aa r X i v : . [ m a t h . N T ] O c t On self-adjointness of Poisson summation
Johannes L¨offler
Abstract
We show that a combination of well-known operators, namely i τ ◦ H ◦ Z is self-adjoint and ad-hoc related to the ζ function. Here τ is an involution appearing in Weil’s positivitycriteria needed for symmetrization, H a regularization operator introduced by Connes [6] and Z essentially Poisson summation. We elaborate on the Hilbert-P´olya conjecture, discuss why theHermite-Biehler theorem, uncertainty relations and cohomologies are interesting in our scenario. Introduction
We hope our exposition contains at least for non-experts new informations and maybe canstimulate other approaches, but we admit that the meaning of some things is still a puzzle.The zeta function ζ is for ℜ ( s ) > ζ ( s ) := Q p ∈ P − p − s = P ∞ n =1 n − s where P isthe set of primes. One way to continue ζ to the complex plane is the following [20]:Let the Mellin transform of f as usual be defined by M [ f ]( s ) := R ∞ tt t s f ( t ). For example wewill in the following considerations sometimes use the so-called Γ function for ℜ ( s ) > e − t , for instance we set Γ( s ) := M [exp( − t )]( s ). The Γ function admitsits continuation by the faculty equation Γ(1 + s ) = s Γ( s ).Let the Fourier transform sending f to b f as usual be defined by b f ( p ) := R ∞−∞ d x f ( x ) e − π i xp .Mellin transform and Fourier transform are related by the equation M [ f ](i s ) = \ ( f ◦ e − id )( s ).For β ∈ R + \ d β by ( d β f )( y ) := f ( βy ) also called scalingoperator. We have [ d β , d β ] = 0 and dilations also commute with the Zeta operators Z α := ∞ X n =1 d n α (0.1)With this notation we define Ψ : R + → R + as usual by Ψ( t ) := Z (cid:0) exp( − πt ) (cid:1) = P ∞ n =1 e − πn t .Notice that we can estimate Ψ by a geometric series for instance we have the inequality Ψ( t ) ≤ exp( − πt ) / (1 − exp( − πt )) and hence Ψ( t ) converges exponentially fast to zero as t → + ∞ .By t → πn t substitutions in the Mellin transform corresponding to the Γ function we have π − s/ ζ ( s )Γ( s/
2) = π − s/ ∞ X n =1 n − s Z ∞ d tt t s/ e − t = Z ∞ d tt t s/ Ψ( t ) (0.2)The most famous analytic continuation due to Riemann is by Ξ( s ) := π − s/ Γ( s/ ζ ( s ) andΞ (cid:18) s (cid:19) = − − s + 2 Z ∞ d tt t / Ψ ( t ) cosh (cid:18) ln( t ) s (cid:19) (0.3)The Poisson summation formula can be considered as a special case of a more geometricone called the Selberg trace formula and briefly is described as follows:Let F ( x ) for x → ±∞ decays faster than 1 / | x | (it is well-known that this assumption can beweakened in several ways). The series P ∞−∞ F ( x + n ) is a well-defined function of the argument x and has period 1, hence can be expanded as a Fourier series and a calculation yields1 X n = −∞ F ( x + n ) = ∞ X n = −∞ b F ( n ) e i2 πnx (0.4)The Theta Θ function is defined as the Poisson summation of a Gaussian, for instance we setΘ( t ) := P ∞ n = −∞ exp( − πn t ). 0.4 for x = 0 combined with d d t f ( p ) = ( d /t b f )( p ) / | t | establishesthe functional equation Θ( t ) = √ t Θ (cid:0) t (cid:1) because the Gaussian is a fixed point of the Fouriertransform. This implies that Ψ( t ) = (Θ( t ) − / t ) = t − / [Ψ (1 /t ) + 1 / − / s ) = Ξ(1 − s )Among other things 0.3 implies that ζ has so-called trivial zeros at the negative even integers − , − , − , − , − , · · · because it is well-known that Γ has simple poles at negative integers.Riemann conjectured [17] that all non trivial zeros of ζ have real part 1 / Acknowledgements:
I am deeply grateful to P. Moree for many inspiring discussions. I wantto thank S. Bhattacharya, R. Friedrich, H. Furusho, D. Radchenko and O. Ramar´e for readingdraft versions, discussions and pointing out references.
