A note on the three dimensional Dirac operator with zigzag type boundary conditions
aa r X i v : . [ m a t h . SP ] J un A NOTE ON THE THREE DIMENSIONAL DIRAC OPERATORWITH ZIGZAG TYPE BOUNDARY CONDITIONS
MARKUS HOLZMANN
Dedicated with great pleasure to my teacher, colleague, and friendHenk de Snoo on the occasion of his 75th birthday.
Abstract.
In this note the three dimensional Dirac operator A m with bound-ary conditions, which are the analogue of the two dimensional zigzag boundaryconditions, is investigated. It is shown that A m is self-adjoint in L (Ω; C ) forany open set Ω ⊂ R and its spectrum is described explicitly in terms of thespectrum of the Dirichlet Laplacian in Ω. In particular, whenever the spec-trum of the Dirichlet Laplacian is purely discrete, then also the spectrum of A m consists of discrete eigenvalues that accumulate at ±∞ and one additionaleigenvalue of infinite multiplicity. Introduction
In the recent years Dirac operators with boundary conditions, which make themself-adjoint, gained a lot of attention. From the physical point of view, they appearin various applications such as in the description of relativistic particles that areconfined in a box Ω ⊂ R ; in this context the MIT bag model is a particularly in-teresting example, cf. [2]. Moreover, in space dimension two the spectral propertiesof self-adjoint massless Dirac operators play an important role in the mathematicaldescription of graphene, see, e.g., [7] and the references therein. On the other hand,from the mathematical point of view, self-adjoint Dirac operators with boundaryconditions are viewed as the relativistic counterpart of Laplacians with Robin typeand other boundary conditions.To set the stage, let Ω ⊂ R be an open set, let σ := (cid:18) (cid:19) , σ := (cid:18) − ii (cid:19) , σ := (cid:18) − (cid:19) , (1.1)be the Pauli spin matrices, and let α j := (cid:18) σ j σ j (cid:19) and β := (cid:18) I − I (cid:19) , (1.2)be the C × -valued Dirac matrices, where I d denotes the identity matrix in C d × d .For m ∈ R we introduce the differential operator τ m acting on distributions by τ m := − i X j =1 α j ∂ j + mβ =: − iα · ∇ + mβ. (1.3) Mathematics Subject Classification.
Primary 81Q10; Secondary 35Q40.
Key words and phrases.
Dirac operator, boundary conditions, spectral theory, eigenvalue ofinfinite multiplicity.
The main goal in this short note is to study the self-adjointness and the spectralproperties of the Dirac operator A m in L (Ω; C ) := L (Ω) ⊗ C which acts as τ m on functions f = ( f , f , f , f ) ∈ L (Ω; C ) which satisfy τ m f ∈ L (Ω; C ) and theboundary conditions f | ∂ Ω = f | ∂ Ω = 0 (1.4)in a suitable sense specified below. Note that no boundary conditions are imposedfor the components f and f . If m >
0, then the solution of the evolution equationwith Hamiltonian A m describes the propagation of a quantum particle with mass m and spin in Ω taking these boundary conditions and relativistic effects intoaccount.The motivation to study the operator A m is twofold. Firstly, in the recent paper[5] Dirac operators in L (Ω; C ) acting as τ m on function satisfying the boundaryconditions ϑ (cid:0) I + iβ ( α · ν ) (cid:1) f | ∂ Ω = (cid:0) I + iβ ( α · ν ) (cid:1) βf | ∂ Ω (1.5)were studied in the case that m > C -domain with compactboundary and unit normal vector field ν ; in (1.5) the convention α · x = α x + α x + α x is used for x = ( x , x , x ) ∈ R . The authors were able to provethe self-adjointness and to derive the basic spectral properties of these operators,whenever the parameter ϑ appearing in (1.5) is a real-valued H¨older continuousfunction of order a > satisfying ϑ ( x ) = ± x ∈ ∂ Ω, and it is shown thatthe domain of definition of these self-adjoint operators is contained in the Sobolevspace H (Ω; C ). For bounded domains Ω this implies, in particular, that thespectrum is purely discrete. The case when ϑ ( x ) = ± x ∈ ∂ Ω remainedopen and it is conjectured that different spectral properties should appear. Wenote that ϑ ≡ ϑ = 1 were studied in 3D in[1, 2, 12] and in 2D in [6, 7, 10, 11].The second main motivation for this study is the paper [14], where the twodimensional counterpart of A m was investigated in the massless case ( m = 0). Itwas shown in [14] that the two dimensional Dirac operator with similar boundaryconditions as in (1.4), which are known as zigzag boundary conditions, is self-adjoint on a domain which is in general not contained in H (Ω) and that for anybounded domain Ω zero is an eigenvalue of infinite multiplicity. In particular, thespectrum of the operator is not purely discrete. Let us mention here that the twodimensional zigzag boundary conditions have a physical relevance, as they appearin the description of graphene quantum dots, when a lattice in this quantum dot isterminated and the direction of the boundary is perpendicular to the bonds [9].The goal in the present note is to prove similar and even more explicit resultsas those in [14] also in the three dimensional setting, which complement then theresults from [5] in the critical case ϑ ≡ ϑ ≡ − − A m and hence this case is also contained in the analysis inthis note. In the formulation of the following main result of the present paper wedenote by − ∆ D the self-adjoint realization of the Dirichlet Laplacian in L (Ω). Theorem 1.1.
