willbe our main argument in proving eigenvalue gap estimates. Theorem 1.
Let α ∈ (0 , . Let V ∈ V α (( a, b )) , −∞ < a < b < ∞ . Then ϕ is symmetric andunimodal in ( a, b ) , i.e., ϕ is nondecreasing in ( a, ( a + b ) / and nonincreasing in (( a + b ) / , b ) . Theorem 2.
Let α ∈ (1 , . Let V ∈ V α (( a, b )) , −∞ < a < b < ∞ . Then for all n ≥ thederivative of ϕ n exists in ( a, b ) . Moreover, if [ c, d ] ⊂ ( a, b ) , then there exists a constant C V,n,α,a,b,c,d such that for all x ∈ [ c, d ] we have (cid:12)(cid:12)(cid:12)(cid:12) ddx ϕ n ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C V,n,α,a,b,c,d (cid:107) ϕ n (cid:107) ∞ . Let us recall that our orthonormal basis { ϕ n } is chosen so that each ϕ n is either symmetric orantisymmetric in ( a, b ). It follows from the fact that { ϕ n } is an orthonormal basis that among ϕ n there are infinitely many antisymmetric functions in ( a, b ). Denote by λ ∗ the smallest eigenvaluecorresponding to the antisymmetric eigenfunction ϕ ∗ . It is a natural hypothesis that λ ∗ = λ . Forthe classical Schr¨odinger operator on the the interval this fact is well known. It is a consequenceof the Courant-Hilbert theorem which states that ϕ has exactly two nodal domains (the intervalconsists of exactly two subintervals on which the sign of ϕ is fixed). In our case this problem ismore complicated. This is due to the fact that no version of the Courant-Hilbert theorem is knownfor operators which are non-local. Despite this fact, this hypothesis was proved by R. Ba˜nuelos andT. Kulczycki [6] for α = 1 and V ≡
0. Very recently, M. Kwa´snicki proved this hypothesis in [45]for α ∈ (1 ,
2) and V ≡ λ ∗ − λ is derived by using Theorem 1 only. Theorem 3.
Assume that α ∈ (0 , . Let V ∈ V α (( a, b )) , −∞ < a < b < ∞ . Let λ ∗ be the smallesteigenvalue corresponding to the antisymmetric eigenfunction ϕ ∗ . Then we have λ ∗ − λ ≥ A − α ( b − a ) α . (5)The next theorem is the main result of this paper. Theorem 4.
Assume that α ∈ (1 , . Let V ∈ V α (( a, b )) , −∞ < a < b < ∞ . Then we have λ − λ ≥ C (3) α ( b − a ) α , (6) where C (3) α = A − α (cid:18)(cid:16) α − α +1) (cid:17) ∨ (cid:16) α − α +1) (cid:17) α +1 ∨ (cid:0) (cid:1) α +1 α − (cid:19) . Since we do not know whether the second eigenfunction is antisymmetric, we also have to considerthe case that it is symmetric in the proof of Theorem 4. Our crucial argument applied to the proofin this case is the following integral inequality which is known to have important consequences inthe embedding theory of Sobolev spaces of fractional order.
PECTRAL GAP FOR FRACTIONAL SCHR ¨ODINGER OPERATORS 5
Proposition 2.
Assume that α ∈ (1 , . Let −∞ < a < b < ∞ and let f be a Lipschitz functionin [ a, b ] . Then (cid:90) ba (cid:90) ba ( f ( x ) − f ( y )) | x − y | α dxdy ≥ C (4) α ( f ( b ) − f ( a )) ( b − a ) α − (7) with C (4) α = (cid:16) α − α +1) (cid:17) ∨ (cid:16) α − α +1) (cid:17) α +1 ∨ (cid:0) (cid:1) α +1 α − . In [39] M. Kassmann showed that the inequalities of the form (7) are the direct corollary fromthe GRR-type inequalities. The constant (( α − / (16( α + 1))) is derived from the classical one-dimensional GRR-lemma [34, Lemma 1.1], while the constant (( α − / (12( α + 1))) α +1 is a con-sequence of the Kassmann’s proof [39, Proof of Theorem 3]. To obtain the inequality with thisconstant he used the multidimensional extension of GRR-Lemma by L. Arnold and P. Imkeller[2]. Note that there are several improvements of the GRR-lemma in various directions. However,according to the author’s knowledge these two constants seem to be the best constants known upto now in the one-dimensional inequality (7). Since our main concern here is to obtain possibly thebest constant in the inequality (7), we propose a completely different proof of the inequality (7)based on some inductive argument. This direct and much more self-contained procedure allows usto obtain the constant 3 − α +1) / ( α − which is better for α near 2 (approximately for α > . C (4) α in Section 5 (see Remark 2 and Figures 1 and 2). Remark 1.
Let 1 < β < p < ∞ . Let −∞ < a < b < ∞ and let f be a continuous function in [ a, b ].More general inequalities of the form (cid:90) ba (cid:90) ba ( f ( x ) − f ( y )) p | x − y | β dxdy ≥ C (5) β,p ( f ( b ) − f ( a )) p ( b − a ) β − (8)and their multidimensional versions are known to be the direct corollaries from the GRR-typeinequalities. Their direct consequence is that if for some function f the double integral on the lefthand side is finite, then the function f is H¨older continuous with exponent ( β − /p . As shown in[39], these inequalities directly imply the embedding theorems from the so-called Sobolev-Slobodeckispaces into the spaces of H¨older continuous functions with exponent smaller or equal to ( β − /p (for more details see [39] and references therein). Simple modification of the inductive part of ourproof of Proposition 2 also gives the inequality (8) with constant C (5) β,p = 3 − p ( β +1) / ( β − for Lipschitzcontinuous functions.The following counterexample shows that in the case α ∈ (0 ,
1) the inequality (7) does not holdwith any positive constant. This is a reason why our method of the proof of bound (6) does notwork in this case. The case α = 1 remains open also. Example 1.
Assume that α ∈ (0 , . Let f be a C ∞ -class function such that f ( x ) = , x < / , ∈ [0 , , / ≤ x < / , , x ≥ / . (9) Consider the sequence of functions of the form f n ( x ) = f ( nx ) , n ≥ . Clearly, each f n is a C ∞ -classfunction such that f n (0) = 0 , f n (1) = 1 . However, (cid:90) (cid:90) ( f n ( x ) − f n ( y )) | x − y | α dxdy → as n → ∞ . (10)The justification of the above example is a very special version of similar one in [31, Section 2]and is given in Section 5.Note that the constants C (3) α and C (4) α obtained in Theorem 4 and Proposition 2, respectively, arenot optimal. As we will see below, the inequality (7) is an important argument used in proving the KAMIL KALETA bound (6). Indeed, we have C (3) α = A − α / C (4) α . It follows that by improving the constant C (4) α inthe inequality (7), one can improve the constant C (3) α in (6). Notice also that in view of Theorem 3another way to improve the constant in Theorem 4 is to show that λ ∗ = λ .A consequence of Theorem 2 is the following fractional version of the weighted Poincar´e inequality. Corollary 1.
