Existence of discrete eigenvalues for the Dirichlet Laplacian in a two-dimensional twisted strip
EExistence of discrete eigenvalues for the Dirichlet Laplacian in atwo-dimensional twisted strip
Rafael T. Amorim, Alessandra A. VerriOctober 2, 2020
Abstract
We study the spectrum of the Dirichlet Laplacian operator in a two-dimensional twisted strip embeddedin R d with d ≥
2. It is shown that a local twisting perturbation can create discrete eigenvalues for theoperator. In particular, we also study the case where the twisted effect “grows” at infinity while the widthof the strip goes to zero. In this situation, we find an asymptotic behavior for the eigenvalues.
Let Ω be a strip in R d , d ≥
2, and denote by − ∆ D Ω the Dirichlet Laplacian operator in Ω. A problemextensively studied in the literature is to find spectral information of − ∆ D Ω . If Ω is a bounded strip, it isknown that the spectrum σ ( − ∆ D Ω ) is purely discrete. Otherwise, the existence of discrete eigenvalues is anon-trivial property and it depends on the geometry of Ω [2, 3, 5, 6, 8, 9, 12, 14, 15, 16, 20, 21, 22, 25].Consider the case where Ω is an unbounded strip obtained as a tubular neighborhood of constant widthalong an infinite curve in R . In the pioneering paper [13], the authors demonstrated the existence of discretespectrum for the operator − ∆ D Ω in curved strips. In several subsequent studies, the results were improvedand generalized [8, 17, 23, 28]. We highlight the paper [23] where the authors make an overview of somenew and old results on spectral properties of − ∆ D Ω , including other boundary conditions in ∂ Ω.In [4], the authors introduced a new, two dimensional model of strip to study the spectral problem of − ∆ D Ω . In that work, Ω is a strip in R which is built by translating a segment oriented in a constant directionalong an unbounded curve in the plane. The spectrum of the operator − ∆ D Ω was carefully studied and themodel covers different effects: purely essential spectrum, discrete spectrum or a combination of both.One can consider strips embedded in a Riemannian manifold instead of the Euclidean space. For example,suppose that Ω is a strip of constant width which is defined as a tubular neighborhood of an infinite curvein a two-dimensional Riemannian manifold. This situation was considered in [20]. In particular, the authorproved that the discrete spectrum of − ∆ D Ω is nonempty for non-negatively curved strips. Now, consider thecase where Ω is a two-dimensional, infinite and twisted strip in R . In [22], assuming that the twisted effectdiverges at infinity, the authors studied the spectral problem of − ∆ D Ω . In particular, they proved that thiskind of geometry can create discrete eigenvalues. In a similar situation, in [30] was proved that the discretespectrum of − ∆ D Ω is nonempty since the twisted effect “grows” at infinity while the width of Ω goes to zero.Some results for strips on ruled surfaces can be extended to higher dimensions. Let Ω be a twisted andbent two-dimensional strip embedded in R d with d ≥
3. In [26], the spectrum of the Dirichlet Laplacian − ∆ D Ω was carefully studied. The authors also proved that, in the limit when the width of the strip tendsto zero, the Dirichlet Laplacian converges in the norm resolvent sense to a one-dimensional Schr¨odingeroperator whose potential depends on the deformations of twisting and bending. An interesting point is thatthe geometric construction of those strips were performed with a relatively parallel adapted frame insteadof a Frenet frame. In fact, it is known that a Frenet frame of a curve does not need to exist. However, theauthors proved that a relatively parallel adapted frame always exists for an arbitrary curve. The goal of1 a r X i v : . [ m a t h . F A ] S e p his work is to find additional information about the spectrum of the Dirichlet Laplacian in this situation.In the next paragraphs, we present the formal construction of the strip and more details of the problem.Let Γ : R → R n +1 , n ≥
1, be a curve of class C , parameterized by its arc-length s , i.e., | Γ (cid:48) ( s ) | = 1,for all s ∈ R . The vector T ( s ) := Γ (cid:48) ( s ) denotes its unitary tangent vector at the point Γ( s ). Note that T is a locally Lipschitz continuous function which is differentiable almost everywhere in R . The number k ( s ) := | Γ (cid:48)(cid:48) ( s ) | , s ∈ R , is called the curvature of Γ at the position Γ( s ). In Appendix A of [26], the authorsproved the existence of a relatively parallel adapted frame for the curve Γ. More exactly, it was shown thatthere exist n almost-everywhere differentiable normal vector fields N , · · · , N n so that TN ... N n (cid:48) = k · · · k n − k · · · − k n · · · TN ... N n , where k j : R → R , j ∈ { , · · · , n } , are locally bounded functions. In particular, the vector ( k , · · · , k n )satisfies k + · · · + k n = k .Now, let Θ j : R → R , j ∈ { , · · · , n } , be functions of class C , so thatΘ + · · · + Θ n = 1 , (1)and define N Θ := Θ N + · · · + Θ n N n . (2)Consider the twisted domainΩ ε := { Γ( s ) + N Θ ( s ) ε t : ( s, t ) ∈ R × ( − , } ; (3)Ω ε is obtained by translating the segment ( − ,