Definition
Let α ∈ C and the operators H α , ∆ α : C ∞ ( R ) → C ∞ ( R ) be defined by H α := id + αt ∂∂t and ∆ α := 2 αt ∂∂t + α (cid:18) t ∂∂t (cid:19) For α = 2 we have essentially H ∝ XP + P X , where X and P are the position and momentumoperators respectively and the proportionality factor is i ~ / ~ is the Planck constant. ThisHamiltonian for the standard inner product was introduced by Connes in his approach towardsa proof of the generalized RH see [4] and [5] and the article [1] of Berry and Keating.A well-known fact is that H α commutes with dilations d β , i.e. [ H α , d β ] = 0 because of thechain rule, hence H α ′ commute with the Poisson summation maps Z α . For any α, α ′ we canexchange H α and H α ′ . By calculation we have H α = id +∆ α and H α also commutes with ∆ α ′ .A crucial regularization property of an operator of the shape H α = id + αt∂ t is that it kills t − /α singularities. Hence a singularity of the shape P ki =1 c i t − /α i gets canceled if we apply Q ki =1 H α i . For a logarithmic singularity of the shape t − /α ln n ( t ) with n ∈ N we have H mα (cid:16) t − /α ln n ( t ) (cid:17) = m ! (cid:18) nm (cid:19) α m t − /α ln n − m ( t ) (1.1)by a small calculation, hence we have the vanishing H n +1 α t − /α ln n ( t ) = 0, in other words H α can also eliminate positive integer powers of logarithmic singularities.This is a warning, the ζ function and Riemann’s original continuation formula are quiteintricate: For an unexperienced reader the l.h.s of Riemann’s original illusive formula41 − z = 2 Z ∞ d tt t / Ψ( t ) cosh (cid:18) ln( t ) z (cid:19) (1.2)for the Ξ ((1 + z ) /
2) = 0 zeros z and also the l.h.s of the formula4 z − /z = 2 Z ∞ d tt t / ( H Ψ)( t ) sinh (cid:18) ln( t ) z (cid:19) (1.3)2ay look like a direction to an algebraic coordinate descriptions of the Ξ zeros, but this is morea Fata Morgana because in the further integration by parts procedure this hints get lost:Let ϕ : [1 , ∞ ] → C be a smooth function, with no singularities at 1 and in addition enjoyingthe property that at ∞ the function ϕ and all of its derivatives are rapidly decaying. Proposition
For ϕ as above and ± , = s ∈ C we have Z ∞ d tt t /α ϕ ( t ) ( coshsinh (cid:18) ln( t ) sα (cid:19) = − s "( αϕ (1) + Z ∞ d tt t /α ( H α ϕ )( t ) ( sinhcosh (cid:18) ln( t ) sα (cid:19) = − − s "( ( αH α ϕ )(1) − sαϕ (1) + Z ∞ d tt t /α (∆ α ϕ )( t ) ( coshsinh (cid:18) ln( t ) sα (cid:19) We will use the full statement of the following auxiliary lemma 1.0.3 to show that it is noteasy to chase the
Fata Morgana , but here it is essentially enough to consider only the case m = 0and α = 4, for instance: If Ψ : R + → C is smooth with Ψ( t ) = t − [Ψ (1 /t ) + C/ − C/ ψ n := ∆ n Ψ satisfies ( H ψ )(1) = − C/ H ψ n )(1) = 0 for all n ∈ N + because ψ n (1 /t ) = t − ψ n ( t ) + δ n C (cid:2) t − − (cid:3) / H ψ n )(1 /t ) = − t − ( H ψ n ) ( t ) − Cδ n (cid:2) t − + 1 (cid:3) / Proposition
Consider a smooth function Ψ ± : R + → C enjoying Ψ ± ( t ) = ± ( − m t − α Ψ ± (1 /t ) + C ln m ( t ) h t − α ∓ i / for some m ∈ N and α ∈ R \ . The series ψ n defined by ψ ± n := ∆ nα Ψ ± respectively satisfy ψ ± n ( t ) = ± ( − m t α ψ ± n (cid:18) t (cid:19) + Cα n n X i =0 (2 n − i )!( α/ i (cid:18) m n − i (cid:19)(cid:18) ni (cid:19) ln m − n + i ( t ) (cid:20) ( − i t α ∓ (cid:21) H α ψ ± n ( t ) = ∓ ( − m t α H α ψ ± n (cid:18) t (cid:19) − Cα n n X i =0 (2 n − i )!