Assume that α ∈ (1 , . Let −∞ < a < b < ∞ and let g : [ a, b ] → R be continuous,nonincreasing and strictly positive in [ a, b ) . Let f be a Lipschitz function on the interval [ a, b ] suchthat f ( a ) = 0 . Then (cid:90) ba (cid:90) ba ( f ( x ) − f ( y )) | x − y | α g ( x ) g ( y ) dxdy ≥ C (4) α ( b − a ) α (cid:90) ba f ( x ) g ( x ) dx, (11) where C (4) α is given in Theorem 2. The paper is organized as follows. In Section 2 we introduce additional notation and collectvarious facts which are used below. In particular, here we justify the variational formulas foreigenvalue gaps. In Section 3 we prove the properties of eigenfunctions. Section 4 contains theproof of Theorem 3. In Section 5 we show the crucial inequality in Proposition 2 and justifyExample 1. Here we also discuss the structure of the constant C (4) α in Proposition 2. Section 6contains the proof of main theorem concerning the spectral gap lower bound.2. Preliminaries
Let α ∈ (0 , C α,κ we always mean a strictly positive and finite constant depending on α and parameter κ . We adopt the convention that constants in some proofs may change their valuefrom one use to another. However, very often, especially in the statements of our results, we write C (1) κ , C (2) κ etc to distinguish between constants.We now summarize the properties of the symmetric α -stable process and some facts from itspotential theory. For further information on the potential theory of stable processes we refer to[40, 23, 16, 19].Let X = ( X t ) t ≥ be the standard one-dimensional symmetric α -stable process with the L´evymeasure ν ( dx ) = A − α | x | − − α dx , where the constant A − α is given by (4). By P x we denote thedistribution of the process starting at x ∈ R . For each fixed t > p ( t, y − x ), t > x, y ∈ R , of the process X is a continuous and bounded function on R × R satisfying thefollowing estimates C − α (cid:18) t | y − x | α ∧ t − /α (cid:19) ≤ p ( t, y − x ) ≤ C α (cid:18) t | y − x | α ∧ t − /α (cid:19) . (12)Denote by q ( t, y − x ) = 1 √ πt exp (cid:18) − | x − y | t (cid:19) , t > , x, y ∈ R , (13)the transition density of the standard one dimensional Brownian motion ( B t ) t ≥ runing at twicethe speed.It is well known that the symmetric α -stable process ( X t ) t ≥ can be represented as X t = B η t , where ( η t ) t ≥ is an α/ B t ) t ≥ (see [16]). Thus p ( t, y − x ) = (cid:90) ∞ q ( s, y − x ) g α/ ( t, s ) ds, (14)where g α/ ( t, s ) is the transition density of η t . PECTRAL GAP FOR FRACTIONAL SCHR ¨ODINGER OPERATORS 7
It is known that when α <
1, the process X is transient with potential kernel [12] K ( α ) ( y − x ) = (cid:90) ∞ p ( t, y − x ) dt = A α | y − x | α − , x, y ∈ R . Whenever α ≥ α > α ≥ K ( α ) ( y − x ) = (cid:90) ∞ ( p ( t, y − x ) − p ( t, x )) dt, where x = 0 for α >
1, and x = 1 for α = 1. In this case K ( α ) ( x ) = 1 π log 1 | x | for α = 1 and K ( α ) ( x ) = (2Γ( α ) cos( πα/ − | x | α − , x ∈ R , for α >
1. Note that K ( α ) ( x ) ≤ α > V : R → R belongs to the Kato class K α corresponding to thesymmetric α -stable process X if V satisfies either of the two equivalent conditions (see [53] and [15,(2.5)]) lim (cid:15) → sup x ∈ R (cid:90) | y − x | <(cid:15) | V ( y ) || K ( α ) ( y − x ) | dy = 0 , lim t → sup x ∈ R E x (cid:20)(cid:90) t | V ( X s ) | ds (cid:21) = 0 . For instance, if V ( x ) = (1 − x ) − β , β >
0, then V ∈ K α and V ∈ V α (( − , β < α ∧
1. It can be verified directly that for every α ∈ (0 , K α ⊂ L ( R ).We denote by p D ( t, x, y ) the transition density of the process killed upon exiting an open boundedset D ⊂ R . It satisfies the relation p D ( t, x, y ) = p ( t, y − x ) − E x [ p ( t − τ D , y − X τ D ); τ D ≤ t ] , x, y ∈ D, t > , where τ D = inf { t > X t / ∈ D } is the first exit time from D . For every t > x, y ∈ D , we have0 < p D ( t, x, y ) ≤ p ( t, y − x ) . (15)Let us recall that the semigroup of the process killed upon leaving set D is given by P Dt f ( x ) = E x [ f ( X t ); τ D > t ] = (cid:90) D f ( y ) p D ( t, x, y ) dy, f ∈ L ( D ) , x ∈ D. The Green operator of an open bounded set D is denoted by G D . We set G D ( x, y ) = (cid:90) ∞ p D ( t, x, y ) dt and call G D ( x, y ) the Green function for D . We have G D f ( x ) = E x (cid:20)(cid:90) τ D f ( X t ) dt (cid:21) = (cid:90) D G D ( x, y ) f ( y ) dy, for non-negative Borel function f on R .We now discuss some selected properties of the Feynman-Kac semigroup for the fractionalSchr¨odinger operator with a potential V on bounded intervals of R , which are needed below. Forthe rest of this section we assume that D is a bounded interval ( a, b ) ⊂ R , a < b , and V ∈ V α (( a, b )).We refer the reader to [14, 15, 22, 23, 27] for more systematic treatment of fractional Schr¨odingeroperators. KAMIL KALETA
The V -Green operator for ( a, b ) is defined by G V ( a,b ) f ( x ) = (cid:90) ∞ T t f ( x ) dt = E x (cid:20)(cid:90) τ ( a,b ) e − (cid:82) t V ( X s ) ds f ( X t ) dt (cid:21) , for non-negative Borel functions f on ( a, b ). The corresponding gauge function is given by (see e.g.[14, p. 58], [23, 27]) u ( a,b ) ( x ) = E x (cid:20) e − (cid:82) τ ( a,b )0 V ( X s ) ds (cid:21) , x ∈ ( a, b ) . When it is bounded in ( a, b ), then (( a, b ) , V ) is said to be gaugeable. It is easy to check that if V ≥ a, b ), then gaugebility holds.The following perturbation type formula will be an important argument in the proof of differ-entiability of eigenfunctions. For the potential V such that (( a, b ) , V ) is gaugable and for boundedfunction f we have (see [14, Formula 9]) G V ( a,b ) f ( x ) = G ( a,b ) f ( x ) − G ( a,b ) ( V G V ( a,b ) f )( x ) , x ∈ ( a, b ) . (16)Since V ∈ V α (( a, b )), inf V > −∞ . Observe that when V ≡
0, then ( T t ) t ≥ is an usual semigroup ofthe symmetric stable process killed upon leaving ( a, b ). That is for each t ≥ T t = P ( a,b ) t .Generally, for f ≥ T t f ( x ) ≤ e − inf V t P ( a,b ) t f ( x ) , t > , x ∈ ( a, b ) . Using this fact, (12), (15), and the Riesz-Thorin interpolation theorem, it can be shown that foreach t > T t : L p (( a, b )) → L q (( a, b )), 1 ≤ p ≤ q ≤ ∞ , are bounded. Denote by (cid:107) T t (cid:107) p,q the norm of a such operators.Recall that the class V α (( a, b )) contains the signed potentials V . So we do not exclude the casethat the operators T t are not sub-Markovian. However, each operator T t can be transformed to besub-Markovian by adding a constant to the potential V . Indeed, if inf V <
0, then we put V = V − inf V . Clearly, V ≥
0, and thus, the corresponding Feynman-Kac semigroup is sub-Markovian.Denote this semigroup by ( T t ) t ≥ . It can be checked directly that its generator is the operator L = L +inf V with purely discrete spectrum of the form {− λ + inf V, − λ + inf V, − λ + inf V, ... } .However, both operators L and L have the same eigenvalue gaps λ n − λ , n ≥
1. We will use thistranslation invariance property of the eigenvalue gaps in the sequel.We now justify the variational formula in Proposition 1.