1) along Γ with respect to a normal field (2). Let − ∆ D Ω ε be the Dirichlet Laplacian operator in Ω ε . More precisely, − ∆ D Ω ε is defined as the self-adjoint operatorassociated with the quadratic form a ε ( ϕ ) = (cid:90) Ω ε |∇ ϕ | d x, dom a ε = H (Ω ε ); (4) ∇ ϕ is the gradient of ϕ . For simplicity, we denote − ∆ ε := − ∆ D Ω ε .Define the function Θ : R → R n , Θ( s ) := (Θ ( s ) , · · · , Θ n ( s )) , and write | Θ (cid:48) ( s ) | := (Θ (cid:48) ( s ) + · · · + Θ (cid:48) n ( s )) / . Note that Θ ∈ C , ( R ; R n ). As in [26], Θ is called twistingvector ; if Θ (cid:48) = 0, Ω ε is called untwisted or purely bent strip; if k · Θ := k Θ + · · · + k n Θ n = 0, Ω ε is called unbent or purely twisted strip. Geometrically, interpreting Γ as a curve in Ω ε , k · Θ is the geodesic curvature ofΓ; −| Θ (cid:48) | / (1 + | Θ (cid:48) | ε t ) is the Gauss curvature of Ω ε . One can see [26] for a detailed geometric descriptionof Ω ε .Let ( π/ be the first eigenvalue of the Dirichlet Laplacian − ∆ D ( − , in L ( − , − ∆ ε . In particular, under the conditions k · Θ , | Θ (cid:48) | ∈ L ∞ ( R ),and ε (cid:107) k · Θ (cid:107) L ∞ ( R ) <
1, they proved that:(i) if ( k · Θ)( s ) → | Θ (cid:48) ( s ) | →
0, as | s | → ∞ , then σ ess ( − ∆ ε ) = [( π/ ε ) , ∞ );(ii) if Θ (cid:48) = 0 and k · Θ (cid:54) = 0, then inf σ ( − ∆ ε ) < ( π/ ε ) .As a consequence of these results, if ( k · Θ)( s ) →
0, as | s | → ∞ , k · Θ (cid:54) = 0, and Θ (cid:48) = 0, then the discretespectrum of − ∆ ε is nonempty. A natural question is whether twisting can be used to create discreteeigenvalues, and this problem is the subject of this work.2n the first part of this paper, we assume thatΘ ∈ C , ( R ; R n ) , ( k · Θ)( s ) = 0 , and | Θ (cid:48) ( s ) | = γ − β ( s ) , ∀ s ∈ R , (5)where γ is a positive number, and β : R → R is a continuous, almost-everywhere differentiable functionwith compact support so that β (cid:48) ∈ L ∞ ( R ). In this situation, we will find information about the spectrumof − ∆ ε . Geometrically, the third condition in (5) means that Ω ε is “locally twisting”. In particular, we willshow that this effect can create discrete eigenvalues for − ∆ ε . Note that the condition k · Θ = 0 does notnecessarily implies that Γ is a straight line.At first, we study the essential spectrum of − ∆ ε . The strategy is based on a direct integral decompositionof the operator; see Section 3. In particular, consider the one-dimensional operator D ε (0) := − ∂ t ε + Y ε ( t ) , dom D ε (0) = H ( − , ∩ H ( − , , (6)where ∂ t := ∂/∂t , Y ε ( t ) := − γ ε t h ε ( t ) + γ h ε ( t ) , h ε ( t ) := (cid:112) γ ε t . (7)Since Y ε ∈ C ∞ [ − , D ε (0) has compact resolvent. Denote by λ ε, (0) its first eigenvalue and by u ε, thecorresponding orthonormal eigenfunction; λ ε, (0) is simple. Take ε > Y ε > , ∀ ε ∈ (0 , ε ) . (8)Thus, for each ε ∈ (0 , ε ), u ε, can be chosen to be real and positive in ( − , Theorem 1.
Assume the conditions (5) and (8). Then, σ ess ( − ∆ ε ) = [ λ ε, (0) , ∞ ) . The proof of Theorem 1 is presented in Section 3. In that same section, Remark 4 gives an asymptoticbehavior for the sequence { λ ε, (0) } ε ; in fact, ε λ ε, (0) → ( π/ , as ε → − ∆ ε below λ ε, (0). Let s > β ⊂ [ − s , s ]. Theorem 2.
Assume the conditions (5) and (8). If (cid:82) s − s ( | Θ (cid:48) ( s ) | − γ )d s < , then there exists ε > sothat, for each ε ∈ (0 , ε ) , inf σ ( − ∆ ε ) < λ ε, (0) , i.e., σ dis ( − ∆ ε ) (cid:54) = ∅ . Theorem 3.
Assume the conditions (5) and (8). If (cid:82) s − s ( | Θ (cid:48) ( s ) | − γ )d s = 0 , then there exists ε > sothat, for each ε ∈ (0 , ε ) , inf σ ( − ∆ ε ) < λ ε, (0) , i.e., σ dis ( − ∆ ε ) (cid:54) = ∅ . Theorems 2 and 3 show that a local twisted effect can create discrete eigenvalues for − ∆ ε . Now, wepresent a different situation where this result can also be obtained. Recently, in [30], the author studied theDirichlet Laplacian restricted in a two-dimensional twisted (straight) strip in R . In that work, the twistedeffect “grows” at infinity while the width of the strip goes to zero. It was shown that the discrete spectrumof − ∆ ε is nonempty and was found an asymptotic behavior for the eigenvalues. In the next paragraphs, wepresent an adaptation the results of [30] for the model of strips treated in this work.In this new situation, for n ≥
1, assume that Θ : R → R n is a C , function, which satisfies (1), and3I) lim | s |→∞ | Θ (cid:48) ( s ) | = ∞ ;(II) | Θ (cid:48) | is decreasing in ( −∞ ,
0) and increasing in (0 , ∞ ).Fix a number 0 < a < /
3. For each ε > ν ( ε ) < ν ( ε ) > | Θ (cid:48) ( ν i ( ε )) | = 1 ε a , i ∈ { , } . (9)Define I ε := ( ν ( ε ) , ν ( ε )) and let Θ ε : R → R n be a function of class C , so that(III) Θ ε ( s ) = Θ( s ), for all s ∈ I ε ;(IV) | Θ (cid:48) ε ( s ) | ≤ | Θ (cid:48) ( s ) | , for all s ∈ R ;(V) there exists K > | Θ (cid:48) ε ( s ) | ≤ Kε a , | Θ (cid:48)(cid:48) ε ( s ) | ≤ Kε b , | Θ (cid:48)(cid:48)(cid:48) ε ( s ) | ≤ Kε c , ∀ s ∈ R , where b, c are real numbers so that b < , a + c < { Θ ε } ε satisfies(VI) | Θ (cid:48) ε ( s ) | ≤ | Θ (cid:48) ε (cid:48) ( s ) | , for all s ∈ R , if ε > ε (cid:48) .Finally, we use the notation Θ ε := (Θ ε , · · · , Θ εn ), and we assume that(VII) (Θ ε ) + · · · + (Θ εn ) = 1.For each ε > ε : R → R n +1 be a curve of class C , whose curvature k ε satisfies(VIII) supp k ε ⊂ I ε , and ( k ε · Θ ε )( s ) = 0, for all s ∈ R .The normal vector fields of Γ ε are denoted by N ε , · · · , N εn , and N ε Θ ε := Θ ε N ε + · · · + Θ εn N εn .Consider the strip ˜Ω ε := { Γ ε ( s ) + N ε Θ ε ( s ) ε t : ( s, t ) ∈ R × ( − , } . Geometrically, ˜Ω ε is a locally twisting (and locally bending) strip which the twisted effect “grows” at infinitywhile its width goes to zero.Let − ∆ D ˜Ω ε be the Dirichlet Laplacian operator in ˜Ω ε , i.e., the self-adjoint operator associated with thequadratic form ˜ a ε ( ϕ ) = (cid:90) ˜Ω ε |∇ ϕ | d x, dom ˜ a ε = H ( ˜Ω ε ) . (10)For simplicity, write − ˜∆ ε := − ∆ D ˜Ω ε . Remark 1.