The operator A m is self-adjoint in L (Ω; C ) and its spectrum is σ ( A m ) = { m } ∪ n ± p λ + m : λ ∈ σ ( − ∆ D ) o . The value m always belongs to the essential spectrum of A m , while for m = 0 thenumber − m is not an eigenvalue of A m . Theorem 1.1 is proved in a series of results in Section 3. It gives a full descriptionof the spectrum of A m in terms of the spectrum of the Dirichlet Laplacian inΩ, which is well-studied in many cases. For bounded domains Ω it follows fromthe Rellich embedding theorem that the spectrum of − ∆ D is purely discrete andtherefore, the spectrum of A m consists of an infinite sequence of discrete eigenvaluesaccumulating at ±∞ and the eigenvalue m , which has infinite multiplicity. Inparticular, the essential spectrum of A m is not empty, which is in contrast to thecase of non-critical boundary values in [5]. Moreover, if Ω is a bounded Lipschitzdomain, then the non-emptiness of the essential spectrum implies that the domainof A m is not contained in the Sobolev space H s (Ω; C ) for any s > A m may look like. On the one hand it isknown that for some special horn shaped domains Ω, which have infinite measure,the spectrum of − ∆ D is purely discrete, cf. [13, 15]. Therefore, by Theorem 1.1also in this case the spectrum of A m consists only of eigenvalues and it followsfrom the spectral theorem that the multiplicity of m is again infinite. On the otherhand, for many unbounded domains it is known that σ ( − ∆ D ) = [0 , ∞ ) and thus, σ ( A m ) = ( −∞ , −| m | ] ∪ [ | m | , ∞ ) for such Ω. The simplest example for this case iswhen Ω is the complement of a bounded domain.Let us finally collect some basic notations that are frequently used in this note.If not stated differently Ω is an arbitrary open subset of R . For n ∈ N we write L (Ω; C n ) := L (Ω) ⊗ C n . The inner product and the norm in L (Ω; C n ) aredenoted by ( · , · ) and k · k , respectively. We use for k ∈ N the symbol H k (Ω) forthe L -based Sobolev spaces of k times weakly differentiable functions and H (Ω)for the closure of the test functions C ∞ (Ω) in H (Ω). For a linear operator A itsdomain is dom A and its Hilbert space adjoint is denoted by A ∗ . If A is a closedoperator, then σ ( A ) is the spectrum of A , and if A is self-adjoint, then its essentialspectrum is σ ess ( A ). 2. Some auxiliary operators
In this section we introduce and discuss two auxiliary operators T min and T max in L (Ω; C ) which will be useful to study the Dirac operator A m with zigzag typeboundary conditions. Let Ω ⊂ R be an arbitrary open set and let σ = ( σ , σ , σ )be the Pauli spin matrices defined by (1.1). In the following we will often use thenotation σ · ∇ = σ ∂ + σ ∂ + σ ∂ . We define the set D max ⊂ L (Ω; C ) by D max := (cid:8) f ∈ L (Ω; C ) : ( σ · ∇ ) f ∈ L (Ω; C ) (cid:9) , where the derivatives are understood in the distributional sense, and the operators T max and T min acting in L (Ω; C ) by T max f := − i ( σ · ∇ ) f, dom T max = D max , (2.1) M. HOLZMANN and T min := T max ↾ H (Ω; C ), which has the more explicit representation T min f := − i ( σ · ∇ ) f, dom T min = H (Ω; C ) . (2.2)In the following lemma we summarize the basic properties of T min and T max . Lemma 2.1.
