The Wave Equation in Spherically Symmetric Spacetimes

Transcription

1 in Spherically Symmetric Spacetimes Department of M University of Michigan

2 Outline 1 Background and Geometry Preliminaries 2 3

3 Introduction Background and Geometry Preliminaries There has been much recent effort devoted to studying the stability of several spacetime metrics coming from GR. In particular, the Minkowski spacetime, the Schwarzschild spacetime, and the Kerr spacetime. The ultimate goal in this study is to prove the nonlinear stability of these metrics. For Minkowski, this was done by Christodoulou and Klainerman in It is a great open problem in GR to prove the nonlinear stability of the Kerr metric; nonlinear stability of the Schwarzschild metric remains open as well (this would of course follow from a result on Kerr).

4 An Apology Background and Geometry Preliminaries This talk will focus on the linear wave equation in a general spherically symmetric spacetime. Why do we care about the linear wave equation? In short, the nonlinear stability of these metrics is too hard. Instead, we first investigate the linear stability of these metrics. We see three obvious reasons for this: of the linear wave equation implies linear stability of the metric under scalar wave perturbations; results on the linear wave equation are important to obtain results in nonlinear regime; and it is worthwhile understanding the linear wave equation.

5 Review of Results Background and Geometry Preliminaries There is a plethora of results concerning the decay of the linear wave equation in the Schwarzschild and Kerr metrics, and the methods break down into roughly two classes. The first class is basically a harmonic analysis approach where the main tool is Strichartz estimates. This is the idea behind Dafermos & Rodnianski (2008), Donninger, Schlag, Soffer (2009), Luk (2010), etc. These results are very technical works even reading the introduction and discerning the results can be a challenge. The second class uses spectral methods to obtain representation formulae, and then analyzes the formulae. This approach is very classical, and we obtain robust, general results.

6 Review of Results, continued Kronthaler showed decay of the linear wave equation in the Schwarzschild metric. Kronthaler, Blue-Sterbenz, Dafermos-Rodnianski get rates: t 3 2l for modes, t 3 2 for full solution Dafermos & Rodnianski obtain pointwise boundedness in Kerr Finster, Kamran, Smoller, Yau showed decay of the linear wave equation in Kerr Dafermos & Rodnianski obtain t 1 rate for Kerr

7 What is Left Background and Geometry Preliminaries It may seem that most of the decay questions have been answered, but we now wish to study more general spacetime metrics. For example, perturbations on Minkowski or perturbatoins on Schwarzschild. Solutions of the Einstein-Yang/MIlls equations fall into this category. Right now we restrict our attention to spherically symmetric cases the positive angular momentum case may be future work. Much of the work by Kronthaler (2006) can be extended to apply in a more general setting, but only after a few key ideas.

8 Outline Background and Geometry Preliminaries 1 Background and Geometry Preliminaries 2 3

9 Schwarzschild: Basic facts Let us recall some facts about the Schwarzschild metric: The Schwarzschild metric is given by ( ds 2 = 1 2m ) ( dt m ) 1 dr 2 + r 2 dω 2, r r where r 0, 0 θ π, 0 φ 2π, and dω 2 = dθ 2 + sin 2 θdφ 2. The wave equation in Schwarzschild is given by [ ( t 2 1 2m ) 1 ( r r r 2 (r 2 ) ] 2mr) r + S 2 ζ = 0. Kronthaler analyzed this PDE in the exterior region (r > 2m) with data compactly supported away from the horizon (r = 2m).

10 The Regge-Wheeler Coordinate The starting point for studying the wave equation in Schwarzschild is change coordinates to u(r) = r + 2m log ( r 2m 1) and let ζ(t, r(u), θ, φ) = r(u)ψ(t, u, θ, φ). Then the wave equation becomes [ ( t 2 u m r ) ( 2m r 3 S 2 r 2 )] ψ = 0 on R R S 2. One then couples this with data (ψ, iψ t )(u, 0, θ, φ) = Ψ 0 (u, θ, φ) C 0 (R S 2 ) 2.

11 The Regge-Wheeler Coordinate, continued Why is the Regge-Wheeler coordinate ubiquitous in Schwarzschild? It yields a beautiful energy, the density given by ( ψt 2 +ψu m ) [ 2m r r 3 ψ2 + 1 ( )] 1 r 2 sin 2 θ ( φψ) 2 + sin 2 θ( cos θ ψ) 2. Unfortunately, when working in a more general spherically symmetric black hole metric, the generalization of the Regge-Wheeler coordinate fails to yield a positive definite energy density. The positive energy density is imperative to using energy/spectral methods.