( α/ i (cid:18) m n − i (cid:19) (cid:18)(cid:18) n + 1 i + 1 (cid:19) + (cid:18) ni + 1 (cid:19)(cid:19) ln m − n + i ( t ) (cid:20) ( − i t α ± (cid:21) If m is odd this implies ψ + n (1) = − Cα m n − m m ! (cid:0) n n − m (cid:1) = − H α ψ − n (1) for all n ∈ N and if m iseven we have H α ψ + n (1) = − Cα m n − m m ! (cid:16)(cid:0) n +12 n − m +1 (cid:1) + (cid:0) n n − m +1 (cid:1)(cid:17) = − ψ − n (1) .Proof. Notice 1.4 is self-consistent and ψ ( t ) = ± t − α ψ (1 /t ) implies H α ψ ( t ) = ∓ t − α ( H α ψ )(1 /t )by differential calculus. As a hint it is easier to evaluate H α ∆ nα t − α ln m ( t ) with help of 1.1 byswitching operator coordinates, for instance put H α = 2 H α/ − id and ∆ α = 4 (cid:0) H α/ − H α/ (cid:1) anduse ( A + B ) n = P ni =0 (cid:0) ni (cid:1) A i B n − i for commuting operators. Now use (cid:0) a +1 b +1 (cid:1) = (cid:0) ab (cid:1) + (cid:0) ab +1 (cid:1) .We consider 1.0.3 for the case α = 4, n = 0: Riemann’s analytic continuation 0.3 combinedwith the previous rewriting 1.0.2 shows the iteration of the integration by parts procedure isΞ (cid:18) s (cid:19) = − − s δ n + (cid:18) − − s (cid:19) n Z ∞ d tt t / (∆ n Ψ) ( t ) cosh (cid:18) ln( t ) s (cid:19) (1.5) ∀ n ∈ N . Formula 1.5 was discovered by P´olya in the Nachlass of Jensen [15].3otice that (∆ Ψ)( t ) ∝ P ∞ n =1 (cid:2) π n t − n t (cid:3) exp( − πn t ) is [1 , ∞ ] strictly positive, thiscontradicts zeros of Ξ on R . The function ( H Ψ)( t ) ∝ P ∞ n =1 (cid:2) − n t (cid:3) exp( − πn t ) is a strictlynegative charged integral kernel restricted to the integration interval [1 , ∞ ].Because of the functional equation (∆ n +14 Ψ)( t ) = t − / (∆ n +14 Ψ)(1 /t ) the functions (∆ n +14 Ψ)( t )behave also nice for t → (cid:18) s (cid:19) = M (cid:2) ∆ n +14 Ψ (cid:3) (cid:18) s (cid:19),(cid:0) s − (cid:1) n +1 (1.6)Formula 1.5 also implies that R ∞ d t t − / (∆ n Ψ) ( t ) cosh (ln( t ) s/
4) decays rapidly for |ℑ ( s ) | → ∞ ,roughly the oscillating in the integral kernel gets fast. The previous formula 1.6 has been discussedby various people, and P´olya’s conjecture that the Tur´an inequalities are satisfied was verified [8]by Csordas, Norfolk and Varga.Let us follow 1.2 approximately with help of 1.0.2, 1.0.3, the calculation is not importantin the following but maybe a bit amusing: We first use the well-known hyperbolic additiontheorem cosh( a + b ) = cosh( a ) cosh( b ) + sinh( a ) sinh( b ) to split up the real and imaginary partof the r.h.s. of the zero equation 1.2. For the imaginary part of 1.2 we have with sinh( x ) = P ∞ m =0 x m +1 / (2 m + 1)! the following formal expansion8 xy i(1 − x + y ) + 4 x y = ∞ X m =0 ( x/ m +1 (2 m + 1)! 2 Z ∞ d tt t Ψ( t ) ln m +1 ( t ) sinh (cid:18) ln( t )i y (cid:19) = ∞ X m =0 ( x/ m +1 (2 m + 1)! " y m +1 X i = m +1 (cid:18) −
11 + y (cid:19) i +1 m +1 i − m − (2 m + 1)! (cid:18) i m + 1 − i (cid:19) (1.7)+ 2(1 + y ) m +2 Z ∞ d tt t ∆ m +24 (cid:0) Ψ( · ) ln m +1 ( · ) (cid:1) ( t ) sinh (cid:18) ln( t )i y (cid:19) where we used 1.0.3 for the + case and (cid:0) ab (cid:1) = (cid:0) aa − b (cid:1) . It would be absurd if we just truncate,approximate one side of 1.7 where the first line represents the only algebraic terms appearingand the second line integrals that decay faster than any negative power of y , because∆ m +24 (cid:0) Ψ( · ) ln m +1 ( · ) (cid:1) ( t ) = − t − / ∆ m +24 (cid:0) Ψ( · ) ln m +1 ( · ) (cid:1) (1 /t ) (1.8)The algebraic line can be rewritten as follows − y y ∞ X m =0 (cid:16) x (cid:17) m +1 2 m +1 X i = m +1 (cid:18) −
41 + y (cid:19) i (cid:18) i m + 1 − i (cid:19) = − y y ∞ X i =0 (cid:18) − x y (cid:19) i ∞ X m =0 (cid:16) x (cid:17) m (cid:18) im (cid:19) (cid:0) − ( − m + i (cid:1) = − y y ∞ X i =0 (cid:18) − x y (cid:19) i (cid:2) ( x/ i − ( − i ( − x/ i (cid:3) = − y y "