The operators T min and T max are both closed and adjoint to eachother, i.e. T ∗ min = T max . Moreover, the inclusion D max ⊂ H (Ω; C ) holds.Proof. The facts that T min and T max are closed and adjoint to each other are simpleto obtain by replacing α · ∇ by σ · ∇ in the proof of [4, Proposition 3.1] or [12,Proposition 2.10], see also [6, Lemma 2.1] for similar arguments. Furthermore, D max ⊂ H (Ω; C ) can be proved similarly as [14, Proposition 1]. (cid:3) In the next lemma it is shown that T min does not have eigenvalues. If Ω is a C -domain with compact boundary, then this would follow from the simplicity of T min , which can be proved in the same way as for the minimal Dirac operator inΩ in [5, Proposition 3.2]. However, for our purposes also the weaker statement ofabsence of eigenvalues is sufficient. Lemma 2.2.
The operator T min does not have eigenvalues.Proof. Assume that λ is an eigenvalue of T min with corresponding eigenfunction f = 0 and let ϕ ∈ C ∞ ( R ; C ) be arbitrary. In the following we will denote by − ∆ the free Laplace operator on R , which is defined on H ( R ; C ), and by e f theextension of f by zero onto R . Note that σ j σ k + σ k σ j = 2 δ jk I holds by thedefinition of the Pauli matrices in (1.1) and hence, ( σ · ∇ ) ϕ = ∆ ϕ . Using this and f, T min f = λf ∈ dom T min = H (Ω; C ) we find that Z R e f · ( − ∆ ϕ )d x = Z Ω f · ( iσ · ∇ )( iσ · ∇ ) ϕ d x = Z Ω ( − iσ · ∇ ) f · ( iσ · ∇ ) ϕ d x = Z Ω ( − iσ · ∇ )( − iσ · ∇ ) f · ϕ d x = Z Ω λ f · ϕ d x = Z R λ e f · ϕ d x, i.e. − ∆ e f = λ e f in L ( R ; C ). Therefore, we conclude that e f is an eigenfunctionof − ∆ and hence, e f = 0. This is a contradiction to the assumption and completesthe proof of this lemma. (cid:3) Eventually, we show that 0 always belongs to the spectrum of T max . This resultwill be of importance to prove that m is in the essential spectrum of A m . Proposition 2.3.
There exists a sequence ( f n ) ⊂ dom T max with k f n k = 1 converg-ing weakly to zero such that T max f n → , as n → ∞ . In particular, ∈ σ ( T max ) .Proof. We distinguish two cases for Ω. First, assume that g n ( x ) := ( x + ix ) n , x = ( x , x , x ) ∈ Ω, belongs to L (Ω; C ) for all n ∈ N . Then, we follow ideas from[14, Proposition 2] and see that the functions f n := 1 k g n k (cid:18) g n (cid:19) ∈ L (Ω; C )fulfil ( σ · ∇ ) f n = 0, i.e. f n ∈ ker T max . Hence, zero is an eigenvalue of infinitemultiplicity, which implies immediately the claim. In the other case, when g n ( x ) = ( x + ix ) n , x = ( x , x , x ) ∈ Ω, does notbelong to L (Ω; C ) for some n ∈ N , we follow ideas from the appendix of [8], whereit is shown that zero always belongs to the essential spectrum of the NeumannLaplacian in L ( G ; C ), when the domain G has infinite measure, to construct thesequence ( f n ). Let k ∈ N be the smallest number such that g k / ∈ L (Ω; C ). Definethe Borel measure µ acting on Borel sets B ⊂ R as µ ( B ) := Z B | g k ( x ) | d x = Z B ( x + x ) k d x and the sets Ω n := { x ∈ Ω : | x | ≤ n } . Then by assumption µ (Ω) = ∞ and µ (Ω n ) ≤ cn k +3 . Next, define for n ∈ N the functions h n ( x ) := g k ( x ) , x ∈ Ω n − , ( n − | x | ) g k ( x ) , x ∈ Ω n \ Ω n − , , x ∈ Ω \ Ω n , and f n := 1 k h n k (cid:18) h n (cid:19) ∈ H (Ω; C ) . Because of ( σ · ∇ ) g k = 0 one has kT max f n k = kT max h n k k h n k ≤ µ (Ω n \ Ω n − ) µ (Ω n − ) =: α n . We claim that lim inf n →∞ α n = 0, which implies that there exists a subsequence of( f n ), that is still denoted by ( f n ), converging weakly to zero (as k h n k → µ (Ω) = ∞ for n → ∞ ) such that kT max f n k →
