Absence of eigenvalues in the continuous spectrum for Klein-Gordon operators
aa r X i v : . [ m a t h - ph ] S e p A B SENCE OF EIGENVALUES IN THE C ONTINUOUS SPEC TRUMFOR K LEIN -G ORDON OPER ATORS
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R. Ferreira
Universidade de São Paulo – USPSão Paulo, SP, 05508-090, Brazil [email protected]
F. N. Lima ∗ GTMCOQ, Instituto Federal do Piauí – IFPISão Raimundo Nonato, Piauí, 64770-000, Brazil [email protected]
A. S. Ribeiro
Instituto Federal do Piauí – IFPISão Raimundo Nonato, Piauí, 64770-000, Brazil [email protected]
October 1, 2020 A BSTRACT
We construct the one-dimensional analogous of von -Neumann Wigner potential to the relativisticKlein-Gordon operator, in which is defined taking asymptotic mathematical rules in order to obtainexistence conditions of eigenvalues embedded in the continuous spectrum. Using our constructedpotential, we provide an explicit and analytical example of the Klein-Gordon operator with pos-itive eigenvalues embedded in the so called relativistic "continuum region". Even so in this notstandard example, we present the region of the "continuum" where those eigenvalues cannot oc-cur. Besides, the absence of eigenvalues in the continuous spectrum for Klein-Gordon operators isproven to a broad general potential classes, including the minimally coupled electric Coulomb poten-tial. Considering known techniques available in literature for Schrodinger operators, we demonstratean expression for Klein-Gordon operator written in Schrodinger’s form, whereby is determined themathematical spectrum region of absence of eigenvalues. K eywords Schrodinger operator · Klein-Gordon operator · continuous spectrum · eigenvalues · von Newmman-Wigner potential.
Theories involving general spectral problems are well-developed of a mathematical point of view for operators fromnon-relativistic quantum mechanics. However, the known relativistic operators that arising in the relativistic quantummechanics still require certain mathematical rigours in terms of indispensable properties that need demonstrated inspectral theory. As example, since of von
Newmman-Wigner work [1], we have know that there is a potential wherebyeigenvalues equations for Schrodinger operator provide an unitary positive energy [2]. Great efforts of scientificcommunity are aimed to investigate potential classes in which eigenvalues equations provide only negative eigenvalues.In particular, in the non-relativistic case of Schrodinger there are general theorems that proven the mathematicaland physical question of non-existence of positive eigenvalues in the continuum or essential spectrum, see Refs. [3–8]. For Schrodinger’s operator the positive spectrum is known as continuum spectrum, and a relevant problem of amathematical point of view it is localize the essential spectrum that for Schrodinger’s operator is exactly the samecontinuum spectrum. ∗ I am corresponding author.
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1, 2020A non-trivial question in mathematical physics is localize the essential spectrum of the relativistic Klein-Gordon (K-G)operator coupled to some interaction potentials. There is a consistent demonstration for some potential classes thatproven that positive relativistic region of the continuum (also namely in literature of positive continuum region [9]),it is the essential spectrum, that is, the region [ m, ∞ ) [10, 11]. The negative continuum region occurs in ( −∞ , − m ] ,being known as Dirac’s sea. Conditions of absence of eigenvalues for Schrodinger’s operators are well addressedproblems in literature, as discussed above (more details can be seen in Refs. [3–8]). However, such conditions forrelativistic operators still need of general theorems, i. e., theorems capable to define with mathematical rigours theabsence of eigenvalues in the located essential spectrum. In particular, in physical problems the continuum spectrum istraditionally taken (without lost of generality), and locating the essential spectrum is commonly treated as an essentiallymathematical problem. In fact, by use of the relativistic Virial theorem, using the version demonstrated by Kalf [12],it is possible proving the theorem of absence of