12 Essential Properties of the Blackhole Geometry: Near the Horizon Consider a metric ds 2 = T 2 (r)dt 2 + K 2 (r)dr 2 + r 2 dω 2. For this to be a generalized Schwarzschild blackhole, we require T, K C (r 0, ) with T, K > 0 for r > r 0 > 0. In addition, we require for r near r 0, T (r) c 1 (r r 0 ) O(1), c 1 > 0 K(r) c 2 (r r 0 ) O(1), c 2 > 0 T (r) c 3 (r r 0 ) O(r r 0 ) 1 2 K (r) c 4 (r r 0 ) O(r r 0 ) 1 2

13 Essential Properties of the Blackhole Geometry: The Far-Field We must also impose conditions on the far-field behavior of the metric. We shall insist that for large r, T (r) 1 + O ( ) 1 r K(r) 1 + O ( ) 1 r T (r) T (r) + K (r) K(r) O ( ) 1 r 2 Remark The above conditions are satisfied by the Schwarzschild metric, the non-extreme Reissner-Nordstrom metric, and black hole solutions of the Einstein-Yang/Mills (EYM) equations.

14 The wave equation in a geometry ds 2 = g ij dx i dx j is given by 0 = ζ := g ij i j ζ = 1 ( ) gg ij g x i x j ζ, where g ij is the inverse of the metric g ij and g = det(g ij ). So in the geometry ds 2 = T 2 (r)dt 2 + K 2 (r)dr 2 + r 2 dω 2, the d Alembert operator takes the form T 2 t ( r 2 ) r r 2 K 2 + T ( ) K K 3 r r + 1 T r 2 S 2, where we have dropped the arguments of T, K and S 2 standard Laplacian on the sphere S 2. is the

15 The Change of Coordinate Upon making the change of coordinate u(r) := r K(α)T (α) α 2 dα, the wave equation ζ = 0 becomes ( r 4 2t + 2u + r 2 ) T 2 S 2 ψ = 0 on R (, 0) S 2. Note: If we set T = K = ( 1 2m r then we find u(r) = 1 2m log ) 1 2 (Schwarzschild coefficients), ( 1 2m r ).

16 Existence/Uniqueness Theorem The Cauchy problem ) {( r 4 t 2 + u 2 + r 2 T 2 S 2 ψ = 0 on R (, 0) S 2 (ψ, iψ t )(0, u, θ, φ) C0 ((, 0) S 2 ) 2 has a global, smooth, unique solution ψ that is compactly supported in (, 0) S 2 for each time t. To prove it, one applies the theory of symmetric hyperbolic systems to the equation for ξ = rψ in the coordinate s(u) := u u r 2 (α)dα. Note: s is a generalization of the Regge-Wheeler coordinate.

17 Energy and the Hamiltonian The solution ψ admits a conserved energy: E(ψ) = 2π r 2 T 2 r 4 (ψ t ) 2 + (ψ u ) 2 ( 1 sin 2 θ ( φψ) 2 + sin 2 θ( cos θ ψ) 2 ) dud(cos θ)dφ. Recast the PDE as i t Ψ = HΨ for Ψ = (ψ, iψ t ) T and ( ) 0 1 H = where A = 1 A 0 r 4 2 u S 2 r 2 T 2.

18 Energy and the Hamiltonian, continued The solution of the original Cauchy problem yields a solution Ψ of the Hamiltonian system with data in C 0 ((, 0) S 2 ) 2. The energy functional induces an inner product, on the space C0 ((, 0) S 2 ) 2, with respect to which the Hamiltonian is symmetric: 0 = d Ψ, Ψ dt = t Ψ, Ψ + Ψ, t Ψ = i HΨ, Ψ i Ψ, HΨ.

19 Energy and the Hamiltonian, continued The above implies that HΨ, Ψ = Ψ, HΨ for a solution Ψ. In particular, HΨ 0, Ψ 0 = Ψ 0, HΨ 0. But the data is arbitrary in C0 ((, 0) S 2 ) 2 a simple polarization argument then yields the symmetry of H on C0 ((, 0) S 2 ) 2.