0, as n → ∞ , and hence the claim of thisproposition also in the second case.If lim inf n →∞ α n = 0, then there exists α > α n ≥ α for almost all n ∈ N . In particular, this implies µ (Ω n ) = µ (Ω n \ Ω n − ) + µ (Ω n − ) µ (Ω n − ) µ (Ω n − ) ≥ (1 + α ) µ (Ω n − )and, by repeating this argument, µ (Ω n ) ≥ e c (1 + α ) n − for a constant e c > µ (Ω n ) ≤ cn k +3 . Thus, lim inf n →∞ α n = 0and all claims have been shown. (cid:3) Definition of A m and its spectral properties This section is devoted to the study of the operator A m and the proof of themain result of this note, Theorem 1.1. First, we introduce A m rigorously and showits self-adjointness, then we investigate its spectral properties.Let Ω ⊂ R be an arbitrary open set and let T max and T min be the operatorsdefined in (2.1) and (2.2), respectively. We define for m ∈ R the Dirac operator A m with zigzag type boundary conditions, which acts in L (Ω; C ), by A m = (cid:18) mI T min T max − mI (cid:19) . (3.1)The operator in (3.1) is the rigorous mathematical definition of the expressionin (1.3) with the boundary conditions (1.4). We note that Lemma 2.1 implies thatdom A m ⊂ H (Ω; C ). M. HOLZMANN
Remark . If Ω is a C -domain with compact boundary, then there exists a Dirich-let trace operator on D max and one can show with the help of [12, Propositions 2.1and 2.16] that the expressions in (1.3)–(1.4) and (3.1) indeed coincide.Before we start analyzing A m we remark that this operator is unitarily equivalentwith the operator − B m , where B m is defined by B m = (cid:18) mI T max T min − mI (cid:19) . Note that B m is the Dirac operator acting on spinors f = ( f , f , f , f ) ∈ L (Ω; C )satisfying the boundary conditions f | ∂ Ω = f | ∂ Ω = 0. In particular, the followingLemma 3.2 shows that all results which are proved in this paper for A m can besimply translated to corresponding results for B m . In order to formulate the lemmawe recall the definition of the Dirac matrix β from (1.2), define the matrix γ = (cid:18) I I (cid:19) , and note that βγ is a unitary matrix. Lemma 3.2.
Set M := βγ . Then B m = −M A m M holds. In particular, B m isunitarily equivalent to − A m . Now we start analyzing A m . With the help of Lemma 2.1 it is not difficult toshow that A m is self-adjoint. In the proof, we use in a similar way as in [14] thatthe operator A (i.e. A m for m = 0) has a supersymmetric structure. Theorem 3.3.
The operator A m is self-adjoint in L (Ω; C ) .Proof. We use for f ∈ L (Ω; C ) the splitting f = ( f , f ) with f , f ∈ L (Ω; C ),that means f and f are the upper and lower two components of the Dirac spinor,respectively. It suffices to consider m = 0, as mβ is a bounded self-adjoint pertur-bation. Moreover, we note that T max = T ∗ min and T ∗ max = T min hold by Lemma 2.1;this will be used several times throughout this proof.First we show that A is symmetric. Indeed for f = ( f , f ) ∈ dom A a simplecalculation shows( A f, f ) = ( T min f , f ) + ( T max f , f ) = 2 Re ( T min f , f ) ∈ R . Next, one has for f = ( f , f ) ∈ dom A ∗ and g = ( g , g ) ∈ dom A ( A ∗ f, g ) = ( f, A g ) = ( f , T min g ) + ( f , T max g ) . (3.2)Choosing g = 0 we get from (3.2) (cid:0) ( A ∗ f ) , g (cid:1) = ( f , T min g )for all g ∈ dom T min and hence f ∈ dom T max and T max f = ( A ∗ f ) . Similarly,choosing g = 0 we obtain from (3.2) that f ∈ dom T min and T min f = ( A ∗ f ) .Therefore, we conclude f ∈ dom A and A ∗ f = A f , that means A ∗ ⊂ A . Thisfinishes the proof of this theorem. (cid:3) In the following theorem we state the spectral properties of A m . We will see thatthey are closely related to the spectral properties of the Dirichlet Laplacian − ∆ D , which is the self-adjoint operator in L (Ω; C ) that is associated to the closed andnon-negative sesquilinear form a D [ f, g ] := Z Ω ∇ f · ∇ g d x, dom a D = H (Ω; C ) . (3.3) Theorem 3.4.