eigenvalues in systems with Coulomb interaction potential to Dirac’soperator. An alternative explanation can be found in Ref. [13].For Klein-Gordon (K-G) operators, general theorems of absence of eigenvalues keep non-demonstrated up to now.A standard reduction procedure can be applied to Schrödinger and Dirac operators, where there is an inner productthat allows reductions to partial wave subspace problems, see for example Refs. [13–15]. This does not occurs toK-G operators that have a non-trivial structure of "energy norm", and probably being one of the reasons for the lackof such standard results. Weyl’s criterion or the asymmetry form method (see ref. [14]) for self-adjointness in thehalf-line cannot be used for K-G operators due to the reduction procedures at each quantum number channel representa non-trivial problem, as already mentioned. As discussed methodologies (Weyl’s criterion or the asymmetry formmethod) are not applied in this case, the self-adjointness property cannot be extend to whole operator in R . Inparticular, even in the level of self-adjointness theorems, only a few number of works (especially taking Coulombsingularities) are available in literature to K-G operators, as discussed by Gitman et al [14]. Although, Coulomb’spotentials have been suitable to apply Weyl’s criterion. We have know that for Schrödinger’s operators there arephysically reasonable examples of potentials providing bound states with positive energy. The first potential presentedin literature is proposed by Wigner and von Newmman [1], with its experimental evidence demonstrated in Ref. [16]. Inthe ordinary framework of the relativistic quantum mechanics there are no potentials providing eigenvalues embeddedin the continuum spectral region of relativistic operators.We provide an interaction potential in which the K-G operator has eigenvalues embedded in the continuum region, i. e.,in ( −∞ , − m ] S [ m, ∞ ) . General results on the absence of eigenvalues in the essential spectrum to some self-adjointK-G operators obtained by Weder (see Refs. [10, 11] are presented taking electric Coulomb interactions, the core ofthis work. We have used the same notation introduced by Schechter [17], and the same formalism extensively takenin Weder’s works, Refs. [10, 11] in K-G operators treatment. Besides, some of Weder’s theorems are useful to achievethe purposes of this paper, being presented always that necessary.This work is organized as follows. In section 2, it is presented a brief review on Klein-Gordon operator written inSchrodinger’s form, as well as is constructed the one-dimensional analogous to the von Neumann-Wigner potential.This potential is still used to demonstrate the existence of eigenvalues in the continuum spectrum to K-G operator. Insection 3, the same methodology of previous section is extended to Coulomb interactions. In addition, the absenceof eigenvalues in the continuum spectrum is demonstrated for some potentials. As it turns, we summarize our mainfindings and draw some perspectives in section 4. Throughout this work, we use units of ~ = c = 1 . We begin this section with a brief review on K-G operators and Weder’s conditions that prove the self-adjointness ofK-G operator. Remembering that general expression to K-G equation is given by (cid:18) i ∂∂t − b (cid:19) ψ ( x, t ) = " n X i =1 ( D i − b i ) + m + q s ψ ( x, t ) , (1)where x ∈ R n , t ∈ R , D j = − i∂/∂ x j , b ( x ) is the electric potential, b i ( x )(1 ≤ i ≤ n ) is the magnetic potential, and q s ( x ) the scalar potential. By implementing the ansatz transformations of Eq. 2 in Eq. 1 f = ψ ( x, t ) , f = i ∂ψ∂t ( x, t ) , (2)2 PREPRINT - O