20 Projecting Onto Spherical Harmonics Projecting onto spherical harmonics yields, for each angular momentum number l = 0, 1, 2,..., the Cauchy problem { i t Ψ lm = H l Ψ lm on R (, 0) where Ψ lm (0, u) = Ψ lm 0 C 0 ( H l = (, 0) r 4 u + l(l+1) 0 r 2 T 2 We also note that H l is symmetric on C0 (, 0)2 with respect to the inner product ). Ψ, Γ l = 0 r 4 ψ 2 γ 2 + ( u ψ 1 )( u γ 1 ) + r 2 T 2 l(l + 1)ψ 1γ 1 du.

21 Essential Self-Adjointness of H l We consider the Hilbert space H := HV 1 l,0 H r 2,0, where V l (u) = l(l+1) and r 2 T 2 HV 1 l,0 is the completion of C 0 (, 0) within the Hilbert space ({ ψ : ψu L 2 (, 0) and r 2 V l ψ L 2 (, 0) } ),, l1 and H r 2,0 is the completion of C0 (, 0) within the Hilbert space ({ ψ : r 2 ψ L 2 (, 0) } ),, l2. Proposition The operator H l with domain D(H l ) = C0 self-adjoint in the Hilbert space H. (, 0)2 is essentially

22 Stone s Theorem Background and Geometry Preliminaries Theorem (Stone s Theorem) Let U(t) be a strongly continuous one-parameter unitary group on a Hilbert space H. Then there is a self-adjoint operator A on H so that U(t) = e ita. Furthermore, let D be a dense domain which is invariant under U(t) and on which U(t) is strongly differentiable. Then i 1 times the strong derivative of U(t) is essentially self-adjoint on D and its closure is A. For proof, we refer to Reed and Simon.

23 Essential Self-Adjointness of H l, continued Let U(t) be the solution operators: U(t)Ψ lm 0 = Ψlm (t). Then the U(t) extend to a one-parameter unitary group on H that is strongly continuous on H and strongly differentiable on C0 (, 0)2 (due to the smoothness of the solution Ψ lm and the energy conservation). Moreover, U(t) leave the dense subspace C0 (, 0)2 invariant for all times t and the strong-derivative of U(t) is just H l. Thus, by Stone s theorem, H l is essentially self-adjoint and U(t) = e ith l.

24 A Representation Formula By Stone s theorem, we can write Ψ lm (t, u) = e ith l Ψ lm 0 (u). To obtain an explicit representation of Ψ lm, we shall express Ψ lm 0 in terms of the spectral projections of H l. For, if we had Ψ lm 0 (u) = R Ψlm 0 (v)dµ(v, u), where dµ are the spectral measures of H l, then the spectral theorem would imply Ψ lm (t, u) = e iωt Ψ lm 0 (v)dµ(v, u). R

25 Stone s Formula Background and Geometry Preliminaries In general, the spectral measures are difficult to compute. But for a self-adjoint operator A, Stone s formula relates the spectral projections to the resolvent: ( ) 1 P(a,b) +P [a,b] = lim ε 0 πi b a (A ω iε) 1 (A ω + iε) 1 dω, where the limit is taken in the strong operator topology. So, in order to derive a useful representation formula, we shall investigate the resolvent of H l.

26 The Resolvent Background and Geometry Preliminaries We will construct the resolvent out of special solutions of the eigenvalue equation H l Ψ = ωψ. The eigenvalue equation is equivalent to the ODE ξ (u) ω 2 r 4 ξ + r 2 l(l + 1)ξ = 0 on (, 0). T 2 Change to the s coordinate and consider η(s) = rξ(u(s)). The equation for η is η (s) ω 2 η(s) + on (, ). ( l(l + 1) r 2 T 2 1 rt 2 K 2 ( T T + K )) η = 0 K Note: s(r) = r r K(r )T (r )dr

27 Jost Solutions Background and Geometry Preliminaries Let us consider Im ω < 0. We seek linearly independent solutions η 1,ω, η 2,ω satisfying the boundary conditions lim s e iωs η 1,ω (s) = 1, and lim s eiωs η 2,ω (s) = 1. These solutions are referred to as the Jost solutions. We have existence, uniqueness, smoothness, and analyticity in ω for Im ω < 0, and we can also extend η 1,ω, η 2,ω continuously up to the real axis. For Im ω > 0, we obtain solutions via the definition η i,ω := η i,ω.

