Deriving a complete set of eigendistributions for a gravitational wave equation describing the quantized interaction of gravity with a Yang-Mills field in case the Cauchy hypersurface is non-compact
aa r X i v : . [ g r- q c ] M a y DERIVING A COMPLETE SET OF EIGENDISTRIBUTIONSFOR A GRAVITATIONAL WAVE EQUATION DESCRIBINGTHE QUANTIZED INTERACTION OF GRAVITY WITH AYANG-MILLS FIELD IN CASE THE CAUCHYHYPERSURFACE IS NON-COMPACT
CLAUS GERHARDT
Abstract.
In a recent paper we quantized the interaction of gravitywith a Yang-Mills and Higgs field and obtained as a result a gravi-tational wave equation in a globally hyperbolic spacetime. Assumingthat the Cauchy hypersurfaces are compact we proved a spectral res-olution for the wave equation by applying the method of separationof variables. In this paper we extend the results to the case when theCauchy hypersurfaces are non-compact by considering a Gelfand tripletand applying the nuclear spectral theorem.
Contents
1. Introduction 12. The nuclear spectral theorem 43. The eigendistributions are smooth functions 84. The positivity of the eigenvalues 11References 151.
Introduction
In a recent paper [3] we quantized the interaction of gravity with a Yang-Mills and Higgs field and obtained as a result a gravitational wave equationof the form(1.1) 132 n n − u − ( n − t − n ∆u − n t − n Ru + α n t − n F ij F ij u + α n t − n γ ab σ ij Φ ai Φ bi u + α n mt − n V ( Φ ) u + nt Λu = 0 , Date : October 4, 2018.2000
Mathematics Subject Classification.
Key words and phrases. unified field theory, quantization of gravity, quantum gravity,Yang-Mills fields, eigendistributions, Gelfand triple, nuclear spectral theorem, mass gap. in a globally hyperbolic spacetime(1.2) Q = (0 , ∞ ) × S describing the interaction of a given complete Riemannian metric σ ij in S with a given Yang-Mills and Higgs field; R is the scalar curvature of σ ij , V isthe potential of the Higgs field, m a positive constant, α , α are positive cou-pling constants and the other symbols should be self-evident. The existenceof the time variable, and its range, is due to the quantization process.1.1. Remark.
For the results and arguments in that paper it was com-pletely irrelevant that the values of the Higgs field Φ lie in a Lie algebra, i.e., Φ could also be just an arbitrary scalar field, or we could consider a Higgsfield as well as an another arbitrary scalar field. Hence, let us stipulate thatthe Higgs field could also be just an arbitrary scalar field.If S is compact we also proved a spectral resolution of equation (1.1) byfirst considering a stationary version of the hyperbolic equation, namely, theelliptic eigenvalue equation(1.3) − ( n − ∆v − n Rv + α n F ij F ij v + α n γ ab σ ij Φ ai Φ bi v + α n mV ( Φ ) v = µv. It has countably many solutions ( v i , µ i ) such that(1.4) µ < µ ≤ µ ≤ · · · , (1.5) lim µ i = ∞ . Let v be an eigenfunction with eigenvalue µ >
0, then we looked at solutionsof (1.1) of the form(1.6) u ( x, t ) = w ( t ) v ( x ) .u is then a solution of (1.1) provided w satisfies the implicit eigenvalue equa-tion(1.7) − n n − w − µt − n w − nt Λw = 0 , where Λ is the eigenvalue.This eigenvalue problem we also considered in a previous paper and provedthat it has countably many solutions ( w i , Λ i ) with finite energy, i.e.,(1.8) Z ∞ {| ˙ w i | + (1 + t + µt − n ) | w i | } < ∞ . More precisely, we proved, cf. [2, Theorem 6.7],
COMPLETE SET OF EIGENDISTRIBUTIONS 3
Theorem.