Let T be a self-adjoint operator that is bounded from below. We denote by { λ j ( T ) } j ∈ N thenon-decreasing sequence of numbers corresponding to the spectral problem of T according to the Min-MaxPrinciple; see, for example, Theorem XIII.1 in [27].Let Π( ε ) be the infimum of the essential spectrum of − ˜∆ ε . Denote by N ( ε ) ≤ ∞ the number ofeigenvalues λ j ( − ˜∆ ε ) of − ˜∆ ε below Π( ε ). Let − ∆ R be the one-dimensional Laplacian and consider theself-adjoint operator − ∆ R + ( | Θ (cid:48) ( s ) | / acting in L ( R ); denotes the identity operator in L ( R ). Dueto the condition (I), this operator has purely discrete spectrum. In Section 5 of this work, we prove thefollowing result. 4 heorem 4. Assume the conditions (I)-(VIII). For ε > small enough, the discrete spectrum σ dis ( − ˜∆ ε ) is nonempty and N ( ε ) → ∞ , as ε → . Furthermore, for each j ∈ N , lim ε → (cid:20) λ j ( − ˜∆ ε ) − (cid:16) π ε (cid:17) (cid:21) = λ j (cid:18) − ∆ R + | Θ (cid:48) ( s ) | (cid:19) . Estimating the number of discrete eigenvalues of the Dirichlet Laplacian operator in unbounded strips isalso an interesting problem. In the case of curved strips in R , in [8], the authors found that this number isfinite and bounded for a constant that does not depend on the width of the strip. However, in the conditionsof Theorem 4, the number of discrete eigenvalues of − ˜∆ ε grows when the width of the strip goes to zero.In [30], a result similar to Theorem 4 was obtained as a consequence of a convergence in the normresolvent sense of the operators associated to the problem. In this text, we present a simpler proof wherethe strategy is based on to find upper and lower bounds for the eigenvalues λ j ( − ˜∆ ε ).We finish this introduction with some remarks and examples of the model presented. Remark 2.
Theorem 4 shows that the locally bending effect imposed by (VIII) does not affect the finalresult.
Example 1.
A simple example of a family of curves { Γ ε } ε satisfying (VIII) is the following. Let Θ be afunction satisfying (I), and let Γ : R → R n +1 be a curve of class C , whose curvature k has compact supportand satisfies ( k · Θ)( s ) = 0, for all s ∈ R . Define Γ ε := Γ, for all ε > ε > Remark 3.
In the case n = 2, conditions (1) and (I) imply that Θ can be written as Θ = (cos( ψ ) , sin( ψ ))for some function ψ ∈ C , ( R ; R ) so that | ψ (cid:48) ( s ) | → ∞ , as | s | → ∞ . In the case n = 3, due to the condition(1), Θ can be written as Θ = (cos( φ ) cos( ψ ) , sin( φ ) cos( ψ ) , sin( ψ )), for functions φ, ψ ∈ C , ( R ; R ). Since | Θ (cid:48) | = (cid:112) ( φ (cid:48) ) cos ( ψ ) + ( ψ (cid:48) ) , the condition (I) is satisfied, for example, if | ψ (cid:48) ( s ) | → ∞ , as | s | → ∞ , or, if ψ ( s ) ∈ ( c , c ] ⊂ (0 , π/ s ∈ R , and | φ (cid:48) ( s ) | → ∞ , as | s | → ∞ . Example 2.
Consider Θ : R → R defined by Θ( s ) = (cos( s ) , sin( s )). Some calculations show that | Θ (cid:48) ( s ) | = 2 | s | , s ∈ R . Fix a number 0 < a < /
3. For each ε > α ε : R −→ R , α ε ( s ) := ( − ε a s − /ε a , s ≤ − /ε a ,s , s ∈ ( − /ε a , /ε a ) , (2 ε a s − /ε a , s ≥ /ε a , and Θ ε ( s ) := (cos( α ε ( s )) , sin( α ε ( s ))), s ∈ R . Taking b = 2 a and c = 3 a , the conditions in (V) are satisfied.Therefore, the sequence { Θ ε } ε satisfies the conditions (III)-(VII). Example 3.
Consider the function Θ : R → R defined byΘ( s ) = (cid:18) cos( s ) cos (cid:18)
11 + s (cid:19) , sin( s ) cos (cid:18)
11 + s (cid:19) , sin (cid:18)
11 + s (cid:19)(cid:19) . Note that the condition (1) is satisfied and the function | Θ (cid:48) ( s ) | = (cid:115) s cos (cid:18)
11 + s (cid:19) + 4 s (1 + s ) , s ∈ R , satisfies (I) and (II). Fix a number 0 < a < /
3. For each ε > ν ( ε ) > | Θ (cid:48) ( ± ν ( ε )) | = 1 /ε a ; then, there exists K > | ν ( ε ) | ≤ K/ε a . Let α ε : R −→ R be the functiondefined by α ε ( s ) := − ν ( ε ) s − ν ( ε ) , s ≤ − ν ( ε ) ,s , s ∈ ( − ν ( ε ) , ν ( ε )) , ν ( ε ) s − ν ( ε ) , s ≥ ν ( ε );5ote that α ε ∈ C , ( R ; R ). Now, defineΘ ε ( s ) := (cid:18) cos( α ε ( s )) cos (cid:18)
11 + s (cid:19) , sin( α ε ( s )) cos (cid:18)
11 + s (cid:19) , sin (cid:18)
11 + s (cid:19)(cid:19) , s ∈ R . The sequence { Θ ε } ε satisfies the conditions (III)-(VII), where b = 2 a and c = 3 a .This paper is organized as follows. In Section 2 we present some details of the construction of the regionin (3) and we make usual change of coordinates in the quadratic form in (4). Sections 3 and 4 are dedicatedto study the essential and discrete spectrum of − ∆ ε , respectively. In Section 5, we study the spectralproblem of − ˜∆ ε . In Appendix 6 are presented results that are useful in this text. Along the text, K is usedto denote different constants. Recall the twisted domain Ω ε given by (3) in the Introduction and the straight strip Λ := R × ( − , ε with the Riemannian manifold (Λ , G ε ), where G ε is given by (11), below. Afterthat, we perform usual changes of coordinates in the quadratic form a ε ( ϕ ).Consider the map L ε : R −→ R n +1 ( s, t ) (cid:55)−→ Γ( s ) + N Θ ( s ) εt . We have Ω ε = L ε (Λ). Define the metric G ε := ∇L ε · ( ∇L ε ) ⊥ . Some calculations show that G ε = (cid:18) f ε ε (cid:19) , f ε ( s, t ) := (cid:112) | Θ (cid:48) ( s ) | ε t . (11)Let J ε be the Jacobian matrix of L ε . One has det J ε = | det G ε | / = εf ε >
0, for all ( s, t ) ∈ Λ. If Γand Θ are smooth functions, the map L ε : Λ → Ω ε is a local smooth diffeomorphism. Then, Ω ε can beidentified with the Riemannian manifold (Λ , G ε ). However, as mentioned in the Introduction of this work,the assumptions about Γ and Θ are more general. At first, one has Proposition 1.