28 Jost Solutions, continued One finds the Jost solutions by converting the ODE for η 1,ω into the integral equation η 1,ω (s) = e iωs + 1 ω s sin[ω(s s)]v ( s)η 1,ω ( s)d s, where V is the potential term in the η ODE. One solves this integral equation by a perturbation series: η 1,ω (s) = 1,ω (s), where η(0) 1,ω (s) = eiωs and n=0 η(n) η (n+1) 1,ω (s) = 1 s sin[ω(s s)]v ( s)η (n) 1,ω ω ( s)d s.

29 Integral Representation of the Resolvent Corresponding to η i,ω, there are solutions ζ i,ω of the ODE ζ (u) ω 2 r 4 ζ + r 2 Let h ω (u, v) := 1 w(ζ 1,ω,ζ 2,ω ) l(l + 1)ζ = 0. T 2 { ζ 1,ω (u)ζ 2,ω (v), u v ζ 1,ω (v)ζ 2,ω (u), u > v. We then have ( H l ω ) 1 Γ(u) = 0 k ω(u, v)γ(v)dv, where ( 0 0 k ω (u, v) = δ(u v) 1 0 ) ( + r 4 ω 1 (v)h ω (u, v) ω 2 ω ).

30 Spectral Projections From Stone s formula, we then get ( ) 1 P[a,b] +P (a,b) Ψ(u) = lim ε 0 πi b 0 a (k ω+iε (u, v) k ω iε (u, v))ψ(v)dvdω. We have k ω (u, v) = k ω (u, v), k can be continuously extended to Im ω = 0, and we consider Ψ C0 (, 0)2. Thus, 1 ( ) 1 b P[a,b] + P 2 (a,b) Ψ(u) = Im(k ω (u, v))ψ(v)dvdω. π a supp Ψ This formula yields P {a} = 0 for any a R, and thus P (a,b) = 1 π b a supp Ψ Im(k ω (u, v))ψ(v)dvdω.

31 A Representation Formula By the spectral theorem, we then have Ψ lm (t, u) = e ith l Ψ lm 0 (u) = 1 e iωt π R supp Ψ (Im(k ω (u, v))ψ(v)dvdω. We next wish to analyze the integrand, so let us note that the pair {ζ 1,ω, ζ 1,ω } form a fundamental set for the ζ ODE. Thus, we have ζ 2,ω (u) = λ(ω)ζ 1,ω (u) + µ(ω)ζ 1,ω (u) where µ(ω) is never zero.

32 A Representation Formula, continued We make the definitions γ 1,ω = Re ζ 1,ω, γ 2,ω = Im ζ 1,ω, and Γ a ω = (γ a,ω, ωγ a,ω ) T, as well as α 11 (ω) = 1 + Re ( ) λ(ω), α 22 (ω) = 1 + Re µ(ω) and α 12 (ω) = α 21 (ω) = Im Some calculation then yields Ψ lm (t, u) = 1 e iωt 1 2π R ω 2 2 a,b=1 ( λ(ω) µ(ω) ( λ(ω) µ(ω) ). ), α ab (ω)γ a ω(u) Γ b ω, Ψ 0 l dω.

33 for the Modal Solutions Theorem For fixed u (, 0), the integrand in the representation formula for Ψ lm is in L 1 (R, C 2 ). This follows from the following facts for large ω : λ(ω) = O(1) and µ(ω) = 1 + O ( ) 1 ω ζ 1,ω (u) C + O ( ) 1 ω Γ b ω, Ψ lm 0 l exhibits arbitrary polynomial decay in ω. The Riemann-Lebesgue lemma guarantees then that for fixed u (, 0), Ψ lm (t, u) 0 as t.

34 of the Full Solution Let Ψ L := l=l m l Ψlm Y lm. For any arbitrary compact subset K of (, 0) S 2, with smooth boundary and any ε > 0, we can find an L N so that Ψ L (t) H 2 (K) < ε for all t. Idea: There exists L 0 so that Ψ L 0 (t) 2 = l=l 0 m l Ψlm 0 2 < ε for all t. Next, the problem with data HΨ 0 = l=0 has the solution HΨ. So there exists an L 1 so that HΨ L 1 (t) < ε for all t. m l (H lψ lm 0 )Y lm Proceeding inductively, for each N N there is an L N so that H n Ψ L N (t) < ε for each t.