Assume n ≥ and S to be compact and let ( v, µ ) be asolution of the eigenvalue problem (1.3) with µ > , then there exist countablymany solutions ( w i , Λ i ) of the implicit eigenvalue problem (1.7) such that (1.9) Λ i < Λ i +1 < · · · < , (1.10) lim i Λ i = 0 , and such that the functions (1.11) u i = w i v are solutions of the wave equation (1.1) . The transformed eigenfunctions (1.12) ˜ w i ( t ) = w i ( λ n n − i t ) , where (1.13) λ i = ( − Λ i ) − n − n , form a basis of L ( R ∗ + , C ) and also of the Hilbert space H defined as thecompletion of C ∞ c ( R ∗ + , C ) under the norm of the scalar product (1.14) h w, ˜ w i = Z ∞ { ¯ w ′ ˜ w ′ + t ¯ w ˜ w } , where a prime or a dot denotes differentiation with respect to t . In this paper we want to extend this spectral resolution to the case when S is non-compact. Denote by A the elliptic differential operator on the left-handside of (1.3), then, assuming that its coefficients are smooth with bounded C m -norms for any m ∈ N , we have a self-adjoint operator in H = L ( S ) anda Gelfand triplet(1.15) S ⊂ H ⊂ S ′ such that we can apply the nuclear spectral theorem of Gelfand-Maurin lead-ing to a complete set of eigendistributions(1.16) f ( λ ) ∈ S ′ , λ ∈ Λ, of A , where Λ is a measure space. For almost every λ ∈ Λ we have 0 = f ( λ )and f ( λ ) is a solution of the eigenvalue equation(1.17) Af ( λ ) = a ( λ ) f ( λ )where(1.18) a : Λ → σ ( A )is a measurable function having σ ( A ) as its essential range. Since the f ( λ )are distributions and A is uniformly elliptic and smooth, the f ( λ ) are alsosmooth, and since they are also tempered distributions we could prove thatthe eigenvalues satisfy(1.19) a ( λ ) > CLAUS GERHARDT for a.e. λ . Hence, the separation of variables, described in (1.6), can beapplied with an eigenfunction v be replaced by an eigendistribution f . Sinceall eigenvalues a ( λ ) are strictly positive this can be considered to be a spectralresolution of the wave equation. The smooth functions(1.20) u i = w i f ( λ )are classical solutions of the wave equation (1.1) with bounded temporalenergy and locally bounded spatial energy.2. The nuclear spectral theorem
We assume that ( S , σ ij ) is complete and that there exists a compactsubset K ⊂ S and a chart ( U , x ) such that(2.1) S \ K ⊂ U and(2.2) Ω = x ( U ) = R n \ ¯ B R (0) . Moreover, we require2.1.
Assumption. (i) The metric σ ij and the lower order coefficients ofthe elliptic operator on the left-hand side of equation of the equation (1.3)on page 2 are smooth with bounded C m -norms for any m ∈ N . We call theelliptic operator A .(ii) The metric σ ij is uniformly elliptic.The last assumption implies that the radial distance from a center x ∈ K ,(2.3) r ( x ) = d ( x, x ) , and the Euclidean distance | x | are equivalent in Ω , i.e., there are constants c , c such that(2.4) r ( x ) ≤ c | x | ≤ c r ( x ) ∀ x ∈ Ω and hence the Schwartz space of rapidly decreasing test functions in S canbe identified with the Schwartz space in R n . We shall denote the Schwartzspace by(2.5) S = S ( S )and its dual space, the tempered distributions, by(2.6) S ′ = S ′ ( S ) . The topology of S is defined by a sequence of norms(2.7) | ϕ | m,k = sup x ∈S (1 + r ( x ) ) k X | α |≤ m | D α ϕ ( x ) | . S is a Fr´echet space and also a nuclear space, cf. [8, Example 5, p. 107]. Thedifferential operator A defined by the left-hand side of (1.3) on page 2 is a COMPLETE SET OF EIGENDISTRIBUTIONS 5 continuous map from S into S in view of Assumption 2.1. A is a self-adjointlinear operator in L ( S , C ) and(2.8) S ⊂ D ( A )a dense subspace. By duality A can also be defined on the dual space S ′ ,namely, let f ∈ S ′ , then(2.9) h Af, ϕ i = h f, Aϕ i ∀ ϕ ∈ S , where the self-adjointness of A has been used.2.2. Definition. f ∈ S ′ is said to be an eigendistribution of A witheigenvalue µ ∈ R , i.e.,(2.10) Af = µf, iff(2.11) h Af, ϕ i = µ h f, ϕ i ∀ ϕ ∈ S , or equivalently, iff(2.12) h f, Aϕ i = µ h f, ϕ i ∀ ϕ ∈ S . A setting where we have a self-adjoint operator A in a separable Hilbertspace H , a dense subspace(2.13) E ⊂ H which is also a nuclear