11 + x +2 x y −
11 + x − x y = 8 xy i(1 − x + y ) + 4 x y ≤ Z ∞ d tt t / f (cid:0) ln( t ) (cid:1) Ψ − H Ψ∆ Ψ ( t ) ( coshsinh (cid:18) ln( t ) x (cid:19) (cid:20) √ ± cos (cid:18) ln( t ) y (cid:19) ± ′ sin (cid:18) ln( t ) y (cid:19)(cid:21) for all f that are positive on [0 , ∞ ] and x ≥ H ∆ Ψ)( t ) changes sign on [1 , ∞ ] is a crucialpoint for the proof of the subtle Tur´an inequalities contained in [8]. Mellin transforms and generic zeros
Formula 1.6 can be considered as a special case of the following more general theorem 2.0.4corresponding to H in the previous picture, in other words the following regularization is notthe operator H compatible with the symmetry Ψ( t ) = t − / [Ψ (1 /t ) + 1 / − / Theorem
Let < λ ∈ R and consider a function f of rapid decay at ∞ and with ( H ( f ◦ | id | λ )) ∈ C ( R + ) . We have an analytic continuation by the formula (1 − s ) ζ ( s ) M [ f ]( s/λ ) = M (cid:2) H λ ◦ Z λ f (cid:3) ( s/λ ) for ℜ ( s ) > , hence in this domain (1 − z ) ζ ( z ) = 0 ⇒ M (cid:2) H λ ◦ Z λ f (cid:3) ( z/λ ) = 0 .Proof. We give a proof for the convenience of the reader, although the arguments are well-known:We define an even function F λ ( t ) := f ( | t | λ ) and our assumptions on f imply F λ (0) + 2 ∞ X m =1 F λ (cid:0) m | t | λ (cid:1) = f (0) + 2 ∞ X m =1 f (cid:0) m λ t (cid:1) = 1 | t | λ c F λ (0) + 2 | t | λ ∞ X m =1 c F λ (cid:18) m | t | λ (cid:19) (2.1)or in other words we assumed that Poisson summation is valid.It is well-known that the Fourier transform b F ( p ) satisfies b F ( p ) ≤ Cp − k for real p → ∞ withsome constant C if g is k times differentiable and if F is smooth b F decays rapidly at ∞ .For a rapidly at ∞ decaying function f ( t ) also ( Z λ ) f ( t ) = P ∞ n =1 f ( n λ t ) is rapidly decaying at ∞ : A finite sum of rapidly decaying functions is of rapid decay and we have by assumption thatfor any m ∈ N exist M m , C m ∈ N with | f ( n λ t ) | < C m t − m /n λm for n ≥ M m hence if m > /λ ∞ X n =1 | f ( n λ t ) | ≤ M m X n =1 | f ( n λ t ) | + C m ζ ( λm ) t − m We apply the rescaled operator H λ = id + λt∂ t on the formula 2.1 for t ∈ R + F λ (0) + 2 ∞ X m =1 ( H F λ ) (cid:0) mt /λ (cid:1) = f (0) + 2 ∞ X m =1 ( H λ f ) (cid:0) m λ t (cid:1) = 2 t /λ ∞ X m =1 (cid:0) \ H F λ (cid:1) (cid:16) mt /λ (cid:17) (2.2)5here we used Leibniz rule, H λ t − /λ = 0, the fact that t∂ t is invariant under dilations, theformula t∂ t ( f ◦ id λ ) = λ ( t∂ t f ) ◦ id λ and usual properties of the Fourier transform, for instancethe identity p∂ p b F ( p ) = − \ ∂ x xF ( p ).Because lim t → f ( t ) does not diverge by assumption we have for ℜ ( s ) > ζ ( s ) Z ∞ d tt t s/λ f ( t ) = Z ∞ d tt t s/λ ∞ X n =1 f ( n λ t ) = − λs Z ∞ d tt t s/λ t∂ t ( Z λ f )( t )where we made the substitution t = n λ t ′ and used P ∞ n =1 f ( n λ t ) is of rapid decay at ∞ .Now consider the linear combination of integrals corresponding to H λ : If f ∈ C ( R + ) thedifferentiation assumption in the theorem is obviously satisfied. Moreover we yield a continuationto ℜ ( s ) > \ H F λ ( p ) decays at least like 1 /p , hence1 | t | /λ ∞ X m =1 (cid:0) \ H F λ (cid:1) (cid:16) mt /λ (cid:17) ≤ Cζ (2) | t | /λ and this shows that the integral kernel on the r.h.s. of 2.0.4 has no singularity as t → F λ ( t ) := f ( | t | λ ). Therestrictions on the test functions in 2.0.4 can be weakened in several ways, for example insteadof introducing H we could continue to ℜ ( s ) > F (0) = 0 and c F λ (0) = Z ∞−∞ d xf ( | x | λ ) = 0 (2.3)hence the reader might wonder why we prefer a less general