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1, 2020it is possible written k-G equation given by expression 1 in its hamiltonian form with electromagnetic and scalarpotentials i ∂f∂t = hf, (3) h = (cid:20) l Q (cid:21) , (4) l = n X i =1 ( D i − b i ) + m + q ( x ) , (5) q ( x ) = q s − q , Q ( x ) = 2 b , (6) D ( h ) = { f ∈ C ∞ ( R n ) , lf ∈ L ( R n ) , Qf ∈ L ( R n ) } . (7)In this case, the energy sesquilinear form associated can be written as ( f, g ) E = n X i =1 (( D i − b i ) f , ( D i − b i ) g ) (8) + (cid:0)(cid:0) m + q (cid:1) f , g (cid:1) + ( f , g ) , (9) f, g ∈ C ∞ ( R n ) . (10)It is possible to verify that h is symmetric in relation that sesquiliner form. In general, the value of sesquilinear formis not positive. To ensure its positivity, Weder [10, 11] introduced the following assumption: I) There is a constant ǫ ≥ such that Z q − | f | d n x ≤ n X i =1 k D i f k + (cid:0) m − ǫ (cid:1) k f k , (11) f ∈ C ∞ ( R n ) (12)where q ± are the positive and negative parts of the measurable function q ( x ) .Notice that if I) holds sesquilinear form, ( · , · ) E , starts to define a norm and H E is defined as the completion of C ∞ ( R n ) with this norm, as can seen in Ref. [11] (Lemma 2.1). It is relevant to emphasize that for a pure electricpotential interaction in the K-G equation q s = b i = 0 and b = e | x | , I) holds if | e | ≤ ( n − / .So far we have discuss details of K-G operators as well as some requirement mathematical conditions related to densityof probability in order to unaltered keep its positivity. Let us now start reviewing the problem of absence of positiveeigenvalues for Scrhodinger operators.We have know that there is an interesting potential in which the eigenvalues equation for Schordinger’s operators has apositive eigenvalue embedded in the continuous spectrum with an unitary and positive value.This potential is namely von Newmann-Wigner potential, and can be written as [1, 8] V NW ( r ) = −
32 sin r (cid:2) ζ ( r ) − ζ ( r ) sin ( r ) + ζ ( r ) r + sin r (cid:3) (1 + ζ ( r ) ) , (13)where ζ ( r ) = 2 r − sin (2 r ) , | x | = r is the distance from the origin in R , and V NW ( r ) is a physical example inwhich there is a square integrable ψ with (∆ + V ) ψ = ψ that provides an unitary eingenvalue. The mathematicalcomputations to obtaining that potential can be extend to the one-dimensional case using (∆ + V ) ψ = ψ . A slightlydifferent square integrable eigenfunction in R when compared to R case can be obtained by construction of thepotential on the real line. In this situation, it is possible to obtain a similar expression to that given by Eq. 13, with theeigenfunction ψ ( x ) = ( sin ( x ))(1 + ζ ( x ) ) − . (14)Notice that the contructed one-dimensional potential is differ to given by Eq. 13 only by argument x ∈ R . Inaddition, this allows constructing a potential in which there is a self-adjoint K-G operator with positive eigenvalue in ( −∞ , − m ] S [ m, ∞ ) . 3 PREPRINT - O
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1, 2020In order to demonstrate the existence of a positive eigenvalue let us introduce some concepts and assumptions used byWeder [11]: N α,δ = sup x Z | x − y | <δ | g ( y ) | ω α ( x − y ) dy, (15)where ω α ( x ) = | x | α − n , α < n − log | x | , α = n , α > n,N α ( g ) ≡ N α, ( g ) and N α the set of all g such that N α ( g ) is bounded. II’)
For ≤ i ≤ n, b i ( x ) ∈ N , and if n ≥ , N ,s ( b i ) → as s → . III’) | q ( x ) | / ∈ , and if n ≥ , N ,s ( q ) → as s → . IV’) C ( x ) = P ni =1 ( ∂∂x i b i ( x )) ∈ N , and if n ≥ , N ,s ( C ) → as s → . V’) q ( x ) ∈ N , and if n ≥ , N ,s ( q ) → as s → , b = P ni =1 ( b i ( x )) ∈ N , and if n ≥ , N ,s ( b ) → as s → . VI’) b ( x ) ∈ N , and if n ≥ , N ,s ( b ) → as s → .These set of conditions is relevant to correctly understand the follow theorems, as well as to achieve the central goalof this section, i. e., demonstrate the existence of positive eigenvalue in the continuous spectrum.Allow us to write the theorem 2 from Weder’s [11] article as Theorem 1 . Theorem 1. If I) and II’) - VI’) are satisfied, h has a self-adjoint extension, H , on H E with domain D ( H ) = H ⊗ H . Here H k ( k = 1 , ) are standard Sobolev spaces W k, . We note that in some cases, a bounded g implies N α ( g ) < ∞ .So far, we demonstrate that the h operator has a self-adjoint extension, used along of this section.We have almost all tools to demonstrate the central theorem of this section, except the Lemmas 1 and that arepresented follow. Particularly, the Lemma 1 is interesting in the treatment of bounded potentials, and its proof istrivial.
Lemma 1.
Let g ( y ) be bounded and α ≥ n , then N α ( g ) is bounded. An alternative form to I is presented in Lemma 2 , whereby the condition I is easly verified. Lemma 2.