Assume that Γ ∈ C , ( R ; R n +1 ) and Θ ∈ C , ( R ; R n ) . Then, the map L ε : Λ −→ Ω ε is alocal C , -diffeomorphism. The proof of this result can be found in [26]. We emphasize that in that work does not necessarily k · Θ = 0,and Proposition 1 is proven under the additional assumptions that k · Θ ∈ L ∞ ( R ) and ε (cid:107) k · Θ (cid:107) L ∞ ( R ) < L ε is a C , -immersion. In addition, assume that L ε is injective.Thus, the strip Ω ε does not self-intersect and it is interpreted as an immersed submanifold in R n +1 . As aconsequence, (Λ , G ε ) is an abstract Riemannian manifold.Now, we perform a change of coordinates so that the quadratic form a ε ( ϕ ) starts to act in the Hilbertspace L (Λ) (with the usual metric of R ) instead of L (Ω ε ). At first, consider the unitary operator U ε : L (Ω ε ) −→ L (Λ , f ε d s d t ) ψ (cid:55)−→ ε / ψ ◦ L ε , and define the quadratic form b ε ( ψ ) := a ε (cid:0) U − ε ψ (cid:1) = (cid:90) Λ (cid:104)∇ ψ, G − ε ∇ ψ (cid:105) f ε d s d t = (cid:90) Λ | ∂ s ψ | f ε d s d t + 1 ε (cid:90) Λ | ∂ t ψ | f ε d s d t, b ε := U ε ( H (Ω ε )); ∂ t = ∂/∂ t and ∂ s = ∂/∂ s . Then, consider V ε : L (Λ) −→ L (Λ , f ε d s d t ) ψ (cid:55)−→ f − / ε ψ , which is also a unitary operator, and, finally, define c ε ( ψ ) := b ε ( V ε ψ ) = (cid:90) Λ f ε (cid:12)(cid:12)(cid:12)(cid:12) ∂ s ψ − ∂ s f ε f ε ψ (cid:12)(cid:12)(cid:12)(cid:12) d s d t + 1 ε (cid:90) Λ | ∂ t ψ | d s d t + (cid:90) Λ V ε | ψ | d s d t, where V ε ( s, t ) := − | Θ (cid:48) ( s ) | ε t f ε ( s, t ) + | Θ (cid:48) ( s ) | f ε ( s, t ) , dom c ε = V − ε ( U ε ( H (Ω ε ))). Due to the conditions in (5), one has dom c ε = H (Λ). Denote by C ε theself-adjoint operator associated with the quadratic form c ε ( ψ ) . This section is dedicated to prove Theorem 1. Recall the functions h ε and Y ε defined by (7) in the Intro-duction. Consider the quadratic form d ε ( ψ ) := (cid:90) Λ | ∂ s ψ | h ε d s d t + 1 ε (cid:90) Λ | ∂ t ψ | d s d t + (cid:90) Λ Y ε | ψ | d s d t, dom d ε := H (Λ) . Denote by D ε the self-adjoint operator associated with d ε ( ψ ). We start with the following result. Proposition 2.
Assume the conditions in (5). Then, σ ess ( C ε ) = σ ess ( D ε ) . The proof of this result is presented in Appendix 6. As a consequence, we start to study the essentialspectrum of D ε .Let F s : L (Λ) → L (Λ) be the Fourier transform with respect to s . F s is a unitary operator and, forfunctions in L (Λ), its action is given by( F s ψ )( p, t ) = 1 √ π (cid:90) R e − ips ψ ( s, t )d s. Then, the operator ˆ D ε := F s D ε F − s admits the direct integral decompositionˆ D ε = (cid:90) ⊕ R D ε ( p ) d p, (12)where, for each p ∈ R , D ε ( p ) is the self-adjoint operator associated with the quadratic form d ε ( p )( v ) := 1 ε (cid:90) − | ∂ t v | d t + (cid:90) − Y pε | v | d t, dom d ε ( p ) = H ( − , , where Y pε ( t ) := p /h ε ( t ) + Y ε ( t ). More precisely, D ε ( p ) = − ∂ t ε + Y pε ( t ) , dom D ε ( p ) = H ( − , ∩ H ( − , p = 0 corresponds to the operator defined by (6) in the Introduction. Since Y pε ∈ C ∞ [ − , D ε ( p ) has compact resolvent. Denote by { λ ε,n ( p ) } n ∈ N the sequence of eigenvalues of D ε ( p ) and by { u ε,n ( p ) } n ∈ N the sequence of the corresponding normalized eigenfunctions, i.e., D ε ( p ) u ε,n ( p ) = λ ε,n ( p ) u ε,n ( p ) , n ∈ N , p ∈ R . σ ( D ε ) = ∪ p ∈ R σ ( D ε ( p )) = ∪ n ∈ N { λ ε,n ( p ) : p ∈ R } . (13)In particular, denote u ε, := u ε, (0). Lemma 1.
For each n ∈ N , λ ε,n ( · ) is a real analytic function in p and lim p →±∞ λ ε,n ( p ) = ∞ . Proof.
At first, note that dom D ε ( p ) = dom D ε (0), for all p ∈ R . One can write D ε ( p ) = D ε (0) + p /h ε .Since h ε ≥
1, it holds the estimate (cid:107) ( p /h ε ) v (cid:107) ≤ p (cid:107) v (cid:107) , for all v ∈ dom D ε (0), and for all p ∈ R . Then, p /h ε is D ε (0)-bounded with relative bound zero. Consequently, { D ε ( p ); p ∈ R } is a type A analytic family.By Theorem 3.9 in [19], λ ε,n ( · ) is a real analytic function in p .Now, for each v ∈ dom d ε ( p ), we have d ε ( p )( v ) = 1 ε (cid:90) − | ∂ t v | d t + (cid:90) − Y pε | v | d t ≥ (cid:18) p γ ε − ε γ (cid:19) (cid:90) − | v | d t. As a consequence, for each n ∈ N , λ ε,n ( p ) = d ε ( p )( u ε,n ( p )) ≥ p γ ε − ε γ , p ∈ R . Thus, we obtain the punctual limit λ ε,n ( p ) → ∞ , as p → ±∞ . Proposition 3.