space (in a finer topology) with dual space E ′ suchthat(2.14) E ⊂ H ⊂ E ′ , where the imbedding of E into H is continuous, a property which we alreadyspecified by speaking of a finer topology, and where, moreover,(2.15) A : E → E is continuous, is known as a rigged Hilbert space setting, though, usually, E ′ is replaced by the space of antilinear functionals. However, since we do notuse Dirac’s ket notation, we shall consider E ′ .In such a framework a nuclear spectral theorem has been proved byGelfand and Maurin, cf. [1, Theorem 5’, p. 126], [5, Satz 2] and [6, Chap.XVIII, p. 333] which we shall formulate and prove for a single self-adjointoperator A and not for a family of strongly commuting operators. The proofclosely follows the one given by Maurin in [5, Satz 2]. Since this paper is writ-ten in German we like to include a proof for the convenience of the reader.2.3. Theorem (Maurin) . Let H be a separable complex Hilbert space, A a densely defined self-adjoint operator, E ⊂ H a dense subspace which alsocarries topology such that it is a nuclear space and assume that the imbedding CLAUS GERHARDT in (2.13) and the map A in (2.15) are continuous, then there exists a locallycompact measure space Λ , a finite positive measure µ , a measurable function (2.16) a : Λ → σ ( A ) ⊂ R , and a unitary operator (2.17) U : H → L ( Λ, C , µ ) such that, if we set (2.18) ˆ u = U u, ∀ u ∈ H, (2.19) ˆ A = U AU − , we have (2.20) u ∈ D ( A ) ⇐⇒ a ˆ u ∈ L ( Λ, µ ) , (2.21) ˆ A ˆ u = a ˆ u ∀ u ∈ D ( A ) and for µ a.e. λ ∈ Λ the mapping (2.22) f ( λ ) : ϕ ∈ E → ˆ ϕ ( λ ) ∈ C is continuous in E and does not vanish identically, i.e., (2.23) 0 = f ( λ ) ∈ E ′ and hence (2.24) ˆ ϕ ( λ ) = h f ( λ ) , ϕ i ∀ ϕ ∈ E. Moreover, (2.21) implies (2.25) h f ( λ ) , Aϕ i = ˆ A ˆ ϕ ( λ ) = a ( λ ) ˆ ϕ ( λ ) = a ( λ ) h f ( λ ) , ϕ i ∀ ϕ ∈ E for a.e. λ ∈ Λ , or equivalently, (2.26) Af ( λ ) = a ( λ ) f ( λ ) for a.e. λ ∈ Λ. The generalized eigenvectors f ( λ ) are complete, since (2.27) k ϕ k = k ˆ ϕ k = Z Λ | ˆ ϕ ( λ ) | dµ ∀ ϕ ∈ E, and hence, (2.28) ˆ ϕ ( λ ) = 0 for a.e. λ ∈ Λ, is equivalent to ϕ = 0 .Proof. The first part of the theorem is due to the multiplicative form of thespectral theorem, cf. [7, Theorem VIII.4, p. 260]. Let us remark that we useda different version of von Neumann’s spectral theorem than Maurin whichsimplifies the proof slightly, especially the completeness part. Note that thespectrum(2.29) σ ( A ) = σ ( ˆ A )is the essential range of a . COMPLETE SET OF EIGENDISTRIBUTIONS 7
To prove (2.22) and the following claims, we observe that the imbedding(2.30) j : E → H is continuous and therefore also nuclear, hence there is a semi-norm k·k p on E sequences u k ∈ H , f k ∈ E ′ such that(2.31) j ( ϕ ) = X k h f k , ϕ i u k ∀ ϕ ∈ E and(2.32) X k k f k k − p k u k k = X k k f k k − p k ˆ u k k < ∞ , where k·k − p is the dual norm in E ′ (2.33) k f k k − p = sup k ϕ k p =1 |h f k , ϕ i| . We shall show that the mapping in (2.30), which, when composed with U ,can now be expressed as(2.34) ϕ → ˆ ϕ ( λ ) = X k h f k , ϕ i ˆ u k ( λ )is continuous in E and not identically 0 for a.e. λ ∈ Λ .Indeed, without loss of generality we may assume(2.35) k u k k = 1to deduce from (2.32)(2.36) X k k f k k − p = X k k f k k − p k u k k = X k k f k k − p k u k k = X k k f k k − p Z Λ | ˆ u k ( λ ) | = Z Λ X k k f k k − p | ˆ u k ( λ ) | < ∞ , hence(2.37) X k k f k k − p | ˆ u k ( λ ) | ≡ c ( λ ) < ∞ for a.e. λ ∈ Λ , and(2.38) c ( · ) ∈ L ( Λ, µ ) . CLAUS GERHARDT
To prove (2.22) we now estimate(2.39) | ˆ ϕ ( λ ) | = (cid:12)(cid:12)(cid:12)X k h f k , ϕ i ˆ u k ( λ ) (cid:12)(cid:12)(cid:12) ≤ (cid:0) X k k f k k − p k ϕ k p | ˆ u k ( λ ) | (cid:1) = (cid:0) X k k f k k − p k ϕ k p k f k k − p | ˆ u k ( λ ) | (cid:1) ≤ (cid:0) X k k f k k − p k ϕ k p (cid:1)(cid:0) X k k f k k − p | ˆ u k ( λ ) | (cid:1) = c ( λ ) X k k f k k − p k ϕ k p < ∞ , in view of (2.36) and (2.37).The fact that the mapping in (2.34) does not vanish identically for a.e. λ ∈ Λ is proved in the lemma below. This completes the proof of the theorem,since the other properties are evident. (cid:3) Lemma.