Let S λ ( q ) = sup x Z | q ( y ) | G ,λ ( x − y ) dy, (16) where G ,λ ( q ) is the inverse Fourier transform of (2 π ) − n ( λ + | η | ) − , λ ≥ , (17) then I) holds if and only if S λ ( q − ) ≤ , for some λ < m . A proof of
Lemma
Theorem von -Newmam-Wigner potential there are eigenvalues embeeded in the esssentialspectrum of the k-G operator. It is relevant to emphasize that this our result can be interpreted as the relativisticversion of the demonstrations already introduced by von -Newmman-Wignner to Schrodinger operators in Ref. [1].This result does not occurs to pure eletric interaction, as can seen in section 3.
Theorem 2. h with a real valued pure scalar interaction q s = V W N , (18) where V W N is Wigner - von Newmman potential, has a self-adjoint extension H F , with D ( H F ) = H ⊗ H , andfurther H F has eigenvalues embedded in ( −∞ , − m ] S [ m, + ∞ ) . PREPRINT - O
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Proof.
Keep n = 1 and the pure scalar interaction given by Eq. 18, then II’) and
IV’) are directly satisfied. By thefact that in this particular case q = q s , and V W N ( x ) is bounded, it is possible to see that | q | / ∈ N by applying Lemma
1, and then we have
III’) . In an analogous way,
V’) and
VI’) hold as well. Let us suppose I) to be valid, andthis will be really demonstrated in the end of this theorem, then, by Weder’s theorem, H F is the necessary self-adjointextension. By using of the eigenvalue problem H F ψ = Eψ , one get ( − ∆ + q s + m ) ψ = E ψ , ψ ∈ H ( R ) , (19)where ∆ n denotes the Laplacian in n dimensions. Pick q = V W N , then we have E = ±√ m due to the presenceof von -Neumann Wigner potential in (19). It is important to stress that to the value E = −√ m the I is notsatisfied. Although this found negative value is in the known Dirac sea region, the presence of eigenvalues in thisregion should be better investigated. Now, let us consider I) . For the choice above of q , and by Lemma
2, we see that I) holds if S λ ( V − W N ) ≤ . Indeed sup x Z | q − ( y ) | G ,λ ( x − y ) dy = (1 / √ λ ) sup x Z | V − W N | e −√ λ | x − y | dy , (20)where the "supression" of the integral ensures the existence of an infinite number of particular values for √ λ < m ,such that I) holds true.Notice that the leading term of V W N for large x is given by − x ) x , (21)hence, lim sup x ( ∂V ) /∂x = 16 , where V is the infinitely differentiable function defined with more details in Theo-rem 3 of section 3. Therefore, it is possible to conclude that E − m < in Theorem ( √
16 + m , ∞ ) .We would be have problems when applying Weder’s theorem in the case n ≥ (see Theorem n = 2 , we do not have III’) completely satisfied since N ,s ( q ) as s → for q (in the case underconsideration). The fact we have used the pure scalar interaction is quite technical. Suppose we would like to findeigenvalues in [ m, + ∞ ) , for h with a pure electric interaction given by − ( b ) + 2 Eb = V W N . (22)Thus, even for the most suitable choice for b , we would find problems with the expressions involving the requirement I) with the map (see expression 22), the stronger inequality S ( q + V − W N ) ≤ , (23)where q = 1 + m takes place. So, Theorem
Theorem
2, we prove that K-G operator with pure eletric interactions does not has eigenvalues embeededin the essential spectrum.