35 of the Full Solution, continued Looking at the energy, there is a constant C 0 (K) > 0 so that for Ψ L N = (ψ L N 1, ψl N 2 )T we have ψ L N 1 H 1 (K) + ψ L N 2 L 2 (K) < C 0 (K) Ψ L N. Similarly, applying this to HΨ L N C 1 (K) > 0 so that = (ψ L N 2, AψL N 1 )T, we find Aψ L N 1 L 2 (K) + ψ L N 2 H 1 (K) < C 1 (K) HΨ L N. The ellipticity of A guarantees that for smooth h and K K (, 0) S 2 h H k+2 (K) < C Ah H K ( K) + C h H k+1 ( K).

36 of the Full Solution, continued Thus, there exist new constants C 0 (K), C 1 (K) > 0, so that ψ L N 1 H 2 (K) + ψ L N 2 H 1 (K) < C 0 (K) Ψ L N + C 1 (K) HΨ L N. We can iterate this argument to obtain constants C 0 (k),..., C k (K) > 0 so that ψ L N 1 H k+1 (K) + ψ LN 2 H k (K) < k C n (K) H n Ψ L N. n=0 In particular, given any ε > 0 there is an L so that Ψ L (t) H 2 (K) < ε for all t.

37 of the Full Solution, continued The Sobolev embedding theorem then yields, for a possibly larger L, that Ψ L (t) L (K) < ε for all t. Coupling this with the decay of the fixed modes guarantees that for any ε > 0, we may find an L N and a t 0 so that Ψ(t, u, θ, φ) for t > t 0. L 1 Ψ lm (t, u)y lm (θ, φ) + Ψ L (t, u, θ, φ) < ε l=0 m l

38 EYM Equations Background and Geometry Preliminaries Coupling the Einstein equations with the Yang-Mills equations with gauge group G yields R ij 1 2 Rg ij = σt ij, d F ij = 0, where T ij is the stress-energy tensor associated to the g-valued curvature 2-form F ij and g is the Lie algebra of G. Letting G = SU(2), restricting to static, spherically symmetric solutions: ds 2 = T 2 (r)dt 2 + A 1 (r)dr 2 + r 2 dω 2 and F = w τ 1 dr dθ+w τ 2 dr (sin θdφ) ( 1 w 2) τ 3 dθ (sin θdφ). Thus, the static, spherically symmetric EYM equations over SU(2) reduce to three unknowns: T, A, w.

39 EYM Equations, continued We obtain three coupled ODE s for the unknowns T, A, w: ( ra ( w ) ) ( 2 1 w 2 ) 2 A = 1 r 2, [ ( 1 w r 2 Aw 2 ) ] 2 + r(1 A) w + w ( 1 w 2) = 0, r 2rA T ( 1 w 2 ) 2 ( T = r ( w ) ) 2 A 1. Smoller et al. proved the existence of black hole solutions: i.e. for any r 0 > 0, there exist smooth solutions of these ODE s on (r 0, ), A > 0 on (r 0, ), A(r 0 ) = 0, and A, T 1 as r.

40 The EYM Metric as a Generalized Schwarzschild Blackhole We make the identification K 2 = A 1. Then, the metric ds 2 = T 2 (r)dt 2 + K 2 (r)dr 2 + r 2 dω 2 is a generalized Schwarzschild blackhole. This follows easily from the facts (shown by Smoller, et al.): lim w 2 (r) = 1, r lim rw (r) = 0, r lim w 2 (r) < 1, r r 0 lim w (r) <, r r 0 ( ( 1 w 2 (r 0 ) ) ) 2 r 0 0. r 0

41 Outline 1 Background and Geometry Preliminaries 2 3

42 Essential Properties: At the Origin Here again we consider a metric of the form ds 2 = T (r) 2 dt 2 + K(r) 2 dr 2 + r 2 dω 2, where 0 θ π and 0 φ 2π, but this time we let r [0, ). We assume that T, K C [0, ) and that the metric is nonsingular: so we assume that T, K > 0 for r [0, ). Since we will assume that T, K 1 as r, this implies that T, K are uniformly bounded and uniformly bounded away from zero. We must also enforce the smoothness of the metric at the origin: K(0) = 1, T (0) = 0 = K (0).