and simple statement with H λ ◦ Z λ and no integration assumptions. The first main advantage of this rewriting may be that wehave an interpretation for the terms: H λ is a neat regularization of Z λ , the composition is well-behaved. Moreover as we will prospect in the following section 3 our regularized combination iseasy to symmetrize while the symmetrization of the condition 2.3 seems from our perspectivesomehow rigid and slightly less flexible considering standard symmetrization procedures, moreprecise the image under Z λ of functions satisfying the two conditions 2.3 does not necessarilysatisfy 2.3, only the first of this conditions, namely the vanishing at zero, is obviously conservedby Z λ . Contrary the vanishing condition 2.3 is satisfied for the reminiscent Lie brackets[ f, g ] λ ( t ) := f ( t ) Z ∞−∞ d xg ( | x | λ ) − g ( t ) Z ∞−∞ d xf ( | x | λ ) (2.4)Skew-symmetry of [ · , · ] λ is obvious and the validity of the Jacobi identity and R ∞−∞ d x [ f, g ] λ ( | x | λ )are just straight forward checks. It is also immediate that the space of functions satisfying 2.3 isan ideal with respect to [ · , · ] λ . The condition 2.3 is one of the main points where our descriptionin subsection 3.1 slightly differs from Connes spectral realization of the ζ zeros.Let us also mention that if we suppose the stronger restriction that f is smooth and of rapiddecay at ∞ we can also continue to C quite analogous to Riemann’s original calculation 0.3 by ζ ( s ) Z ∞ d tt t sλ f ( t ) = − λ " F λ (0) s + b F λ (0)1 − s + Z ∞ d tt ∞ X n =1 h t sλ F λ (cid:0) nt λ (cid:1) + t − sλ c F λ (cid:0) nt λ (cid:1)i (2.5)6n consideration of the Fata Morgana H α = id +∆ α and technically curethe a priori quite ill substitution H α ! = ±√ α . Notice the Taylor expansion of √ x = P ∞ i =0 (cid:0) / i (cid:1) x i for | x | < t ν if | αµ + α µ | < F ( x ) to P ∞−∞ F ( x + n ) can be rewritten as the sum id + R +( − id) ◦ R ◦ ( − id ) where the operator R is defined by the formula ( R F )( x ) := P ∞ n =1 F ( x + n )and where we just set as usual (( − id ) f )( x ) = f ( − x ). By this we want to capture that the map R defines a Rota-Baxter operator of weight 1, i.e. we have (cid:0) R ( F · G ) (cid:1) ( x ) = ( R F )( x ) · ( R G )( x ) − (cid:0) R (( R F ) · G ) (cid:1) ( x ) − (cid:0) R ( F · ( R G )) (cid:1) ( x ) (2.6)Inspecting singularities this is paradox compared with 2.1: Poisson summation ( Z f )( t ) = ( R ◦ d t f )(0) produces t singularities on the l.h.s. while on the r.h.s. of 2.6 a priori a squared singularityof the shape t appears and therefore should cancel with the other two more involved terms. It seems interesting to try to extract out of 2.0.4 information about the location of the ζ zerosby variation methods. The appearance of ζ in form Z λ is somehow responsible that there can bestable “points”: ∀ s ∈ C the integral functional ϕ → R ∞ tt t s/λ ( ϕ + λtϕ ′ ) does not admit stable“points” ϕ ∈ C ( R + ), the Euler-Lagrange equations dd t ∂L∂ϕ ′ − ∂L∂ϕ = 0 would imply (1 − s ) t s/λ − =0 ∀ t ∈ R + hence a contradiction if s = 1. However, in the previous variation we neglect themore interesting non-local deformation Z λ representing the ζ function and just considered H λ .Clearly Z λ is not a local operation we only have for the support of H λ ◦ Z λ f the implicationsupp( f ) ⊂ [0 , δ ] ⇒ supp( H λ ◦ Z λ f ) ⊂ [0 , δ ] (2.7)Obviously for some Λ ∈ R the symmetrized or skew-symmetrized integral functionals ϕ → R ∞ tt t Λ /λ ( t s/λ ± t − s/λ )( H λ ◦ Z λ ϕ )( t ) respectively admit a stable vanishing imaginary or realpart respectively if ℜ ( s ) = 0 and clearly a similar statement holds if ℑ ( s ) = 0.Moreover P´olya claims that the Euler product representation and the Hermite-Biehlertheorem imply the observation ξ (1 + s ) + ξ (1 − s ) = 0 ⇒ s ∈ i R where ξ (cid:18) s (cid:19) := (1 − s )Ξ (cid:18) s (cid:19) = − Z ∞ d tt t / (∆ Ψ)( t ) cosh (cid:18) ln( t ) s (cid:19) (2.8)In fact he proves a more general statement about how the zeros react if we impose quite theusual symmetric or also a skew-symmetric functional equation by brute force: Theorem