One has σ ( D ε ) = [ λ ε, (0) , ∞ ) . Proof.
Since λ ε, ( p ) is a real analytic function in p , by (13), we have [ λ ε, (0) , ∞ ) ⊂ σ ( D ε ) . Now, we need toshow that ( −∞ , λ ε, (0)) ∩ σ ( D ε ) = ∅ . (14)Take ψ ∈ C ∞ (Λ). Since u ε, is positive, we can write ψ ( s, t ) = φ ( s, t ) u ε, ( t ), with φ ∈ C ∞ (Λ). Somecalculations show that d ε ( ψ ) − λ ε, (0) (cid:90) Λ | ψ | d s d t = (cid:90) Λ (cid:18) | ∂ s φ | h ε + | ∂ t φ | ε (cid:19) | u ε, | d s d t + 2 ε R (cid:90) Λ φ ∂ t φ u ε, ∂ t u ε, d s d t + 1 ε (cid:90) Λ | φ | | ∂ t u ε, | d s d t + (cid:90) Λ | φ | ( Y ε | u ε, | − λ ε, (0) | u ε, | )d s d t. An integration by parts, and since D ε (0) u ε, = λ ε, (0) u ε, , one has d ε ( ψ ) − λ ε, (0) (cid:90) Λ | ψ | d s d t = (cid:90) Λ (cid:18) | ∂ s φ | h ε + | ∂ t φ | ε (cid:19) | u ε, | d s d t ≥ . Then, we find (14).
Proof of Theorem 1.
Apply Propositions 2 and 3.
Remark 4.
For the sequence { λ ε, (0) } ε , we have the estimate: there exists K > (cid:90) − | ∂ t v | d t ≤ ε d ε (0)( v ) = (cid:90) − (cid:0) | ∂ t v | + ε Y ε | v | (cid:1) d t ≤ (cid:90) − | ∂ t v | d t + Kε (cid:90) − | v | d t, for all v ∈ H ( − , ε > (cid:16) π (cid:17) ≤ ε λ ε, (0) ≤ (cid:16) π (cid:17) + O ( ε ) . Discrete spectrum
Based in [11] and [17], the main idea in the proofs of Theorems 2 and 3 is to find a function ψ ∈ dom c ε sothat ( c ε ( ψ ) − λ ε, (0) (cid:107) ψ (cid:107) ) / (cid:107) ψ (cid:107) < . Proof of Theorem 2.
Take δ >
0. Define ψ δ ( s, t ) := φ ( s ) u ε, ( t ), where φ ( s ) := e δ ( s + s ) , s ≤ − s , , − s ≤ s ≤ s ,e − δ ( s − s ) , s ≥ s . Note that ψ δ ∈ dom c ε . Some calculations show that c ε ( ψ δ ) − λ ε, (0) (cid:90) Λ | ψ δ | d s d t = (cid:90) Λ | ∂ s ψ δ | f ε d s d t + 14 (cid:90) Λ ( ∂ s f ε ) f ε | ψ δ | d s d t − R (cid:90) Λ ∂ s f ε f ε ψ δ ∂ s ψ δ d s d t + 1 ε (cid:90) Λ | ∂ t ψ δ | d s d t − λ ε, (0) (cid:90) Λ | ψ δ | d s d t + (cid:90) Λ V ε | ψ δ | d s d t = (cid:90) Λ | ∂ s ψ δ | f ε d s d t + 14 (cid:90) Λ ( ∂ s f ε ) f ε | ψ δ | d s d t − R (cid:90) Λ ∂ s f ε f ε ψ δ ∂ s ψ δ d s d t + (cid:90) Λ ( V ε − Y ε ) | ψ δ | d s d t. Now, note that f ε → ∂ s f ε →
0, ( V ε − Y ε ) → ( | Θ (cid:48) | − γ ) /
2, uniformly, as ε →
0. Then, c ε ( ψ δ ) − λ ε, (0) (cid:107) ψ δ (cid:107) → δ + 12 (cid:90) s − s (cid:0) | Θ (cid:48) ( s ) | − γ (cid:1) d s, as ε →
0. Since (cid:107) ψ δ (cid:107) = 2 s + δ − , one has c ε ( ψ δ ) − λ ε, (0) (cid:107) ψ δ (cid:107) (cid:107) ψ δ (cid:107) → O ( δ ) + δ (cid:90) s − s (cid:0) | Θ (cid:48) ( s ) | − γ (cid:1) d s, (15)as ε → (cid:82) s − s ( | Θ (cid:48) ( s ) | − γ )d s <
0, we can choose δ small enough so that the limit in (15) is negative.Consequently, there exists ε > c ε ( ψ δ ) − λ ε, (0) (cid:107) ψ δ (cid:107) (cid:107) ψ δ (cid:107) < , for all ε ∈ (0 , ε ). Proof of Theorem 3.
Given δ > η >
0, define ψ δ,η ( s, t ) := φ η ( s ) u ε, ( t ) , where φ η ( s ) := e δ ( s + s ) , s ≤ − s , η ( γ − | Θ (cid:48) ( s ) | ) , − s ≤ s ≤ s ,e − δ ( s − s ) , s ≥ s . Note that ψ δ,η ∈ dom c ε . Similarly as in the proof of Theorem 2, we can show that c ε ( ψ δ,η ) − λ ε, (0) (cid:107) ψ δ,η (cid:107) → δ + O ( η ) − η (cid:90) s − s (cid:0) | Θ (cid:48) ( s ) | − γ (cid:1) (cid:0) | Θ (cid:48) ( s ) | + γ (cid:1) d s, ε → . Since (cid:107) ψ δ,η (cid:107) = 2 s + δ − + O ( η ) , one has c ε ( ψ δ,η ) − λ ε, (0) (cid:107) ψ δ,η (cid:107) (cid:107) ψ δ,η (cid:107) → O ( δ ) + δO ( η ) − δη (cid:90) s − s (cid:0) | Θ (cid:48) ( s ) | − γ (cid:1) (cid:0) | Θ (cid:48) ( s ) | + γ (cid:1) d s, (16)as ε → . Taking η = √ δ, again we can choose δ small enough so that the limit in (16) is negative. Then,there exists ε > c ε ( ψ δ,η ) − λ ε, (0) (cid:107) ψ δ,η (cid:107) (cid:107) ψ δ,η (cid:107) < , for all ε ∈ (0 , ε ). In this section we present the proof of Theorem 4 stated in the Introduction. The strategy will be to establishupper and lower bounds for the eigenvalues λ j ( − ˜∆ ε ). Recall − ˜∆ ε is the self-adjoint operator associatedwith the quadratic form ˜ a ε ( ϕ ); see (10). Then, the analysis will be based on estimates for ˜ a ε ( ϕ ).Define ˜ f ε ( s, t ) := (cid:112) | Θ (cid:48) ε ( s ) | ε t , and consider the Hilbert space H ε := L (Λ , ˜ f ε d s d t ); the norm in this space is denoted by (cid:107) · (cid:107) H ε . Performinga change of coordinates similar to that in Section 2, ˜ a ε ( ϕ ) becomes˜ b ε ( ψ ) := (cid:90) Λ | ∂ s ψ | ˜ f ε d s d t + 1 ε (cid:90) Λ | ∂ t ψ | ˜ f ε d s d t, dom ˜ b ε = H (Λ) ⊂ H ε . Upper bound.