Proof of Proposition 1.
By using the remark on the translation invariance of the eigenvalue gaps,we may and do assume that V ≥ T t ) t ≥ issub-Markovian. For every t > (cid:101) T t f = e λ t ϕ − T t ( ϕ f ) , f ∈ L (( a, b ) , ϕ ) . It is easy to see that operators (cid:101) T t , t >
0, form a semigroup of symmetric Markov operators on L (( a, b ) , ϕ ) such that (cid:101) T t (cid:18) ϕ n ϕ (cid:19) = e − ( λ n − λ ) t ϕ n ϕ , n ≥ . Let (cid:101) E ( f, f ) = lim t → + t ( f − (cid:101) T t f, f ) L (( a,b ) ,ϕ ) (17)for f ∈ L (( a, b ) , ϕ ). It is known that the form (cid:101) E with its natural domain D ( (cid:101) E ) = (cid:110) f ∈ L (( a, b ) , ϕ ) : (cid:101) E ( f, f ) < ∞ (cid:111) PECTRAL GAP FOR FRACTIONAL SCHR ¨ODINGER OPERATORS 9 is the Dirichlet form corresponding to the semigroup ( (cid:101) T t ) t> [33, p. 23]. By the standard variationalformula for eigenvalues we have λ n − λ = inf f ∈F n (cid:101) E ( f, f ) , n ≥ , and the infimum is achieved for f = ϕ n /ϕ . Thus to complete the proof of Proposition 1 it isenough to see that (cid:101) E ( f, f ) = A − α (cid:90) ba (cid:90) ba ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy. (18)However, the equality (18) is a special case of more general results in [21, Theorems 2.6 and 2.8]. (cid:3) Properties of eigenfunctions
For our convenience we work with the symmetric interval ( − a, a ), 0 < a < ∞ , in the proof ofTheorem 1. First we need the following auxiliary lemmas. Let us note that our arguments in theproofs of Lemma 1, Lemma 2 and Theorem 1 extend some earlier ideas from [7, 9] and [10]. Lemma 1.
Assume that α ∈ (0 , . Let < a < ∞ and let V be a bounded, symmetric andcontinuous function on ( − a, a ) such that V (cid:48) ( x ) exists in ( − a, a ) except for at most finite points and V is nonincreasing in ( − a, . Let n be a natural number and let s i , t i , i = 1 , ..., n , be arbitrarypositive parameters. For x ∈ [ − a, a ] we define Φ n ( x ; s , ..., s n , t ,..., t n ) = (cid:90) a − a ... (cid:90) a − a exp (cid:32) − n (cid:88) i =1 t i V ( x i ) (cid:33) n (cid:89) i =1 q ( s i , x i − x i − ) dx ...dx n , where x = x and q is given by (13) . Then for every n ≥ and parameters s i , t i , i = 1 , ..., n , Φ n ( x ; s , ..., s n , t , ..., t n ) is nondecreasing on ( − a, and nonincreasing on (0 , a ) as a function of x .Proof. First note that for every n ≥ s i , t i , i = 1 , ..., n , Φ n ( x ; s , ..., s n , t , ..., t n )is symmetric and positive on [ − a, a ]. Without loss of generality we assume that a = 1. We use theinduction. Integrating by parts and using the fact that V is symmetric, we have √ πs ddx Φ ( x ; s , t ) = − (cid:90) − e − t V ( y ) ∂∂y e − ( y − x )24 s dy = e − (1+ x )24 s lim y →− e − t V ( y ) − e − (1 − x )24 s lim y → e − t V ( y ) + (cid:90) − e − ( y − x )24 s ddy e − t V ( y ) dy = lim y → e − t V ( y ) (cid:18) e − (1+ x )24 s − e − (1 − x )24 s (cid:19) + (cid:90) (cid:18) e − ( y − x )24 s − e − ( y + x )24 s (cid:19) ddy e − t V ( y ) dy. (19)Since e − ( y − x )24 s ≥ e − ( y + x )24 s , x, y, s > , (20)and ddy e − tV ( y ) = − te − tV ( y ) V (cid:48) ( y ) ≤ , t > , for almost all y ∈ (0 , x ∈ (0 , s , t > ( x ; s , t ) is nonincreasing on (0 , s , ...s n − , t , ..., t n − the function Φ n − is non-increasing on (0 , ddx Φ n − ( x ; s , ..., s n − , t , ..., t n − ) ≤ x ∈ (0 , ddx Φ n ( x ; s , ..., s n , t , ..., t n ) ≤ x ∈ (0 ,
1) and for every positive parameters s , ...s n , t , ..., t n .We observe thatΦ n ( x ; s , ..., s n , t , ..., t n ) = 1 √ πs (cid:90) − e − ( y − x )24 s e − t V ( y ) Φ n − ( y ; s , ..., s n , t , ..., t n ) dy . From now on, for more simplicity, we omit the parameters s i , t i in our notation. Integrating byparts, similarly as before, we have √ πs ddx Φ n ( x ) = − (cid:90) − ∂∂y (cid:18) e − ( y − x )24 s (cid:19) e − t V ( y ) Φ n − ( y ) dy = Φ n − (1) lim y → e − t V ( y ) (cid:18) e − (1+ x )24 s − e − (1 − x )24 s (cid:19) + (cid:90) − e − ( y − x )24 s ddy (cid:16) e − t V ( y ) Φ n − ( y ) (cid:17) dy. (21)Observe that by (20) the first term on the right hand side of (21) is nonpositive. Thus it is enoughto show that for all x ∈ (0 ,
1) the last integral also is nonpositive. Denote h ( y ) = e − t V ( y ) Φ n − ( y ).Recall that V (cid:48) ( y ) ≥ y ∈ (0 , h (cid:48) ( y ) = ddy (cid:16) e − t V ( y ) Φ n − ( y ) (cid:17) = − t V (cid:48) ( y ) e − t V ( y ) Φ n − ( y ) + e − t V ( y ) ddy Φ n − ( y ) ≤ h (cid:48) ( y ) = − h (cid:48) ( − y ) , for almost all y ∈ (0 , x ∈ (0 , y ∈ (0 ,
1) and t , s > e − ( y − x )24 s h (cid:48) ( y ) + e − ( y + x )24 s h (cid:48) ( − y ) = e − ( y − x )24 s h (cid:48) ( y ) − e − ( y + x )24 s h (cid:48) ( y ) = (cid:18) e − ( y − x )24 s − e − ( y + x )24 s (cid:19) h (cid:48) ( y ) ≤ . Integrating this inequality on [0 , (cid:90) − e − ( y − x )24 s ddy (cid:16) e − t V ( y ) Φ n − ( y ) (cid:17) dy ≤ , for x ∈ (0 , (cid:3) Lemma 2.