The mapping (2.34) does not vanish identically in E fora.e. λ ∈ Λ .Proof. We argue by contradiction and assume that there exists a measurableset Λ ⊂ Λ with positive measure such that(2.40) ˆ ϕ ( λ ) = 0 ∀ ( λ, ϕ ) ∈ Λ × E. Let χ be the characteristic function of Λ and set(2.41) u = U − χ , then(2.42) 0 = u ∈ H. Let ϕ k ∈ E be sequence converging to u , then(2.43) k u k = lim k h ϕ k , u i = lim k Z Λ ¯ˆ ϕ k χ = 0 , a contradiction. (cid:3) The eigendistributions are smooth functions
In our case E = S and A is a uniformly elliptic linear differential operatorwith smooth coefficients. Hence, we can prove:3.1. Theorem.
Let A satisfy the Assumption on page , then thesolutions f ( λ ) ∈ S ′ of the eigenvalue problem (3.1) Af ( λ ) = µf ( λ ) belong to C ∞ ( S ) and for each m ∈ N and R > f ( λ ) can be estimated by (3.2) | f ( λ ) | m,B R ( x ) ≤ c m R N k f ( λ ) k − p , COMPLETE SET OF EIGENDISTRIBUTIONS 9 where k·k p is one of the defining norms in S such that (3.3) k f ( λ ) k − p = sup k ϕ k p =1 |h f ( λ ) , ϕ i| and N depends on n , k·k p , A and S , while c m depends on m , A theeigenvalue µ and on S . B R ( x ) is a geodesic ball of radius R for a fixed x ∈ K ⊂ S .Proof. First we note that we can absorb the right-hand side of the eigenvalueequation into the left-hand side and simply consider the equation(3.4) Af ( λ ) = 0 . Hence, it is well-known that the distributional solutions is smooth and equa-tion (3.4) can be understood in the classical sense, see e.g., [4, Theorem 3.2,p.125].The important estimate (3.2) is due to the fact that f ( λ ) is a tempereddistribution. Since f ( λ ) ∈ S ′ we have(3.5) |h f ( λ ) , ϕ i| ≤ c sup x ∈S (1 + r ( x ) ) k X | α |≤ m | D α ϕ ( x ) | ≡ c k ϕ k p and the dual norm(3.6) k f ( λ ) k − p = c. To prove (3.2) we fix m ∈ N and assume that(3.7) | f ( λ ) | m,B R ( x ) ≤ c , for some sufficiently large radius R such that we only have to prove theestimate in the domain(3.8) B R (0) \ ¯ B R (0) , where we now consider Euclidean balls, cf. the assumptions in (2.1) and (2.2)on page 4. Hence we may consider equation (3.4) to be a uniformly ellipticequation in an exterior region of Euclidean space with smooth coefficients.Let R > R , then we first prove a priori estimates for f ( λ ) in smalls balls(3.9) B ρ ( y ) ⋐ B R (0) \ B R (0) , where(3.10) 2 ρ < ρ ≤ ρ is fixed.Let(3.11) H m, ( Ω ) , m ∈ N , be the usual Sobolev spaces, where(3.12) Ω ⊂ R n is an open set, to be defined as the completion of C ∞ c ( Ω ) under the norm(3.13) k ϕ k m, = Z Ω X | α |≤ m | D α ϕ | .H m, ( Ω ) is a Hilbert space. Its dual space is denoted by(3.14) H − m, ( Ω )and its elements are the distributions f ∈ D ′ ( Ω ) which can be written in theform(3.15) f = X | α |≤ m D α u α , where(3.16) u α ∈ L ( Ω )and the dual norm of f is equal to(3.17) k f k − m, = (cid:0) X | α |≤ m k u α k (cid:1) . The Sobolev imbedding theorem states