Denote by χ ( t ) := cos( πt/
2) the first eigenfunction of the Dirichlet Laplacian − ∆ D ( − , in L ( − , π/ is the eigenvalue associated with χ . Consider the closed subspace A ε := { ϕ w := w ( s ) χ ( t )( ˜ f ε ( s, t )) − / : w ∈ H ( R ) } of the Hilbert space H ε . The identification w (cid:55)→ ϕ w , w ∈ H ( R ), motivates the definition of the one-dimensional quadratic form m ε ( w ) := ˜ b ε ( ϕ w ) − ( π/ ε ) (cid:107) ϕ w (cid:107) H ε , dom m ε := H ( R ). Denote by M ε the self-adjoint operator associated with m ε ( w ). In particular, for each j ∈ N , λ j ( − ˜∆ ε ) − (cid:16) π ε (cid:17) ≤ λ j ( M ε ) . (17)We are going to get upper bounds for the values λ j ( M ε ).Define the function W ε ( s, t ) := − | (Θ (cid:48) ε · Θ (cid:48)(cid:48) ε )( s ) | ε t ˜ f ε ( s, t ) + (2 | Θ (cid:48)(cid:48) ε ( s ) | + 2(Θ (cid:48) ε · Θ (cid:48)(cid:48)(cid:48) ε )( s ) − | Θ (cid:48) ε ( s ) | ) ε t f ε ( s, t ) + | Θ (cid:48) ε ( s ) | f ε ( s, t ) . Recall that we have the condition (V) in the Introduction, we get the estimates (cid:107) (1 / ˜ f ε ) − (cid:107) L ∞ (Λ) ≤ ε − a , (cid:107) W ε − | Θ (cid:48) ε | / (cid:107) L ∞ (Λ) ≤ K ( ε − a + b ) + ε − b + ε − ( a + c ) + ε − a ) , K >
0, for all ε > m ε ( w ) = (cid:90) Λ | w (cid:48) χ | ˜ f ε d s d t + (cid:90) Λ W ε | wχ | d s d t ≤ (cid:90) R (cid:18) | w (cid:48) | + | Θ (cid:48) ε ( s ) | | w | (cid:19) d s + O ( ε d ) (cid:90) R | w | d s, for all w ∈ H ( R ), for all ε > d = min { − a + b ) , − b, − ( a + c ) , − a } . As aconsequence, for each j ∈ N , λ j ( M ε ) ≤ λ j (cid:18) − ∆ R + | Θ (cid:48) ε ( s ) | (cid:19) + O ( ε d ) . (18)By (17) and (18), for each j ∈ N , λ j ( − ˜∆ ε ) − (cid:16) π ε (cid:17) ≤ λ j (cid:18) − ∆ R + | Θ (cid:48) ε ( s ) | (cid:19) + O ( ε d ) . (19) Lower bound.
For each ε ≥ S ε v )( t ) := − v (cid:48)(cid:48) ( t ) − εt εt v (cid:48) ( t ) , dom S ε = H ( − , , acting in the Hilbert space L (( − , , √ εt d t ). The particular case ε = 0 corresponds to the DirichletLaplacian operator − ∆ D ( − , in L ( − , ε ) the first eigenvalue of S ε . By the analytic perturbation theory, we can writeΣ( ε ) = (cid:16) π (cid:17) + δ ( ε ) ε + O ( ε ) , where δ ( ε ) := − (cid:90) − t √ εt χ (cid:48) ( t ) χ ( t )d t ;see [18] for more details. Consequently, for each ψ ∈ H (Λ), we have the estimate˜ b ε ( ψ ) ≥ (cid:90) Λ (cid:18) | ∂ s ψ | ˜ f ε + Σ( ε | Θ (cid:48) ε ( s ) | ) ε ˜ f ε | ψ | (cid:19) d s d t. More exactly, ˜ b ε ( ψ ) − (cid:16) π ε (cid:17) (cid:107) ψ (cid:107) H ε ≥ (cid:90) Λ (cid:18) | ∂ s ψ | ˜ f ε + | Θ (cid:48) ε ( s ) | δ ( ε | Θ (cid:48) ε ( s ) | ) ˜ f ε | ψ | (cid:19) d s d t (20)+ O ( ε | Θ (cid:48) ε ( s ) | )Now, define the quadratic form n ε ( ψ ) := (cid:90) Λ (cid:18) | ∂ s ψ | ˜ f ε + | Θ (cid:48) ε ( s ) | δ ( ε | Θ (cid:48) ε ( s ) | ) ˜ f ε | ψ | (cid:19) d s d t, dom n ε = H (Λ). Denote by N ε the self-adjoint operator associated with n ε ( ψ ). For each j ∈ N , inequality(20) implies λ j ( N ε ) + O ( ε − a ) ≤ λ j ( − ˜∆ ε ) − (cid:16) π ε (cid:17) . (21)The next step is to find lower bounds for the values λ j ( N ε ).11 emma 2. There exists a number
K > so that (cid:13)(cid:13)(cid:13)(cid:13) | Θ (cid:48) ε ( s ) | δ ( ε | Θ (cid:48) ε ( s ) | ) ˜ f ε − | Θ (cid:48) ε ( s ) | (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (Λ) ≤ Kε − a , for all ε > small enough.Proof. At first, note that (cid:12)(cid:12)(cid:12)(cid:12) | Θ (cid:48) ε ( s ) | δ ( ε | Θ (cid:48) ε ( s ) | ) ˜ f ε − | Θ (cid:48) ε ( s ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤ | Θ (cid:48) ε ( s ) | (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) δ ( ε | Θ (cid:48) ε ( s ) | ) − (cid:12)(cid:12)(cid:12)(cid:12) | ˜ f ε | + 12 | ˜ f ε − | (cid:19) , for all ( s, t ) ∈ Λ.Some calculations show that δ ( ε | Θ (cid:48) ε ( s ) | ) −
12 = π (cid:90) − (cid:18) f ε − (cid:19) t sin (cid:18) πt (cid:19) cos (cid:18) πt (cid:19) d t. Since (cid:107) (1 / ˜ f ε ) − (cid:107) L ∞ (Λ) ≤ ε − a , we have the estimate (cid:107) δ ( ε | Θ (cid:48) ε ( s ) | ) − / (cid:107) L ∞ (Λ) ≤ ε − a /