For any m ∈ R the following is true: (i) All eigenvalues of A m have even multiplicity. (ii) m ∈ σ ess ( A m ) . (iii) If m = 0 , then − m is not an eigenvalue of A m . (iv) Let − ∆ D be the Dirichlet Laplacian on Ω . Then σ ( A m ) = { m } ∪ n ± p λ + m : λ ∈ σ ( − ∆ D ) o . In particular, σ ( A m ) ∩ ( −| m | , | m | ) = ∅ . We note that Theorem 3.4 applied for m = 0 shows that the spectrum of A issymmetric w.r.t. λ = 0. This observation would also follow from the stronger factthat A = − βA β , i.e. A is unitarily equivalent to − A . Proof of Theorem 3.4. (i) Consider the nonlinear time reversal operator
T f := − iγ α f , f ∈ L ( R ; C ) , γ := (cid:18) I I (cid:19) . One has f ∈ dom A m if and only if T f ∈ dom A m and T f = − f . Let λ ∈ σ p ( A m )and let f λ be a corresponding eigenfunction. Then, one can show in the same wayas in [3, Proposition 4.2 (ii)] that also T f λ is a linearly independent eigenfunctionof A m for the eigenvalue λ . This shows the claim of statement (i).(ii) According to Proposition 2.3 there exists a sequence ( g n ) ⊂ dom T max , whichconverges weakly to zero, such that k g n k = 1 and T max g n →
0, as n →
0. Define f n := ( g n , ∈ L (Ω; C ), i.e. the first two components of f n ∈ L (Ω; C ) are g n ∈ L (Ω; C ) and the last two components are zero. Then f n ∈ dom A m , k f n k = 1,( f n ) converges weakly to zero, and( A m − m ) f n = (cid:18) T min T max − m (cid:19) (cid:18) g n (cid:19) = (cid:18) T max g n (cid:19) → , as n → ∞ . Hence ( f n ) is a singular sequence for A m and λ = m and thus, m ∈ σ ess ( A m ).(iii) Assume that − m ∈ σ p ( A m ) and that f is a nontrivial eigenfunction. Ac-cording to the definition of A m we can write f = ( f , f ) with f ∈ D max and f ∈ dom T min = H (Ω; C ). Then (cid:18) (cid:19) = ( A m + m ) f = (cid:18) mf T min f T max f (cid:19) , i.e. T max f = 0 and T min f = − mf . With Lemma 2.1 this implies − m k f k = ( T min f , f ) = ( f , T max f ) = 0 , i.e. f = 0. Thus, f ∈ ker T min and since T min has no eigenvalues by Lemma 2.2,we conclude that also f = 0. Therefore, we have shown f = ( f , f ) = 0, i.e. − m / ∈ σ p ( A m ). M. HOLZMANN (iv) First, we prove the inclusion σ ( A m ) ⊂ { m } ∪ n ± p λ + m : λ ∈ σ ( − ∆ D ) o . For this, due to the results from items (ii) and (iii), it suffices to prove σ ( A m ) = { m } ∪ (cid:8) λ + m : λ ∈ σ ( − ∆ D ) (cid:9) . (3.4)A simple direct calculation shows that A m = (cid:18) m + T min T max m + T max T min (cid:19) . Hence, σ ( A m ) = σ ( m + T min T max ) ∪ σ ( m + T max T min )= { m + λ : λ ∈ σ ( T min T max ) ∪ σ ( T max T min ) } . Since T max = T ∗ min , we have σ ( T min T max ) ∪ { } = σ ( T max T min ) ∪ { } and thus, as m ∈ σ ( A m ) by (ii), σ ( A m ) = (cid:8) m + λ : λ ∈ σ ( T max T min ) ∪ { } (cid:9) . We claim that σ ( T max T min ) = σ ( − ∆ D ). To see this, we note that T max T min is theunique self-adjoint operator corresponding to the quadratic form a [ f ] := kT min f k , f ∈ dom a = dom T min = H (Ω; C ) . Since σ j σ k + σ k σ j = 2 δ jk I holds by the definition of the Pauli matrices in (1.1),we have for f ∈ C ∞ (Ω; C ) a [ f ] = (cid:0) − i ( σ · ∇ ) f, − i ( σ · ∇ ) f (cid:1) = (cid:0) f, − ( σ · ∇ ) f (cid:1) = (cid:0) f, − ∆ f (cid:1) = k∇ f k , which extends by