If an entire real non-constant function f ( s ) with Hadamard product f ( s ) = ce − γs + βs z q Y (cid:16) − sz (cid:17) e sz (cid:16) − sz (cid:17) e sz (where c, γ, β are real constants, γ ≥ and q is an integer) has no zeros for ℜ ( s ) > λ then the twoat the Λ ≥ λ axis respectively symmetrized or skew-symmetrized functions f (Λ + i s ) ± f (Λ − i s ) respectively vanish identically or admit only real zeros.Proof. Roughly the punch line of P´olya’s argumentation goes as follows, for details we refer tothe original article [15]. First we can approximate f by Hadamard factorization by polynomials,then the Hermite-Biehler theorem deduces the statement for the roots of the polynomials:7he Hermite-Biehler theorem states that a polynomial p ( x ) = e( x ) + x o( x ) with real evenpart e( x ) ∈ R [ x ] and real odd part o( x ) ∈ R [ x ] is stable, i.e. f ( x ) = 0 ⇒ ℜ ( x ) <
0, if andonly if e( − x ) and o( − x ) have simple real interlacing roots and ℜ (cid:0) e( x ) /x o( x ) (cid:1) > x with ℜ ( x ) >
0, it is well-known that this condition can also replaced by the condition that[ e, o ] ′ ( s ) ≥ ∀ s ∈ i R , here [ · , · ] ′ is the so called Wronskian defined by [ f, g ] ′ := f ′ g − f g ′ , see4.0.11. Finally we conclude the theorem by Hurwitz theorem on the limit.For example if ξ has a zero free region ℜ ( s ) > λ for some 1 / ≤ λ ∈ R then ∀ Λ ≥ λ we have ξ (Λ + s ) ± ξ (Λ − s )= − Z ∞ d tt t (∆ Ψ)( t ) ( coshsinh (cid:18) ln( t )(2Λ − (cid:19)( coshsinh (cid:18) ln( t ) s (cid:19) = 0 ⇒ s ∈ i R (2.9)with the obvious exception that ξ (cid:0) + s (cid:1) − ξ (cid:0) − s (cid:1) identically vanishes, we refer to [11] and [19]. In this context also other results of for example P´olya [15], [16], Brujin [2], Newman [13] andOdlyzko [14] on the reality of the roots of certain trigonometric integrals are interesting, we referfor more detailed considerations of the universal factors to [10]. Briefly they showed that certainclasses of trigonometric integrals admit only real roots and that there exist universal integralkernel multiplication factors, in our notation the functions e − ρ ln ( t ) with ρ ∈ R + , conserving andeven improving the reality of the roots of trigonometric integrals R ∞ tt t i s/ ϕ ( t ) with the integralkernel symmetry ϕ ( t ) = ϕ (1 /t ).It is not difficult to project on solutions of this symmetry, namely for smooth f of rapid decayat 0 and ∞ the functions f → ϕ ± f ( t ) := (cid:0) f ( t ) ± f (1 /t ) (cid:1) / ϕ ± ( t ) = ± ϕ ± (1 /t ) respectively. If f is real and we have f = ϕ + f + ϕ − f . The functions e − ρ ln ( t ) are symmetric under t ↔ /t and it isagain easy to project on the solutions, for instance for f as above f → φ ± f ( t ) := ( f ( t ) ± f (1 /t )) / φ ± ( t ) = ± φ ± (1 /t ) and we have the decomposition f = φ + f + φ − f .Following P´olya we have the Mellin transform interpretation M [ e − ρ ln ( t ) ]( s ) = M [ e − ρ ln ( t ) ](0) exp (cid:18) s ρ (cid:19) = r πρ exp (cid:18) s ρ (cid:19) (2.10)because we have by integration by parts the well-known differential equation ∂ s M [ e − ρ ln ( t ) ]( s ) = Z ∞ d tt t s ln( t ) e − ρ ln ( t ) = − ρ Z ∞ d t t s ∂ t e − ρ ln ( t ) = s ρ M [ e − ρ ln ( t ) ]( s )Consider the functions Ξ f ( s ) := M [Ψ · f ]( s/