Assume that α ∈ (0 , . Let < a < ∞ and V ∈ V α (( − a, a )) . For t > denote g t ( x ) = E x (cid:20) exp (cid:18) − (cid:90) t V ( X s ) ds (cid:19) ; t < τ ( − a,a ) (cid:21) , x ∈ ( − a, a ) . Then for every fixed t > the functions g t ( x ) is nondecreasing in ( − a, and nonincreasing in (0 , a ) .Proof. Let V k ( x ) = V ( x ) ∧ k , x ∈ ( − a, a ), k ≥
1. First note that for every x ∈ ( − a, a ) and t > (cid:90) t V k ( X s ) ds → (cid:90) t V ( X s ) ds as k → ∞ , P x -almost surely, by the monotone convergence theorem. Thus for all x ∈ ( − a, a ) and t > E x (cid:20) exp (cid:18) − (cid:90) t V ( X s ) ds (cid:19) ; t < τ ( − a,a ) (cid:21) = lim k →∞ E x (cid:20) exp (cid:18) − (cid:90) t V k ( X s ) ds (cid:19) ; t < τ ( − a,a ) (cid:21) as a consequence of the fact thatexp (cid:18) − (cid:90) t V k ( X s ) ds (cid:19) ≤ exp( − (inf V ∧ t ) , t > , k ≥ , PECTRAL GAP FOR FRACTIONAL SCHR ¨ODINGER OPERATORS 11 and the bounded convergence theorem. Thus, it suffices to show that for any k ≥ E x (cid:20) exp (cid:18) − (cid:90) t V k ( X s ) ds (cid:19) ; t < τ ( − a,a ) (cid:21) is nondecreasing in ( − a,
0) and nonincreasing in (0 , a ) as a function of variable x , for each fixed t > k ≥
1. By using the fact that P x ( X τ ( − a,a ) ∈ {− a, a } ) = 0 for x ∈ ( − a, a ) and the pathsare c`adl`ag, the boundedness and continuity of V k , and the Markov property of the process ( X t ) t ≥ ,we have E x (cid:20) exp (cid:18) − (cid:90) t V k ( X s ) ds (cid:19) ; t < τ ( − a,a ) (cid:21) = E x (cid:20) exp (cid:18) − (cid:90) t V k ( X s ) ds (cid:19) ; X s ∈ ( − a, a ) , ∀ ≤ s ≤ t (cid:21) = lim n →∞ E x (cid:34) exp (cid:32) − t n n (cid:88) i =1 V k ( X it/ n ) (cid:33) ; X it/ n ∈ ( − a, a ) , i = 1 , , ..., n (cid:35) = lim n →∞ (cid:90) a − a ... (cid:90) a − a n (cid:89) i =1 exp (cid:18) − t n V k ( x i ) (cid:19) p ( t/ n , x i − x i − ) dx ...dx n , where x = x . Moreover, by the subordination formula (14) and Fubini’s theorem, the last multipleintegral can be rewritten as (cid:90) ∞−∞ ... (cid:90) ∞−∞ (cid:32)(cid:90) a − a ... (cid:90) a − a n (cid:89) i =1 exp (cid:18) − t n V k ( x i ) (cid:19) q ( s i , x i − x i − ) dx ...dx n (cid:33) × n (cid:89) i =1 g α/ ( t/ n , s i ) ds ...ds n = (cid:90) ∞−∞ ... (cid:90) ∞−∞ Φ n ( x ; s , ..., s n , t/ n , ..., t/ n ) n (cid:89) i =1 g α/ ( t/ n , s i ) ds ...ds n , (22)where Φ n ( x ; s , ..., s n , t , ..., t n ) := (cid:90) a − a ... (cid:90) a − a n (cid:89) i =1 exp ( − t i V k ( x i )) q ( s i , x i − x i − ) dx ...dx n , with x = x . By Lemma 1 we obtain that for every natural n and for any positive parameters t i , s i , i = 1 , ..., n , the function Φ n is nondecreasing in ( − a,
0) and nonincreasing in (0 , a ). Letting now n → ∞ in (22), we conclude that the same is true for E x (cid:104) exp (cid:16) − (cid:82) t V k ( X s ) ds (cid:17) ; t < τ ( − a,a ) (cid:105) , forany k ≥ t >
0. Thus the proof is complete. (cid:3)
Proof of Theorem 1.
Let 0 < a < ∞ and V ∈ V α (( − a, a )) be fixed. First we show a symmetry.Recall that ϕ is strictly positive in ( − a, a ) and suppose contrary that ϕ is not symmetric. Thus (cid:98) ϕ ( x ) := ϕ ( x )+ ϕ ( − x ) is also an eigenfunction of − L corresponding to the eigenvalue λ such that (cid:98) ϕ / ∈ span( ϕ ). This gives a contradiction, because λ has muliplicity one. Thus ϕ is symmetricin ( − a, a ). Let now P ϕ : L (( − a, a )) → L (( − a, a )) be the projection onto span( ϕ ). By the fact that e λ t T t P ϕ = P ϕ e λ t T t = P ϕ , we have for t > x ∈ ( − a,a ) (cid:12)(cid:12)(cid:12) ( e λ t T t − P ϕ ) ( − a,a ) ( x ) (cid:12)(cid:12)(cid:12) ≤ a (cid:13)(cid:13)(cid:13) e λ t T t − P ϕ (cid:13)(cid:13)(cid:13) , ∞ ≤ ae λ (cid:107) T (cid:107) , ∞ (cid:13)(cid:13)(cid:13) e λ ( t − T t − − P ϕ (cid:13)(cid:13)(cid:13) (cid:107) T (cid:107) , . Since by the spectral theorem (cid:13)(cid:13)(cid:13) e λ t T t − P ϕ (cid:13)(cid:13)(cid:13) ≤ e − ( λ − λ ) t , we obtain ϕ ( x ) = lim t →∞ C V e λ t T t ( − a,a ) ( x ) = lim t →∞ C V e λ t E x (cid:20) exp (cid:18) − (cid:90) t V ( X s ) ds (cid:19) ; t < τ ( − a,a ) (cid:21) , uniformly in x ∈ ( − a, a ), with C V = ( (cid:107) ϕ (cid:107) ) − . Now the assertion of the theorem follows fromLemma 2. (cid:3) We need the following auxiliary lemma. It is a version of [18, Lemma 5.2] and [17, Lemma 10].