that(3.18) m > n ⇒ H m, ( Ω ) ֒ → C ( Ω )such that(3.19) | u | ≤ c k u k m, ∀ u ∈ H m, ( Ω ) , where c only depends on m and n .As a corollary we deduce(3.20) m > n ⇒ H m + m , ( Ω ) ֒ → C m , ( Ω )with a corresponding estimate(3.21) | u | m , ≤ c k u k m + m , , where c = c ( n, m, m ).Hence, for any ball(3.22) B ρ ( y ) ⊂ B R (0) f ( λ ) can be considered to belong to(3.23) f ( λ ) ∈ H − ( m + n ) , ( B ρ ( y ))with norm(3.24) k f ( λ ) k − ( n + m ) , ≤ cR k in view of the estimate (3.5), where we also assume R >
1; the constant c depends on n , m , k and the constant in (3.5).From the proofs of [4, Theorem 3.1, p. 123] and [4, Theorem 3.2, p. 125]we then deduce that for any m ∈ N there exists ρ < ρ , ρ depending only on COMPLETE SET OF EIGENDISTRIBUTIONS 11 the Lipschitz constant of the metric σ ij , m, n and m such that the C m -normof the solution f ( λ ) of equation (3.4) can be estimated by(3.25) | f ( λ ) | m,B ρ ( y ) ≤ c ρ R k , where c ρ also depends on the C m -norms of the coefficients of A and on theellipticity constants.Now(3.26) (4 R ) n n ρ − n balls(3.27) B ρ ( y ) ⊂ B R (0)cover the closed ball ¯ B R (0), hence we conclude(3.28) | f ( λ ) | m,B R (0) \ K ≤ cR k + n , where c = c ( ρ, m, m , n, A ). (cid:3) The positivity of the eigenvalues
To apply the separation of variables method to find a complete set ofeigensolutions for the wave equation the eigenvalues of the elliptic operatorhave to be positive. In this section we shall prove that eigenvalues of theeigenvalue equation (3.1) on page 8 are always strictly positive provided somerather weak assumptions are satisfied.Let us start with the following lemma:4.1.
Lemma.
Let A be the differential operator on the left-hand side of (1.3) on page and let us write the operator in the form (4.1) Av = − ( n − ∆v − n Rv + Gv + α n mV ( Φ ) v, where (4.2) 0 ≤ G = α n F ij F ij + α n γ ab σ ij Φ ai Φ bj . Assume there are positive constants ǫ , δ , m and R such that (4.3) − n R + G + α n m V ≥ ǫ r − δ ∀ x / ∈ B R ( x ) , where x ∈ K is fixed and r is the geodesic distance to x , then there exists m ≥ m such that for all m ≥ m the quadratic form of A satisfies (4.4) Z B R ( x ) k u k ≤ h Au, u i ∀ u ∈ H , ( S ) , provided (4.5) V ( Φ ) > S . Proof.
The ball B R ( x ) is bounded, hence the imbedding of(4.6) H , ( B R ( x )) ֒ → L ( B R ( x ))is compact and we can apply a compactness lemma to conclude that for any ǫ > c ǫ such that(4.7) Z B R ( x ) | u | ≤ ǫ Z B R ( x ) | Du | + c ǫ Z B R ( x ) V ( Φ ) | u | for all u ∈ H , ( B R ( x )), in view of the assumption (4.5), cf. [3, Lemma7.5].Hence, we deduce(4.8) Z B R ( x ) | u | ≤ ( n − Z B R ( x ) | Du | + Z B R ( x ) ( − n R + G ) | u | + α n m Z B R ( x ) V ( Φ ) | u | for all u ∈ H , ( B R ( x )) provided m is sufficiently large(4.9) m ≥ m . Choosing m ≥ m completes the proof of the lemma because of the assump-tion (4.3). (cid:3) Theorem.