2) and ˜Ξ f ( s ) := M [( H Ψ) · f ]( s/
2) where weassume f decays rapidly at 0 and ∞ to avoid integration convergence issues, hence in this caseΞ f and ˜Ξ f are obviously entire functions.If f is real we have Ξ f ( s ) = Ξ f ( s ). We can decompose Ξ f ( s ) = Ξ φ + f ( s ) + Ξ φ − f ( s ) and if φ ± ( t ) = ± φ ± (1 /t ) we can absorb the inhomogeneity ( t − / − / t ) = t − / [Ψ (1 /t ) + 1 / − / φ ± =Λ φ ± + Ω φ ± withΛ φ ± ( s ) := Z ∞ d tt (cid:16) t s ± t − s (cid:17) Ψ( t ) φ ± ( t ) and Ω φ ± ( s ) := Z d tt t s − − t s φ ± ( t )clearly Λ φ ± ( s ) = ± Λ φ ± (1 − s ) but this symmetry is in general broken for the error summand Ω φ ± :8 roposition Let Ξ mφ ± ( s ) := P ml =0 Ξ φ ± ( s + l ) and ˜Ξ mφ ± ( s ) := P ml =0 ( − l ˜Ξ φ ± ( s + l ) . Ξ mφ ± ( s ) ∓ Ξ mφ ± (1 − m − s ) = M (cid:20) φ ± (cid:21)(cid:18) s − (cid:19) − M (cid:20) φ ± (cid:21)(cid:18) s + m (cid:19) (2.11)˜Ξ mφ ± ( s ) ± ( − m ˜Ξ mφ ± ( s )(1 − m − s ) = −M (cid:20) φ ± (cid:21)(cid:18) s − (cid:19) − ( − m M (cid:20) φ ± (cid:21)(cid:18) s + m (cid:19) (2.12) Proof.
The telescope identity Ω mφ ± ( s ) := P ml =0 Ω φ ± ( s + l ) = R
10 d tt (cid:16) t s − − t s + m (cid:17) φ ± ( t ) holds.We also have the two parity equations M [ φ ± ] ( s ) = ±M [ φ ± ] ( − s ) respectively.The functions Ξ me − ρ ln2( t ) and ˜Ξ me − ρ ln2( t ) solve (cid:0) ∂ ρ + 4 ∂ s (cid:1) Ξ me − ρ ln2( t ) ( s ) = 0, notice the similaritywith the so-called heat equation. We have Ξ e − ρ ln2( t ) ( s ) = P ∞ n =0 ( − ρ ) n n ! ∂ n Ξ( s ) and ˜Ξ e − ρ ln2( t ) ( s ) =(1 − e − ρ ln2( t ) ( s ) + 16 ρ∂ s Ξ e − ρ ln2( t ) ( s ) holds. 2.1.2 and 2.10 imply for ∀ m ∈ N for exampleΩ me − ρ ln2( t ) ( s ) − Ω me − ρ ln2( t ) (1 − m − s ) = M (cid:2) e − ρ ln ( t ) (cid:3) (0) (cid:0) e ( s − / ρ − e ( s + m ) / ρ (cid:1)(cid:14) ρ > me − ρ ln2( t ) ( s ) − Ξ me − ρ ln2( t ) (1 − m − s ) = 0 ⇔ s ∈ − m ρπ i Z m (2.13)˜Ξ me − ρ ln2( t ) ( s ) + ( − m ˜Ξ me − ρ ln2( t ) (1 − m − s ) = 0 ⇔ s ∈ − m ρπ i (cid:18) −
12 + Z m (cid:19) Hence ∀ ρ > − m + i R , if and only if m is odd both symmetrieson the l.h.s of 2.13 are satisfied simultaneous on the points − m + 16 ρ π i Z m . Approximating bypolynomials we also see that ∂ ns Ξ me − ρ ln2( t ) ( s ) − ( − n ∂ ns Ξ me − ρ ln2( t ) (1 − m − s ) = 0 ⇒ ℜ ( s ) = 1 / ρ → Ξ e − ρ ln2( t ) = Ξ and in the limit 2.13 vanishesbecause of Ξ(1 − s ) = Ξ( s ). The previous considerations seem a bit bizarre in consideration of2.1.1 and the remarkable result of Speiser that a zero of ζ is equivalent to a zero of the derivative ζ ′ inside the critical strip [18].In the sequel to the present paper we will discuss the more general family of integralsΞ( ρ, ~s ) := Z ∞ d t t · · · Z ∞ d t n t n n Y i =1 t s i / i Ψ( t i ) ! e − P ≤ i,j ≤ n ρ ij ln( t i ) ln( t j ) where ρ denotes a symmetric n × n matrix underlying certain convergence restrictions and ~s ∈ C n . Some comments on the Hilbert-P´olya conjecture
The