2. Thus, alongwith the condition | Θ (cid:48) ε ( s ) | ≤ K/ε a and the estimate (cid:107) ˜ f ε − (cid:107) L ∞ (Λ) ≤ ε − a , we get the result.Using Lemma 2 and the estimate (cid:107) / ˜ f ε − (cid:107) L ∞ (Λ) ≤ ε − a , one has n ε ( ψ ) ≥ (1 + O ( ε − a )) (cid:90) Λ | ∂ s ψ | d s d t + (cid:90) Λ | Θ (cid:48) ε ( s ) | | ψ | d s d t + O ( ε − a ) (cid:90) Λ | ψ | d s d t. Since | Θ (cid:48) ε ( s ) | ≤ K/ε a , for all s ∈ R , it follows that n ε ( ψ ) ≥ (1 + O ( ε − a )) (cid:90) Λ (cid:18) | ∂ s ψ | + | Θ (cid:48) ε ( s ) | | ψ | (cid:19) d s d t, for all ψ ∈ H (Λ), for all ε > j ∈ N ,(1 + O ( ε − a )) λ j (cid:18) − ∆ R + | Θ (cid:48) ε ( s ) | (cid:19) ≤ λ j ( N ε ) . (22)Inequalities (21) and (22) ensure that, for each j ∈ N ,(1 + O ( ε − a )) λ j (cid:18) − ∆ R + | Θ (cid:48) ε ( s ) | (cid:19) + O ( ε − a ) ≤ λ j ( − ˜∆ ε ) − (cid:16) π ε (cid:17) . (23)Due to inequalities (19) and (23), we will study the spectral problem of the operator − ∆ R +( | Θ (cid:48) ε ( s ) | / . Proposition 4.
For each j ∈ N , lim ε → λ j (cid:18) − ∆ R + | Θ (cid:48) ε ( s ) | (cid:19) = λ j (cid:18) − ∆ R + | Θ (cid:48) ( s ) | (cid:19) . (24) Proof.
Since | Θ (cid:48) ε ( s ) | ≤ | Θ (cid:48) ( s ) | , for all s ∈ R , we have, for each j ∈ N , λ j (cid:18) − ∆ R + | Θ (cid:48) ε ( s ) | (cid:19) ≤ λ j (cid:18) − ∆ R + | Θ (cid:48) ( s ) | (cid:19) . (25)12ow, for each ε > y ε ( w ) := (cid:90) R (cid:18) | w (cid:48) | + | Θ (cid:48) ε ( s ) | | w | (cid:19) d s, dom y ε = H ( R ) , and z ε ( w ) = z ε, ( w ) ⊕ z ε, ( w ) ⊕ z ε, ( w ), dom z ε = dom z ε, ⊕ dom z ε, ⊕ dom z ε, , where z ε, ( w ) := 0 , dom z ε, = H ( −∞ , ν ( ε )) ,z ε, ( w ) := (cid:90) ν ( ε ) ν ( ε ) (cid:18) | w (cid:48) | + | Θ (cid:48) ( s ) | | w | (cid:19) d s, dom z ε, = H ( ν ( ε ) , ν ( ε )) ,z ε, ( w ) := 0 , dom z ε, = H ( ν ( ε ) , ∞ ); ν ( ε ) , ν ( ε ) are defined by (9) in the Introduction. Note that y ε ( w ) ≥ z ε ( w ) , ∀ w ∈ H ( R ) . Consequently, for each j ∈ N , λ j ( Z ε ) ≤ λ j (cid:18) − ∆ R + | Θ (cid:48) ε ( s ) | (cid:19) , (26)where Z ε is the self-adjoint operator associated with z ε ( w ). Now, let us show that, for each j ∈ N ,lim ε → λ j ( Z ε ) = λ j (cid:18) − ∆ R + | Θ (cid:48) ( s ) | (cid:19) . (27)The condition (III) in the Introduction implies that | Θ (cid:48) ε ( s ) | → | Θ (cid:48) ( s ) | pointwise, as ε →
0. Thus, for each w ∈ C ∞ ( R ), Z ε w → ( − ∆ R + ( | Θ (cid:48) ( s ) | / ) w , as ε →
0. Since C ∞ ( R ) is a core of − ∆ R + ( | Θ (cid:48) ( s ) | / ,then Z ε → ( − ∆ R + ( | Θ (cid:48) ( s ) | / ) in the strong resolvent sense, as ε →
0. Now, fix a real number δ > ν ( ε ) < − δ and δ < ν ( ε ), for every ε > z ( w ) = z ( w ) ⊕ z ( w ) ⊕ z ( w ),dom z = dom z ⊕ dom z ⊕ dom z , where z ( w ) := 0 , dom z = L ( −∞ , − δ ) ,z ( w ) := (cid:90) δ − δ | w (cid:48) | d s, dom z = H ( − δ, δ ) ,z ( w ) := 0 , dom z = L ( δ, ∞ ) . Then, for all ε > z ε ( w ) ≥ z ( w ) , ∀ w ∈ dom z ε . Hence, Z − ε ≤ Z − , where Z is the self-adjoint operator associated with z ( w ) . Since Z has compact resolvent, the result followsby Theorem 2.16 of [29]. In particular, that theorem says that if { L ε } ε is a family of compact, self-adjointoperators in a Hilbert space H so that L ε → L strongly, as ε →
0, and there exists a compact, self-adjointoperator ˜ L so that 0 ≤ L ε ≤ ˜ L , for all ε , then (cid:107) L ε − L (cid:107) →
0, as ε → Proof of Theorem 4.
It just to apply inequalities (19) and (23), and Proposition 4.13
Appendix
Stability of the essential spectrum
The results of this appendix are simple adaptations of Lemma 4.1 and Proposition 4.2 of [4], and Lemma4.2 of [7].
Lemma 3.
A real number λ belongs to the essential spectrum of C ε if and only if there exists a sequence { ψ n } n ∈ N ⊂ dom c ε satisfying the following conditions: ( i ) (cid:107) ψ n (cid:107) = 1 , for all n ∈ N ;( ii ) ( C ε − λ ) ψ n → , as n → ∞ , in the norm of the dual space (dom c ε ) ∗ ;( iii ) supp ψ n ⊂ Λ \ ( − n, n ) × ( − , , for all n ∈ N . Proof.