Under the assumptions of the preceding lemma and thegeneral provisions in (2.1) , (2.2) and Assumption on page the eigenvalueequation (3.1) on page is only solvable if µ > .Proof. Since the quadratic form of A is positive we immediately infer(4.10) σ ( A ) ⊂ R + , hence the eigenvalue µ in (3.1) has to satisfy(4.11) 0 ≤ µ so that we have to exclude the case(4.12) µ = 0 . We argue by contradiction. Let(4.13) f ∈ S ′ ∩ C ∞ ( S )be a solution of(4.14) Af = 0 , then we shall prove(4.15) f = 0 . Let k ∈ N and R > R be large and let η be defined by(4.16) η ( x ) = ( R − k , | x | ≤ R, | x | − k , | x | > R, COMPLETE SET OF EIGENDISTRIBUTIONS 13
Then(4.17) f η ∈ H , ( S ) , in view of the estimate (3.2) on page 8, which can be rephrased to(4.18) X | α |≤ m | D α f ( x ) | ≤ c m | x | N ∀ | x | > R . Here, we use the Euclidean distance.Multiplying (4.14) by f η and integrating by parts yields(4.19) 0 ≥ Z S { ( n − | Df | η − n R | f | + G | f | + α n m | f | } η − ( n − Z S \ B R ( x ) | Df || f | η | Dη |≥ Z B R { ( n − | Df | − n R | f | + G | f | + α n m | f | } R − k + Z S \ B R { ǫ | x | − δ − c | x | − }| f | η , where c is a fixed constant depending only on the metric σ ij and k .The first integral is strictly positive unless f vanishes in B R ( x ), and thedifference in the braces is also strictly positive if R is large enough. Hencewe conclude(4.20) f ≡ . (cid:3) We can now prove a spectral resolution of the hyperbolic equation (1.1) onpage 1 by choosing an eigendistribution f = f ( λ ) with eigenvalue µ = a ( λ )and look at solutions of (1.1) of the form(4.21) u ( x, t ) = w ( t ) f ( x ) .u is then a solution of (1.1) provided w satisfies the implicit eigenvalue equa-tion(4.22) − n n − w − µt − n w − nt Λw = 0 , where Λ is the eigenvalue.This eigenvalue problem we also considered in a previous paper and provedthat it has countably many solutions ( w i , Λ i ) with finite energy, i.e.,(4.23) Z ∞ {| ˙ w i | + (1 + t + µt − n ) | w i | } < ∞ , cf. [2, Theorem 6.7].We can then extend the spectral resolution which we proved in [3, The-orem 1.7] for a compact Cauchy hypersurface S to the case when S isnon-compact: Theorem.
Assume n ≥ and let S and the elliptic differential oper-ator A satisfy the assumptions of the Theorem . Pick any solution ( f, µ ) of the eigenvalue problem (3.1) , then there exist countably many solutions ( w i , Λ i ) of the implicit eigenvalue problem (4.22) such that (4.24) Λ i < Λ i +1 < · · · < , (4.25) lim i Λ i = 0 , and such that the functions (4.26) u i = w i f are solutions of the wave equations (1.1) on page . The transformed eigen-functions (4.27) ˜ w i ( t ) = w i ( λ n n − i t ) , where (4.28) λ i = ( − Λ i ) − n − n , form a basis of L ( R ∗ + , C ) and also of the Hilbert space H defined as thecompletion of C ∞ c ( R ∗ + , C ) under the norm of the scalar product (4.29) h w, ˜ w i = Z ∞ { ¯ w ′ ˜ w ′ + t ¯ w ˜ w } , where a prime or a dot denotes differentiation with respect to t . Remark.
This result is the best we can achieve under the presentassumptions. In order to prove a mass gap, i.e., prove an estimate of theform(4.30) 0 < ǫ ≤ µ for ally eigenvalues µ of the eigenvalue equation (3.1) on page 8 we have tostrengthen our assumptions on the zero order terms: Instead of the assump-tion (4.3) we have to require(4.31) − n R + G + α n m V ≥ ǫ > ∀ x / ∈ B R ( x ) , then we immediately would derive a mass gapAn even stronger estimate of the form(4.32) − n R + G + α n m V ≥ ǫ r δ ∀ x / ∈ B R ( x ) , with δ >
0, would yield that the operator A would have a pure point spectrumsince the quadratic form(4.33) h Au, u i would then be compact relative to the L -scalar product and we would be inthe same situation as if S would be compact. COMPLETE SET OF EIGENDISTRIBUTIONS 15
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Ruprecht-Karls-Universit¨at, Institut f¨ur Angewandte Mathematik, Im Neuen-heimer Feld 205, 69120 Heidelberg, Germany
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