43 Essential Properties: The Far-field Interestingly, the far-field behavior we require is the same as in the black hole case. To remind ourselves, we insist for large r that T (r) 1 + O ( ) 1 r K(r) 1 + O ( ) 1 T (r) T (r) + K (r) K(r) O ( ) 1 r. 2 r

44 The Boundary at r = 0 In the black hole case, due to the horizon being pushed to and the compact support of the solution, we didn t consider a boundary condition at r = r 0. In the particle-like case, the boundary r = 0 must be considered. To figure out what boundary condition to impose, we change to Cartesian coordinates (t, x, y, z) where the boundary disappears. We use the theory of symmetric hyperbolic systems in these coordinates to find a smooth solution of ζ = 0 that is compactly supported for each time t.

45 The Boundary at r = 0, continued A straight-forward application of the divergence theorem then shows that ζ r (t, 0, θ, φ) = 0. So this is the boundary condition we impose. Changing coordinates, we have a solution of ζ = 0 in the coordinates (t, r, θ, φ). This solution is not compactly supported (since it need not vanish at the origin), but for each t it vanishes for large r.

46 The Change of Coordinates As in the black hole case, we change variables to the coordinate u(r) = T (α)k(α) r dα. α 2 This maps the interval (0, ) to (, 0). Some asymptotics: r /u 0 yields u = 1 r + O ( ) 1 r 2 = r + O(1) 1 u r 0/u yields u = 1 r + O(1) 1 u = r + O(r 2 ) Letting ψ(t, u, θ, φ) = ζ(t, r(u), θ, φ) we find ( r 4 2t + 2u + r 2 T 2 S 2 ) ψ = 0 on R (, 0) S 2

47 The Cauchy Problem One can then show that ψ is the global, smooth solution of the Cauchy problem ( ) r 4 t 2 + u 2 + r 2 T 2 S 2 ψ = 0 on R (, 0) S 2 ψ u = O ( ) 1 u as u 3 (ψ, iψ t )(0, rθ, φ) = Ψ 0 (r, θ, φ) B 2 B is the set of smooth functions on (, 0) that are supported away from zero, ψ u = O ( ) 1 u as u, and ψ 3 and all its derivatives have finite limits at

48 Energy & Hamiltonian Reformulation As in the black hole case, the solution to this has a conserved energy. Also as in the black hole case, this problem can be recast as a Hamiltonian system (the form of the Hamiltonian H is the same), and there exists a Hilbert space H on which H is essentially self-adjoint due to energy conservation. We also project to spherical harmonics, as before. The analysis to find a representation formula and show decay is similar, but there is a marked difference.

49 The Resolvent We wish to construct the resolvent; to that end we consider the eigenvalue equation H l Γ = ωγ. Changing to the variable s(u) = u r 2 (u )du and letting η(s) = r(u(s))gamma(u(s)), this is equivalent to the ODE ( l(l + 1) η (s) ω 2 η(s) + r 2 T 2 1 ( T rt 2 K 2 T + K K )) η = 0 on (0, ). The form of the ODE is the same as in the black hole case, but the domain is different. We can construct a solution to this with boundary conditions at s = as before, but we need to construct a solution with boundary conditions at s = 0. Fortunately, this can be done and the rest of the analysis can also be done (with a couple of nontrivial modifications).

50 References [1] [Smoller et al., 1993] Existence of Black Hole Solutions for the Einstein-Yang/Mills Equations. Communications in Mathematical Physics, 154: [2] [Kronthaler, 2006] The Cauchy problem for the wave equation in the Schwarzschild geometry. Journal of Mathematical Physics, 47(4): , April [3] [Finster et al., 2006] of Solutions of the Wave Equation in the Kerr Geometry. Communications in Mathematical Physics, 264: , June 2006.

51 Summary & Future Work Summary: We have demonstrated that in a general class of spherically symmetric spacetimes (including both black hole and particle like cases) that scalar waves decay as t. Future Work: Obtain sharp rates of decay Extend to less regular coefficients and data Extend to axially symmetric case (generalize Kerr)

52 Acknowledgments Thanks to Joel Smoller for introducing this problem and his helpful discussions. Special acknowledgment to Johann Kronthaler, whose (2006) work served as a roadmap in solving these problems.

53 Thanks for your attention!