Lemma 3.
Let α ∈ (1 , . Let −∞ < a < b < ∞ and let f ∈ L (( a, b )) be a function such that forevery interval [ c, d ] ⊂ ( a, b ) we have sup x ∈ [ c,d ] | f ( x ) | < ∞ . Then ddx G ( a,b ) f ( x ) = (cid:90) ba ∂∂x G ( a,b ) ( x, y ) f ( y ) dy, x ∈ ( a, b ) , (23) and for every interval [ c, d ] ⊂ ( a, b ) there is a constant C α,f,a,b,c,d < ∞ such that (cid:12)(cid:12)(cid:12)(cid:12) ddx G ( a,b ) f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C α,f,a,b,c,d , x ∈ [ c, d ] . (24) Proof.
Recall that α ∈ (1 , G ( a,b ) ( x, y ) = K ( α ) ( x − y ) − H ( x, y ) , x, y ∈ ( a, b ) , x (cid:54) = y, where K ( α ) ( x − y ) = (2Γ( α ) cos( πα/ − | x − y | α − and H ( x, y ) = E x (cid:104) K ( α ) ( X τ ( a,b ) − y ) (cid:105) . In viewof this equality we have ddx G ( a,b ) f ( x ) = lim h → (cid:90) ba K ( α ) ( x + h − y ) − K ( α ) ( x − y ) h f ( y ) dy − lim h → (cid:90) ba H ( x + h, y ) − H ( x, y ) h f ( y ) dy, x ∈ ( a, b ) . First notice that both partial derivatives ∂∂x K ( α ) ( x − y ), x (cid:54) = y , and ∂∂x H ( x, y ) exist (see (10) in[18]). For x ∈ ( a, b ) denote δ ( x ) = ( b − x ) ∧ ( x − a ). From [18, Lemma 3.2] we have (cid:12)(cid:12) ∂∂x H ( x, y ) (cid:12)(cid:12) ≤ C α,a,b δ ( x ) − . It follows that (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x G ( a,b ) ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x K ( α ) ( x − y ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x H ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C α | x − y | α − + C α,a,b δ ( x ) − , x, y ∈ ( a, b ) , x (cid:54) = y. (25)This inequality and properties of f imply that for each fixed x ∈ ( a, b ) the integral on the righthand side of (23) is absolutely convergent. Thus, to obtain (23) it is enough to show thatlim h → (cid:90) ba | F (1) h ( x, y ) || f ( y ) | dy + lim h → (cid:90) ba | F (2) h ( x, y ) || f ( y ) | dy = 0 , x ∈ ( a, b ) , (26) PECTRAL GAP FOR FRACTIONAL SCHR ¨ODINGER OPERATORS 13 where F (1) h ( x, y ) = K ( α ) ( x + h − y ) − K ( α ) ( x − y ) h − ∂∂x K ( α ) ( x − y ) ,F (2) h ( x, y ) = H ( x + h, y ) − H ( x, y ) h − ∂∂x H ( x, y ) . Fix now x ∈ ( a, b ). Let | h | < δ ( x ) /
4. From [18, Lemma 3.2] and Lagrange’s theorem we obtain | F (2) h ( x, y ) | ≤ C α,a,b δ ( x ) − . (27)This estimate and the fact that f ∈ L (( a, b )) give thatlim h → (cid:90) ba | F (2) h ( x, y ) || f ( y ) | dy = 0by the dominated convergence theorem.It suffices to show that lim h → (cid:90) ba | F (1) h ( x, y ) || f ( y ) | dy = 0(28)for each fixed x ∈ ( a, b ). Let β ∈ (0 , /
2) and (cid:90) ba | F (1) h ( x, y ) || f ( y ) | dy = (cid:90) ( x − βδ ( x ) ,x + βδ ( x )) | F (1) h ( x, y ) || f ( y ) | dy + (cid:90) ( a,b ) ∩ ( x − βδ ( x ) ,x + βδ ( x )) c | F (1) h ( x, y ) || f ( y ) | dy. (29)Fix x ∈ ( a, b ) and ε >
0. We will show that for sufficiently small | h | the left hand side of (29) issmaller than ε . Let [ c, d ] ⊂ ( a, b ) be such that x ∈ ( c, d ). Denote M = sup x ∈ [ c,d ] | f ( x ) | . Let β besmall enough so that ( x − βδ ( x ) , x + βδ ( x )) ⊂ [ c, d ]. It is known (see [18, proof of Lemma 5.2]) that | F (1) h ( x, y ) | ≤ C α (cid:0) | x + h − y | α − ∨ | x − y | α − (cid:1) , y ∈ ( a, b ) , y (cid:54) = x, y (cid:54) = x + h. (30)Hence for any h ∈ R (cid:90) x + βδ ( x ) x − βδ ( x ) | F (1) h ( x, y ) || f ( y ) | dy ≤ M C α (cid:90) x + βδ ( x ) x − βδ ( x ) (cid:0) | x + h − y | α − + | x − y | α − (cid:1) dy ≤ M C α (cid:90) βδ ( x ) − βδ ( x ) | y | α − dy. It is clear that there exists β small enough so that the above integral is smaller than ε/
2. Let usfix such β . Clearly, F (1) h ( x, y ) → h → x (cid:54) = y . By (30) | F (1) h ( x, y ) | ≤ C α (2 β − δ ( x ) − ∨ , for y ∈ ( a, b ) ∩ ( x − βδ ( x ) , x + βδ ( x )) c and h ∈ ( − βδ ( x ) / , βδ ( x ) / f ∈ L (( a, b )), thesecond integral on the right hand side of (29) tends to 0 as h tends to 0 by the bounded convergencetheorem. Hence for | h | sufficiently small the second integral on the right hand side of (29) is smallerthan ε/
2. This finishes the proof (28). Thus (23) is proved. The boundedness property (24) is asimple consequence of the estimate (25) and the properties of f . (cid:3) Proof of Theorem 2.