It is known that λ ∈ σ ess ( C ε ) if and only if there exists a sequence { ξ n } n ∈ N ⊂ dom c ε satisfying ( i ) , ( ii ), and ( iii (cid:48) ) ξ n (cid:42) L (Λ), as n →
0; see, for example, Theorem 5 in [24]. Let { ψ n } n ∈ N be a sequencesatisfying the conditions ( i ) , ( ii ) and ( iii ). Consequently, it satisfies ( i ) , ( ii ) and ( iii (cid:48) ).Now, let { ξ n } n ∈ N ⊂ dom c ε be a sequence satisfying ( i ), ( ii ), and ( iii (cid:48) ). Take η ∈ C ∞ ( R ; R ), 0 ≤ η ≤ η = 0 in [ − , η = 1 in R \ ( − , { η k } k ∈ N ⊂ C ∞ (Λ), where η k ( s, t ) := η ( s/k ).Since (1 − η k )( C ε + ) − is compact in L (Λ), by ( iii (cid:48) ), we have (1 − η k )( C ε + ) − ξ n → L (Λ), as n → ∞ ,for all k ∈ N . Then there exists a subsequence { ξ n k } k ∈ N of { ξ n } n ∈ N so that (1 − η k )( C ε + ) − ξ n k → L (Λ), as k → ∞ . By writing ξ n k = ( C ε + ) − ( C ε − λ ) ξ n k + ( λ + 1)( C ε + ) − ξ n k and using ( ii ), it follows that (1 − η k ) ξ n k → L (Λ), as k → ∞ . Thus, we can assume that (cid:107) η k ξ n k (cid:107) ≥ / , for all k ∈ N . Finally, define ψ k := η k ξ n k (cid:107) η k ξ n k (cid:107) , k ∈ N . The sequence { ψ k } k ∈ N ⊂ dom c ε satisfies the conditions ( i ) and ( iii ). It remains to verify ( ii ), i.e.,sup φ ∈ H (Λ) φ (cid:54) =0 | c ε ( φ, ψ k ) − λ (cid:104) φ, ψ k (cid:105)|(cid:107) φ (cid:107) + −→ , (28)as k → ∞ , where (cid:107) φ (cid:107) := c ε ( φ ) + (cid:107) φ (cid:107) .Some calculations show that c ε ( φ, η k ξ n k ) − λ (cid:104) φ, η k ξ n k (cid:105) = c ε ( η k φ, ξ n k ) − λ (cid:104) η k φ, ξ n k (cid:105) + (cid:90) Λ f ε φ ∂ s η k ξ n k d s d t + 2 (cid:90) Λ f ε (cid:18) ∂ s φ − ∂ s f ε f ε φ (cid:19) ∂ s η k ξ n k d s d t. Since { ξ n k } k ∈ N satisfies ( ii ), we havesup φ ∈ H (Λ) φ (cid:54) =0 | c ε ( η k φ, ξ n k ) − λ (cid:104) η k φ, ξ n k (cid:105)|(cid:107) φ (cid:107) + ≤ sup φ ∈ H (Λ) η k φ (cid:54) =0 | c ε ( η k φ, ξ n k ) − λ (cid:104) η k φ, ξ n k (cid:105)|(cid:107) η k φ (cid:107) + −→ , as k → ∞ . By H¨older’s Inequality and by the estimates (cid:107) φ (cid:107) ≤ (cid:107) φ (cid:107) + and c ε ( φ ) ≤ (cid:107) φ (cid:107) , we getsup φ ∈ H (Λ) φ (cid:54) =0 (cid:26) (cid:107) φ (cid:107) + (cid:90) Λ f ε | φ || ∂ s η k || ξ n k | d s d t (cid:27) ≤ (cid:107) ∂ s η k (cid:107) L ∞ (Λ) = k − (cid:107) η (cid:48)(cid:48) (cid:107) L ∞ ( R ) −→ , φ ∈ H (Λ) φ (cid:54) =0 (cid:26) (cid:107) φ (cid:107) + (cid:90) Λ f ε (cid:12)(cid:12)(cid:12)(cid:12) ∂ s φ − ∂ s f ε f ε φ (cid:12)(cid:12)(cid:12)(cid:12) | ∂ s η k || ξ n k | d s d t (cid:27) ≤ (cid:107) ∂ s η k (cid:107) L ∞ (Λ) = k − (cid:107) η (cid:48) (cid:107) L ∞ ( R ) −→ , as k → ∞ . Finally, since β, β (cid:48) ∈ L ∞ ( R ), we havesup φ ∈ H (Λ) φ (cid:54) =0 (cid:26) (cid:107) φ (cid:107) + (cid:90) Λ (cid:12)(cid:12)(cid:12)(cid:12) ∂ s f ε f ε (cid:12)(cid:12)(cid:12)(cid:12) | φ || ∂ s η k || ξ n k | d s d t (cid:27) ≤ (cid:107) ∂ s η k (cid:107) L ∞ (Λ) = k − (cid:107) η (cid:48) (cid:107) L ∞ ( R ) −→ , as k → ∞ . Then, (28) holds true. Remark 5.
The same conclusion of Lemma 3 holds true for the operator D ε . Proof of Proposition 2.
Let λ ∈ σ ess ( D ε ), then there exists a sequence { ψ n } n ∈ N ⊂ dom d ε so that (cid:107) ψ n (cid:107) = 1 , supp ψ n ⊂ Λ \ ( − n, n ) × ( − , , for all n ∈ N , and ( D ε − λ ) ψ n → , as n → ∞ , in the norm ofthe dual space (dom d ε ) ∗ . Recall supp β ⊂ [ − s , s ], for some s >
0. Take n ∈ N so that n > s . Then,for each φ ∈ H (Λ), c ε ( φ, ψ n ) − λ (cid:104) φ, ψ n (cid:105) = d ε ( φ, ψ n ) − λ (cid:104) φ, ψ n (cid:105) , for all n ≥ n . Consequently, sup φ ∈ H (Λ) φ (cid:54) =0 | c ε ( φ, ψ n ) − λ (cid:104) φ, ψ n (cid:105)|(cid:107) φ (cid:107) + −→ , as n → ∞ . The Lemma 3 implies that λ ∈ σ ess ( C ε ). In a similar way, it is possible to show that σ ess ( C ε ) ⊂ σ ess ( D ε ). Acknowledgments
Rafael T. Amorim was supported by CNPq (Brazil) through the process: 141842/2019-9.
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