Now, we extend same methotodology of previous section in order to determine the absence of eigenvalues in theessential spectrum of K-G operators with Coulomb interactions.To do that, we list some important results about self-adjointness, and on the essential spectrum location, using a sequence of complementary assumptions introduced byWeder in Ref. [11]. In short, these assumptions are related to the self-adjointness; the essential spectrum location; andthe absence of eigenvalues theorems concerning to the K-G operators.Weder’s hypothesis (see Ref. [11]) are listed in I) - V) . In addition, we present here assumption VI) . Note that hypotesis I) is already presented in section 2. II) PREPRINT - O
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1, 2020(1) b i ( x ) ∈ N , ≤ i ≤ n , and if n ≥ , N ,δ ( b i ) δ → −−−→ ,(2) q ( x ) = q ( x ) + q c ( x ) , | q | / ∈ N , and if n ≥ , N ,δ ( | q | / ) δ → −−→ · q c ( x ) ≡ if n ≤ , and q c ( x ) = − e | x | , | e | ( n − / if n > . III) ( N ,x ( b i ) + N ,x ( | q | / )) δ → −−−→ , IV) b = b + b c ∈ N and if n ≥ N ,j ( b ) δ → −−→ . q c ( x ) ≡ if n ≤ .If n ≥ , b c ( x ) = e | x | , where | e | < n − √ . V) N ,x ( b ) | x |→∞ −−−−→ . VI) b c ( x ) is such that, for some real k , − b c ( x ) + kb c ( x ) is a real function in R n so that:(1) Multiplication by V + V = − b c ( x ) + kb c ( x ) is − ∆ -bounded with relative bound less than .(2) There is a closed set S of measure zero so that R n \ S is connected and so that V + V is bounded on any compactsubset of R n \ S .(3) − V is bounded outside some ball { x || x | < r } , and | x | b ( x ) → as | x | → ∞ .(4) V is bounded outside some ball { x || x | < r } , and kb ( x ) → as | x | → ∞ . (This assumption can be replaced by kb ( x ) < for | x | > r ).(5) If b c maps r in b c ( r, · ) from (0 , ∞ ) to L ∞ ( S n − ) where S n − is the ( n − -dimensional sphere, then for | x | = r > r , b c ( x ) is differentiable as an L ∞ -valued function and for some suitable k , klim r →∞ r ∂b c ∂r ≤ .If I) and II) are satisfied, the norm of the completion H E is equivalent to the norm of H ( R n ) ⊗ L ( R n ) , and theycoincide as sets (this is the Lemma 1.4 by Weder [11]). By this lemma and assumption II) , l (remember Eq. (7) above)has a self-adjoint bounded below extension, denoted by L (this is the Lemma 1.5 in Ref. [11]).Now, let H = H L + V, D ( H ) = D ( L ) ⊗ H (24) H L = (cid:20) L (cid:21) , V = (cid:20) Q (cid:21) then if I) - V) are satisfied, then H is a self-adjoint extension of h see (7), with domain D ( H ) = D ( L ) ⊗ H ( R n ) andits essential spectrum coincides with ( −∞ , − m ] S [ m, ∞ ) (this reult is presented in theorem 2 in Ref. [10]).So far, using the formalism introduced by Weder in Refs. [10] and [11], it is possible to obtain the Theorem
3. Thisresult is immediately presented below, and opportunetely demonstrated after presenting an important appilication ofthat.
Theorem 3.
The self-adjoint extension H of the K-G operator, with a pure electric field b c satisfying I) - VI) (we callthis operator ˜ H ), has no eigenvalues embedded in its essential spectrum. As an application of
Theorem
3, we can prove the R case, in which includes the non-central electric Coulombinteraction. Even though the following result is stated here as the Theorem
4, it is a corollary of
Theorem
3. Noticethat for the important case S = { } , we don’t have the necessary connectedness present in assumption VI) - , andin this case we shall directly apply other Theorem (see details in Ref. [8]) with appropriate modifications, presentedhere as the
Theorem 5 below. 6
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Theorem 4.
The self-adjoint extension, H , of the K-G operator with a pure b c as a Coulomb interaction, e | x | , in R (called here ˜ H C ), has no eigenvalues embedded in its essential spectrum in the following sense: for the attractive case( b c < ) this is valid for [ m, ∞ ) , and for the repulsive ( b c > ), it holds in ( −∞ , − m ] . Proof of the Theorem 3:
We can use to K-G operator, and so demonstrate
Theorem
3, the same technique already used to Schrödinger’soperator (as in
Theorem ˜ H , of the K-G operator without scalar potential, magnetic potential and b = 0 ,becomes ˜ H = (cid:20) − ∆ + m − ( b c ) (cid:21) D ( ˜ H ) = D ( L ) ⊗ H . It is clear that the absence of magnetic and scalar potentials, and the choice b = 0 , are all in accordance with Weder’sassumptions I) - V) . Then, we write the eigenvalue equation ˜ Hψ = Eψ , ψ = (cid:20) ψ ψ (cid:21) ∈ D ( ˜ H ) . Hence, in terms of ψ , one can see the compatibility of the above K-G eigenvalue equation with the Schrödinger typeone, ( − ∆ + V eff ) ψ = ˜ Eψ , V eff ≡ − ( b c ) (
2) + 2 Eb c , ˜ E ≡ E − m . Pick V ( x ) = − ( b c ) and V ( x ) = 2 Eb c . See that by VI) , D ( L ) = H , which is the domain of the free Schrödingeroperator, and this is enough to the use of known Kato’s theorem (see more details in Ref. [19]). Hence, V eff satisfiesall the hypothesis of Kato-Agmon-Simon’s theorem [2]. Thus, the Theorem
Proof of the Theorem 4:
As mentioned, the proof of
Theorem
Theorem Theorem 5.