Let −∞ < a < b < ∞ and V ∈ V α (( a, b )). The starting point of the proof arethe eigenequations T t ϕ n = e − λ n t ϕ n , n ≥ . (31) Since we do not exclude the case that V is signed potential, it may happen that λ n < n . Put η = , if inf V > , , if inf V = 0 , − V, if inf V < . Denote V η = V + η . Then V η > a, b ) , V η ) is gaugeable (see p. 7). By (31) we clearly have e − ( λ n + η ) t ϕ n ( x ) = e − ηt T t ϕ n ( x ) = E x (cid:104) e − (cid:82) t V η ( X s ) ds ϕ n ( X t ); τ ( a,b ) > t (cid:105) , x ∈ ( a, b ) . Using the fact that λ n + η > n ≥
1, and integrating over t the above equations we obtain ϕ n ( x ) = ( λ n + η ) G V η ( a,b ) ϕ n ( x ) , x ∈ ( a, b ) , n ≥ . Applying now the perturbation formula (16) to this equality we get ϕ n ( x ) = ( λ n + η ) G ( a,b ) ϕ n ( x ) − G ( a,b ) ( V η ϕ n )( x ) , x ∈ ( a, b ) , n ≥ , which can be rewritten as ϕ n ( x ) = ( λ n + η ) (cid:90) ba G ( a,b ) ( x, y ) ϕ n ( y ) dy − (cid:90) ba G ( a,b ) ( x, y ) V η ( y ) ϕ n ( y ) dy. Since (cid:107) ϕ n (cid:107) ∞ < ∞ , V η ∈ L (( a, b )) and is continuous in ( a, b ), the assumptions of Lemma 3 aresatisfied. Thus for x ∈ ( a, b ) we have ddx ϕ n ( x ) = ( λ n + η ) (cid:90) ba ∂∂x G ( a,b ) ( x, y ) ϕ n ( y ) dy − (cid:90) ba ∂∂x G ( a,b ) ( x, y ) V η ( y ) ϕ n ( y ) dy. A direct consequence of Lemma 3 is that also for any interval [ c, d ] ⊂ ( a, b ) there is a constant C V,α,n,a,b,c,d such that for all x ∈ [ c, d ] we have (cid:12)(cid:12)(cid:12)(cid:12) ddx ϕ n ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C V,α,n,a,b,c,d . (cid:3) Lower bound for λ ∗ − λ Proof of Theorem 3.
Let 0 < a < ∞ . With no loss of generality we provide the arguments for thesymmetric interval ( − a, a ) only. Let V ∈ V α (( − a, a )). Recall that our orthonormal basis { ϕ n } ischosen so that ϕ n are either symmetric or antisymmetric. Let n be the smallest natural numbersuch that ϕ n is antisymmetric in ( − a, a ). Thus ϕ ∗ = ϕ n . Let f = ϕ ∗ /ϕ = ϕ n /ϕ . For every ε ∈ (0 , a ) we have (cid:90) a − a (cid:90) a − a ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy ≥ (cid:90) aε (cid:90) aε ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy + (cid:90) aε (cid:90) − ε − a ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy + (cid:90) − ε − a (cid:90) − ε − a ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy + (cid:90) − ε − a (cid:90) aε ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy. Simple changes of variables in the last three integrals and the fact that f is antisymmetric give thatthe last sum can be transformed to2 (cid:90) aε (cid:90) aε (cid:18) ( f ( x ) − f ( y )) | x − y | α + ( f ( x ) + f ( y )) ( x + y ) α (cid:19) ϕ ( x ) ϕ ( y ) dxdy. PECTRAL GAP FOR FRACTIONAL SCHR ¨ODINGER OPERATORS 15
Clearly, this is bigger or equal to2 (cid:90) aε (cid:90) aε ( f ( x ) − f ( y )) + ( f ( x ) + f ( y )) ( x + y ) α ϕ ( x ) ϕ ( y ) dxdy = 4 (cid:90) aε (cid:90) aε f ( x ) + f ( y )( x + y ) α ϕ ( x ) ϕ ( y ) dxdy, which, by symmetry, is equal to8 (cid:90) aε (cid:90) aε f ( x )( x + y ) α ϕ ( x ) ϕ ( y ) dxdy. Thus, by Theorem 1, we have (cid:90) a − a (cid:90) a − a ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy ≥ (cid:90) aε (cid:90) aε f ( x )( x + y ) α ϕ ( x ) ϕ ( y ) dydx ≥ (cid:90) aε (cid:90) xε f ( x )( x + y ) α ϕ ( x ) ϕ ( y ) dydx ≥ (cid:90) aε (cid:90) xε dy f ( x )(2 x ) α ϕ ( x ) dx ≥ a ) α (cid:90) aε x − εx f ( x ) ϕ ( x ) dx. Now, letting ε →
0, we obtain (cid:90) a − a (cid:90) a − a ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy ≥ a ) α (cid:90) a f ( x ) ϕ ( x ) dx = 2(2 a ) α (cid:90) a − a f ( x ) ϕ ( x ) dx = 2(2 a ) α . Since f = ϕ ∗ /ϕ is antisymmetric, the assertion of Theorem 3 follows simply from Proposition 1. (cid:3) Crucial inequality
Proof of Proposition 2.
For more clarity we divide the proof into the two parts. (Part 1)
In this part we show that in fact it is enough to prove Proposition 2 for a = 0 and b = 1,i.e., for [ a, b ] = [0 , f is a Lipschitz function on an arbitrary interval [ a, b ], −∞ < a < b < ∞ , then the function ˜ f ( x ) := f (( b − a ) x + a ) is a Lipschitz function in [0 ,
1] suchthat ˜ f (0) = f ( a ) and ˜ f (1) = f ( b ). Suppose that the claimed inequality (7) is true for all Lipschitzfunctions on the interval [0 ,
1] with constant C (4) α . Thus, by trivial change of variables, we have (cid:90) ba (cid:90) ba ( f ( x ) − f ( y )) | x − y | α dxdy = (cid:90) ba (cid:90) ba (cid:16) ˜ f (cid:16) x − ab − a (cid:17) − ˜ f (cid:16) y − ab − a (cid:17)(cid:17) | x − y | α dxdy = 1( b − a ) α − (cid:90) (cid:90) ( ˜ f ( x ) − ˜ f ( y )) | x − y | α dxdy. Since (cid:90) (cid:90) ( ˜ f ( x ) − ˜ f ( y )) | x − y | α dxdy ≥ C (4) α ( ˜ f (1) − ˜ f (0)) = C (4) α ( f ( b ) − f ( a )) , the proof of Part 1 is complete. (Part 2) In this part we prove the proposition for a = 0 and b = 1. By Part 1 this will give ourresult for arbitrary −∞ < a < b < ∞ . First note that the constant (( α − / (16( α + 1))) in the inequality (7) for ( a, b ) = (0 ,
1) can be obtained by taking ψ ( x ) = | x | and p ( x ) = | x | ( α +1) / in theclassical GRR-Lemma [34, Lemma 1.1.]. Indeed, from the inequality (1.2) in [34] we clearly get | f (1) − f (0) | ≤ (cid:18)(cid:90) (cid:90) ( f ( x ) − f ( y )) | x − y | α dxdy (cid:19) α + 12 (cid:90) u α − − du ≤ (cid:18) α + 1 α − (cid:19) (cid:18)(cid:90) (cid:90) ( f ( x ) − f ( y )) | x − y | α dxdy (cid:19) . The second constant (( α − / (12( α + 1))) α +1 was derived in a similar manner by M. Kassmann(see [39, Proof of Theorem 3]) from the multidimensional improvement of the GRR-Lemma in [2].We now find the third constant of the form 9 ( − ( α +1) / ( α − in the inequality (7). We use somenew inductive argument, which is completely independent of the GRR-Lemma. If f (0) = f (1),then the claimed inequality is trivial. Without loss of generality we may assume that f (0) = 1 and f (1) >