density to all f ∈ dom a = H (Ω; C ). Therefore, a is thequadratic form associated to − ∆ D and hence, by the first representation theoremwe conclude T max T min = − ∆ D . This implies (3.4).To prove the converse inclusion { m } ∪ n ± p λ + m : λ ∈ σ ( − ∆ D ) o ⊂ σ ( A m ) , (3.5)we note first that m ∈ σ ( A m ) and, if m = 0, that − m / ∈ σ p ( A m ) by (ii) and (iii).Assume that 0 = λ ∈ σ ( − ∆ D ). Then there exists a sequence ( u n ) ⊂ dom ( − ∆ D ) ⊂ H (Ω; C ) such that k u n k = 1 for all n ∈ N and ( − ∆ D − λ ) u n →
0, as n → ∞ . Set g n := ( u n , u n ) ∈ L (Ω; C ) and define f n := (cid:18) m + √ m + λ − iσ · ∇− iσ · ∇ − m + √ m + λ (cid:19) (cid:18) g n (cid:19) = (cid:18) − i ( σ · ∇ ) g n ( − m + √ m + λ ) g n (cid:19) . (3.6)Since ( − iσ · ∇ )( − iσ · ∇ ) g n = − ∆ D g n ∈ L (Ω; C ) holds in the distributional senseand g n ∈ dom ( − ∆ D ) ⊂ H (Ω; C ), we conclude that f n ∈ dom A m . Moreover, k f n k ≥ √ m + λ − m ) ≥ c > n ∈ N and a direct calculation shows (cid:0) A m − p m + λ (cid:1) f n = (cid:18) − ∆ D − λ ) g n (cid:19) → , as n → ∞ . Hence, √ m + λ ∈ σ ( A m ). In a similar manner, one verifies that thesequence ( e f n ) defined by e f n := (cid:18) m − √ m + λ − iσ · ∇− iσ · ∇ − m − √ m + λ (cid:19) (cid:18) g n (cid:19) = (cid:18) − i ( σ · ∇ ) g n − ( m + √ m + λ ) g n (cid:19) (3.7) fulfils k e f n k ≥ c > n and ( A m + √ m + λ ) e f n →
0, as n → ∞ .Hence, also −√ m + λ ∈ σ ( A m ). This shows that also (3.5) is true.Finally, since − ∆ D is a nonnegative operator, we conclude from (3.4) that σ ( A m ) ⊂ [ m , ∞ ) and hence, σ ( A m ) ∩ ( −| m | , | m | ) = ∅ . (cid:3) Let us end this note with a short discussion of the spectral properties of A m for some special domains Ω and some consequences of that. In many situationsit is known that the Dirichlet Laplacian has purely discrete spectrum. Then, byTheorem 3.4 (iv) also the spectrum of A m consists only of eigenvalues and, as aconsequence of the spectral theorem, m is an eigenvalue of infinite multiplicity.Moreover, in a similar way as in (3.6) and (3.7) one can construct eigenfunctionsof A m . The spectrum of the Dirichlet Laplacian is purely discrete, e.g., when Ωis a bounded subset of R , as then the space H (Ω; C ) is compactly embeddedin L (Ω; C ) by the Rellich embedding theorem, and hence the Dirichlet Laplacian − ∆ D associated to the sesquilinear form a D in (3.3) has a compact resolvent. Inthis situation let us denote by 0 < µ D ≤ µ D ≤ µ D ≤ . . . the discrete eigenvalues of − ∆ D , where multiplicities are taken into account. Then one immediately has thefollowing result. Corollary 3.5.
Let Ω ⊂ R be such that σ ( − ∆ D ) is purely discrete. Then σ ( A m ) = { m } ∪ n ± q m + µ Dk : k ∈ N o and m is an eigenvalue of infinite multiplicity. If the Sobolev space H s (Ω; C ) is compactly embedded in L (Ω; C ) for some s > A m can not be contained in H s (Ω; C ) forany s >
0. This is, e.g., the case, when Ω is a bounded Lipschitz domain.
Corollary 3.6.
Assume that Ω is a bounded subset of R with a Lipschitz-smoothboundary. Then dom A m H s (Ω; C ) for all s > . Acknowledgments.
The author thanks Jussi Behrndt, Andrii Khrabustovskyi,and Peter Schlosser for helpful discussions.
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