Let V be a real valued function in R n \ with the following properties:(a) V ∈ L + L ∞ ; V = V + V with(b) for some R , V and V are C in M = { r | r > R } ,(c) lim r →∞ V ( r ) = 0 ,(d) for r > R , V ( r ) < ,(e) for r > R , − ∂V ∂r ≤ − r V .Then − ∆ + V has no eigenvalues in (0 , ∞ ) . We have that the self-adjoint extension of the K-G operator ˜ H c , without scalar and magnetic potentials, b = 0 , and q c ( x ) = − e | x | , | e | ≤ ( n − / (if n > ), and b c ( x ) = e | x | , | e | ≤ ( n − / √ (if n > ), becomes7 PREPRINT - O
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1, 2020 ˜ H C = (cid:20) − ∆ + m − e | x | e | x | (cid:21) D ( ˜ H C ) = D ( L ) ⊗ H . It is easy to see that the lack of magnetic and scalar potentials, and the choice b = 0 , are turning I) - V) conditionssatisfied. Hence, we get the eigenvalue equation for ˜ H C , ˜ H C ψ = Eψ , ψ = (cid:20) ψ ψ (cid:21) ∈ D ( ˜ H C ) . Then, in terms of the component ψ ∈ H , one can see the compatibility of the above K-G eigenvalue equation withthe following Schrödinger equation, ( − ∆ + V eff ) ψ = ˜ Eψ , V eff ≡ − e | x | + 2 E e | x | , ˜ E ≡ E − m . In this case, pick V ( x ) = − e | x | and V = 2 E e | x | , hence V eff holds for all hypothesis of Simon’s theorem or itsmodification given by Theorem 5 , which means that the components ψ ∈ H are ensuring the use of Kato’s theorem,and so it is possible directly to apply the Theorem 5 , with exception that for the attractive case, (d) holds only if
E > m , and that for the repulsive case it holds only if
E < m . Thus, the Theorem 4 is proven.
We found a class of physical potentials in which there is no eigenvalues embedded in the continous region of relativisticspectrum. It is possible to see that the fact associated to the non-existence of eigenvalues embedded in the essentialspectrum can be intepreted as an expected result. We clearly show that with the correct location of continuous spectrumtogether with the result of theorem 3, we can rigour confirm if thresholds ± m belong, or no, to the point spectrum.Here, we emphasize that essential spectrum coincides with ( −∞ , − m ] S [ m, ∞ ) , and so ± m dont belong to the pointspectrum. Generally, these results are not so clear by use of the computational calculations of theoretical physics.It is worthy of emphasis that the present paper demonstrates that relativistic K-G operator contains eigenvalues in itsessential spectrum, as occurs to Schrodinger’s operators. This is a new and interesting result, undemonstrated beforein literature to relativistic operators, and introduced here. It is remarkable that the potential in wich this occurs is von -Neumann Wigner potential, as expected. In particular, this potential provides similar results for Schodinger’soperators, i. e., the existence of an unitary eigenvalue in the essential spectrum.We have expect to be possible extend the same methodology of theorem 2 in order to investigate the existence ofeigenvalues in the essential spectrum for K-G operator to higher-dimensional order. However, in this case, the self-adjointness properties must be better investigated. The authors gratefully acknowledge the support provided by Brazilian Agencies CAPES, CNPQ. We would like tothank the following for their kind support: IFPI and USP; for the great professors D. M. Gitman, C. R. Oliveira and R.Weder for our private communications.
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