0. In fact, by the scaling property of the claimed inequality (7), we may and do assume that f (1) = 1. Let c < / (Step 1) Denote (cid:26) a = 0 b = 1 , (cid:26) f a = 0 f b = 1 , (cid:26) x = a + cy = b − c , (cid:26) f x = f a + f y = f b − , and define m = min { x ∈ ( a , b ) : f ( x ) = f x } and M = max { x ∈ ( a , b ) : f ( x ) = f y } . Clearly, f y − f x = 1 /
3. If m / ∈ ( a , x ) and M / ∈ ( y , b ), then we have (cid:90) (cid:90) ( f ( x ) − f ( y )) | x − y | α dxdy ≥ (cid:90) x a (cid:90) b y ( f ( x ) − f ( y )) | x − y | α dxdy ≥ ( f y − f x ) ( b − y )( x − a ) | b − a | α = (cid:16) c (cid:17) . If m ∈ ( a , x ) or M ∈ ( y , b ), we consider the next step. (Step 2) If m ∈ ( a , x ) let us take (cid:26) a = a b = x , (cid:26) f a = f a f b = f x , (cid:26) x = a + c y = b − c , (cid:40) f x = f a + (cid:0) (cid:1) f y = f b − (cid:0) (cid:1) . If m / ∈ ( a , x ) and M ∈ ( y , b ) let us take (cid:26) a = y b = b , (cid:26) f a = f y f b = f b , (cid:26) x = a + c y = b − c , (cid:40) f x = f a + (cid:0) (cid:1) f y = f b − (cid:0) (cid:1) . Define m = min { x ∈ ( a , b ) : f ( x ) = f x } and M = max { x ∈ ( a , b ) : f ( x ) = f y } . Clearly, f y − f x = 1 / , b − a = c . When m / ∈ ( a , x ) and M / ∈ ( y , b ), we have (cid:90) (cid:90) ( f ( x ) − f ( y )) | x − y | α dxdy ≥ (cid:90) x a (cid:90) b y ( f ( x ) − f ( y )) | x − y | α dxdy ≥ ( f y − f x ) ( b − y )( x − a ) | b − a | α = c c α = (cid:16) c (cid:17) c α − . If m ∈ ( a , x ) or M ∈ ( y , b ), we consider the next step. PECTRAL GAP FOR FRACTIONAL SCHR ¨ODINGER OPERATORS 17
Suppose that after n − m n − ∈ ( a n − , x n − ) or M n − ∈ ( y n − , b n − ). Then let us considerthe next step. (Step n) If m n − ∈ ( a n − , x n − ) let us take (cid:26) a n = a n − b n = x n − , (cid:26) f a n = f a n − f b n = f x n − , (cid:26) x n = a n + c n y n = b n − c n , (cid:26) f x n = f a n + (cid:0) (cid:1) n f y n = f b n − (cid:0) (cid:1) n . If m n − / ∈ ( a n − , x n − ) and M n − ∈ ( y n − , b n − ) let us take (cid:26) a n = y n − b n = b n − , (cid:26) f a n = f y n − f b n = f b n − , (cid:26) x n = a n + c n y n = b n − c n , (cid:26) f x n = f a n + (cid:0) (cid:1) n f y n = f b n − (cid:0) (cid:1) n . Define m n = min { x ∈ ( a n , b n ) : f ( x ) = f x n } and M n = max { x ∈ ( a n , b n ) : f ( x ) = f y n } . Of course, f y n − f x n = 1 / n , b n − a n = c n − . If m n / ∈ ( a n , x n ) and M n / ∈ ( y n , b n ), then we have (cid:90) (cid:90) ( f ( x ) − f ( y )) | x − y | α dxdy ≥ (cid:90) x n a n (cid:90) b n y n ( f ( x ) − f ( y )) | x − y | α dxdy ≥ ( f y n − f x n ) ( b n − y n )( x n − a n ) | b n − a n | α = c n n c ( n − α ) = (cid:16) c (cid:17) (cid:18) c α − (cid:19) n − . If m n ∈ ( a n , x n ) or M n ∈ ( y n , b n ), then we consider the n + 1 step.Recalling that α ∈ (1 ,
2) and choosing c to be 9 − α − , we obtain that (cid:18) c α − (cid:19) n = 1 for all n ≥ . It is enough to see that there exists n ≥ m n / ∈ ( a n , x n ) and M n / ∈ ( y n , b n ). Indeed,in this case we have (cid:90) (cid:90) ( f ( x ) − f ( y )) | x − y | α dxdy ≥ (cid:16) c (cid:17) (cid:18) c α − (cid:19) n − = (cid:16) c (cid:17) = (cid:18) (cid:19) α +1 α − . Suppose contrary that for all n ≥ m n ∈ ( a n , x n ) or M n ∈ ( y n , b n ). This means thatthere exists a decreasing sequence of intervals { ( a n , b n ) } n ≥ such that | b n − a n | = (cid:0) (cid:1) n − α − . By thefact that f is a Lipschitz function on [ a, b ], there is a constant C such that (cid:18) (cid:19) n = | f ( M n ) − f ( m n ) | ≤ C | M n − m n | ≤ C | b n − a n | = C (cid:18) (cid:19) n − α − , n ≥ , which gives a contradiction. Thus the inequality (7) is true for a = 0 and b = 1 with constant C (4) α = (cid:0) (cid:1) α +1 α − , and the proof of the proposition is complete. (cid:3) Remark 2.
The three subintervals of the stability parameter α on which the constant C (4) α inProposition 2 takes the three different forms can be estimated as follows: for α ∈ (1; 1 . C (4) α = (cid:16) α − α +1) (cid:17) α +1 , for α ∈ (1 . . C (4) α = (cid:16) α − α +1) (cid:17) , while for α ∈ (1 . C (4) α = (cid:0) (cid:1) α +1 α − (see Figures 1 and 2). Justification of Example 1.
We clearly have (cid:90) (cid:90) ( f n ( x ) − f n ( y )) | x − y | α dxdy ≤ (cid:90) (cid:90) n ( f n ( x ) − f n ( y )) | x − y | α dxdy + (cid:90) n (cid:90) ( f n ( x ) − f n ( y )) | x − y | α dxdy. Figure 1.
Graphs of constants in Proposition 2 for α ∈ (1 , (cid:0) (cid:1) α +1 α − ;(b) (cid:16) α − α +1) (cid:17) ; (c) (cid:16) α − α +1) (cid:17) α +1 . Figure 2.
Graphs of constants in Proposition 2 for α ∈ (1 , . (cid:16) α − α +1) (cid:17) ;(c) (cid:16) α − α +1) (cid:17) α +1 .By symmetry, the right hand side of the above inequality is equal to2 (cid:90) (cid:90) n ( f n ( x ) − f n ( y )) | x − y | α dxdy. PECTRAL GAP FOR FRACTIONAL SCHR ¨ODINGER OPERATORS 19
Denote the last double integral by J n . We have J n ≤ (cid:90) n (cid:90) n ( f n ( x ) − f n ( y )) | x − y | α dxdy + (cid:90) n (cid:90) n ( f n ( x ) − f n ( y )) | x − y | α dxdy = I n, + I n, . Recall that f n ( x ) = f ( nx ), where f is a C ∞ -class function. Observe that for x, y ∈ [0 ,
2] we have | f n ( x ) − f n ( y ) | ≤ sup z ∈ [0 , | f (cid:48) n ( z ) || x − y | = n sup z ∈ [0 , | f (cid:48) ( z ) || x − y | ≤ Cn | x − y | . By this we obtain I n, = (cid:90) n (cid:90) n ( f n ( x ) − f n ( y )) | x − y | α dxdy ≤ C n (cid:90) n (cid:90) y + n y − n | x − y | − α dxdy ≤ C α n α − . Similarly, I n, ≤ (cid:90) n (cid:90) n | x − y | α dxdy ≤ n (cid:90) ∞ n (cid:12)(cid:12)(cid:12)(cid:12) n − y (cid:12)(cid:12)(cid:12)(cid:12) − − α dy = C α n α − , which ends the proof. (cid:3) Proof of Corollary 1.
Let b ∈ ( a, b ] be such that f ( b ) g ( b ) = max x ∈ ( a,b ] f ( x ) g ( x )We have (cid:90) ba (cid:90) ba ( f ( x ) − f ( y )) | x − y | α g ( x ) g ( y ) dxdy ≥ g ( b ) (cid:90) b a (cid:90) b a ( f ( x ) − f ( y )) | x − y | α dxdy, which, by Theorem 2, is larger than C (4) α ( b − a ) α − f ( b ) g ( b ) ≥ C (4) α ( b − a ) α − b − a ) (cid:90) ba f ( x ) g ( x ) dx = C (4) α ( b − a ) α (cid:90) ba f ( x ) g ( x ) dx. (cid:3) Spectral gap estimate
Proof of Theorem 4.
With no loss of generality we provide the arguments for the symmetric interval( − a, a ), 0 < a < ∞ , only. Let V ∈ V α (( − a, a )). Recall that the orthonormal basis { ϕ n } is chosenso that ϕ n are either symmetric or antisymmetric. If ϕ is antisymmetric, then Theorem 4 followsfrom Theorem 3. Assume now that ϕ is symmetric. We directly deduce from Theorem 2 that thefunction ϕ /ϕ has a bounded derivative in each interval [ a , b ], − a < a < b < a . Hence ϕ /ϕ is a Lipschitz function in each interval [ a , b ] ⊂ ( − a, a ). Thus, by Proposition 1, it is enough toestimate from below the double integral (cid:90) a − a (cid:90) a − a ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy, with f = ϕ /ϕ . Note that f is symmetric on ( − a, a ), f changes the sign in ( − a, a ) and (cid:82) a − a f ( x ) ϕ ( x ) dx = 1.Let a = min { x ∈ [0 , a ) : f ( x ) = 0 } . Consider the following two cases. (Case 1) Assume that (cid:90) aa f ( x ) ϕ ( x ) dx ≥ / . We have (cid:90) a − a (cid:90) a − a ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy ≥ (cid:90) aa (cid:90) aa ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy + (cid:90) − a − a (cid:90) − a − a ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy = 2 (cid:90) aa (cid:90) aa ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy. Let now b ∈ [ a , a ) be such that f ( b ) ϕ ( b ) = max x ∈ ( a ,a ) f ( x ) ϕ ( x ). We have (cid:90) aa (cid:90) aa ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy ≥ ϕ ( b ) (cid:90) b a (cid:90) b a ( f ( x ) − f ( y )) | x − y | α dxdy, which, by Theorem 2, is larger than C (4) α ( b − a ) α − f ( b ) ϕ ( b ) ≥ C (4) α ( b − a ) α − a − a ) (cid:90) aa f ( x ) ϕ ( x ) dx ≥ C (4) α ( a − a ) α . It follows that (cid:90) a − a (cid:90) a − a ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy ≥ C (4) α (2 a ) α , which ends the proof in the first case. (Case 2) Suppose now that (cid:90) a f ( x ) ϕ ( x ) dx ≥ / . Notice that (cid:18)(cid:90) aa f ( x ) ϕ ( x ) dx (cid:19) ≤ (cid:90) aa f ( x ) ϕ ( x ) dx (cid:90) aa ϕ ( x ) dx ≤ ϕ ( a )( a − a ) (cid:90) aa f ( x ) ϕ ( x ) dx (32)by Schwarz inequality and Theorem 1, and − (cid:90) aa f ( x ) ϕ ( x ) dx = (cid:90) a f ( x ) ϕ ( x ) dx (33)by the fact that f is symmetric and (cid:82) a − a f ( x ) ϕ ( x ) dx = 0. Observe that without loosing generalitywe may and do assume that f ≥ , a ]. Let a ∗ ∈ [0 , a ) be such that f ( a ∗ ) = max x ∈ [0 ,a ) f ( x ).Note that (cid:82) a f ( x ) ϕ ( x ) dx = 1 /
2. By (32) and (33), we have1 / ≥ (cid:90) aa f ( x ) ϕ ( x ) dx ≥ (cid:16)(cid:82) aa f ( x ) ϕ ( x ) dx (cid:17) ϕ ( a )( a − a ) = (cid:0)(cid:82) a f ( x ) ϕ ( x ) dx (cid:1) ϕ ( a )( a − a )= f ( a ∗ ) (cid:0)(cid:82) a f ( x ) ϕ ( x ) dx (cid:1) f ( a ∗ ) ϕ ( a )( a − a ) ≥ (cid:0)(cid:82) a f ( x ) ϕ ( x ) dx (cid:1) f ( a ∗ ) ϕ ( a )( a − a ) , which implies that f ( a ∗ ) ϕ ( a ) ≥ / (4( a − a )) . (34) PECTRAL GAP FOR FRACTIONAL SCHR ¨ODINGER OPERATORS 21
We have (cid:90) a − a (cid:90) a − a ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy ≥ (cid:90) a (cid:90) a ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy + (cid:90) − a (cid:90) − a ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy = 2 (cid:90) a (cid:90) a ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy ≥ ϕ ( a ) (cid:90) a a ∗ (cid:90) a a ∗ ( f ( x ) − f ( y )) | x − y | α dxdy. Now, using Theorem 2 and (34), we obtain (cid:90) a − a (cid:90) a − a ( f ( x ) − f ( y )) | x − y | α ϕ ( x ) ϕ ( y ) dxdy ≥ ϕ ( a ) C (4) α ( a − a ∗ ) α − f ( a ∗ ) ≥ C (4) α (2 a ) α , which completes the proof. (cid:3) Acknowledgements.
I am deeply indebted to Prof. T. Kulczycki, my supervisor, for his help andguidance in investigating the theory and preparing this paper, which is part of my Ph.D. thesis.Special thanks go to Dr. B. Dyda for his reading of the manuscript, illuminating discussion andpointing me out the papers [34, 39] and the connections between the GRR-Lemma, the inequalities(8) and the embedding theory of Sobolev spaces. Also, I would like to thank Dr. M. Kwa´snicki forhis careful reading of the manuscript and helpful suggestions and comments.
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Kamil Kaleta, Institute of Mathematics and Computer Science, Wroc(cid:32)law University of Technol-ogy, Wyb. Wyspia´nskiego 27, 50-370 Wroc(cid:32)law, Poland
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