Hilbert-P´olya conjecture goes back to a discussion of P´olya with Landau and it statesthat the Ξ zeros should be interpreted as the energy levels of a quantum system. More precise theconjecture is the existence of a hermitian operator H with the property that if Ξ(1 / E ) = 0then there is a eigenstate ψ E with eigenvalue E , i.e. H ψ E = Eψ E . Because the absolute value ofnon-trivial zeros can be arbitrary big by [15] the Hilbert-P´olya operator H can only be realizedas an unbounded operator. Works of Berry, Keating and Connes gave various evidence that H should be some quantization of H = XP + P X . One of the ideas in [1] is to get a discrete spectrumby imposing boundary conditions for the space of functions on which H acts. A keystone in [4]is to incorporate a topology related to prime numbers and from this point of view our basicconsiderations might be naive. 9t is well-known that the Schwartz space is a dense subspace of the Hilbert space L ( R ) ofsquare integrable functions. The completion of pre-Hilbert spaces and the extension of symmetricoperators are an intensive studied research topic where the Cayley transform, i.e. the map A → U A := ( A − i) ◦ ( A + i) − taking a symmetric operator A to the isometry U A : Ran( A + i) → Ran( A − i), provides useful reformulations. In the following we work with an adequate not Cauchycomplete space H equipped with an inner product: Definition
Consider the pre-Hilbert space H of rapidly at and ∞ decaying smoothfunctions equipped with the inner product h f , f i ~ := R ∞ d t t ~ f ( t ) f ( t ) for ~ ∈ R . Consider a function V that is in addition real on the real axis, i.e. V ( t ) = [ v ( t )+ v ( t )] / V ( t )holds ∀ t ∈ R + , it is well-known that for such a real V the operator m V is hermitian where Definition
For a function V with at worst t λ singularities at and ∞ let as usual themultiplication operator m V : H → H operate on states f ∈ H by f ( t ) → ( m V f ) ( t ) := V ( t ) · f ( t ) . Definition
Let g := { A : H → H|∃ A ∗ : H → H : h f, Ag i ~ = h A ∗ f , g i ~ ∀ f, g ∈ H} denotethe (Lie-)algebra of operators admitting an adjoint and s the sub Lie-algebra of Hamiltonians. Z λ begins with the identity, by the previous reasonings is related to ζ and in 3.1.2 we introducea non-local deformation of H λ with help of Z λ , for the cook we just need one more ingredient: Definition By τ ~ we denote τ − ~ where for µ ∈ R \ the operators τ µ ~ are defined by ( τ µ ~ f )( t ) := (cid:0) ( m t ~ f ) ◦ id µ (cid:1) ( t ) = t µ (1+ ~ ) f (cid:0) t µ (cid:1) Notice that for ~ = 0 we have defined the involution appearing in Weil’s positivity criteria3.23 and that the previous considerations 1.0.3 and 2.1.1 can shortly be rewritten with thisdefinition, we have for example the formula ( τ µ ~ ◦ τ ~ f )( t ) = ( f ◦ id − µ )( t ) = ( τ − µ − f )( t ) = f (1 /t µ ).It is clear that t µ (1+ ~ )1 − µ is a fixed point of τ µ ~ and some of the following reasonings in summaryshow that τ ~ for the test functions H is a hermitian involution, i.e. τ ~ ∗ = τ ~ and τ ~ ◦ τ ~ = id.In 3.0.6 we have not defined an honest group action of R \
0, the composition just satisfies (cid:0) τ µ ~ ◦ τ µ ~ f (cid:1) ( t ) = (cid:0) m t µ ~ ) ◦ τ ~ µ µ f (cid:1) ( t ) = (cid:16) τ µ µ ~ ◦ m t ~ µ f (cid:17) ( t ) = t µ (1+ ~ ) (cid:0) τ µ µ ~ f (cid:1) ( t ) (3.1)and hence only a bona fide group action of R \ ~ = −
1. The inverse ( τ µ ~ ) − of τ µ ~ is given by (cid:0) ( τ µ ~ ) − f (cid:1) ( t ) := (cid:16) m t − (1+ ~ )(1+ µ ) µ ◦ τ /µ ~ f (cid:17) ( t ) = t ~ f (cid:0) t /µ (cid:1) . A reason for the definition 3.0.6 canbe seen in the adjunction h τ µ ~ f , f i ~ = h f , ( τ µ ~ ) ∗ f i ~ with the adjoint ( τ µ ~ ) ∗ given by (cid:0) τ µ ~ (cid:1) ∗ = τ /µ ~ / | µ | where we restrict to test functions in H and with respect to 3.0.3.Let the adjoint conjugation be defined by Con ∗ B ( A ) := B ◦ A ◦ B ∗ . It is clear by essentiallyiteration of the adjoint conjugation A → B ◦ A ◦ B ∗ of A ∈ s with B ∈ g that for a sequence µ , · · · , µ n ∈ R \ τ ~ µ ◦ · · · ◦ τ ~ µ n ◦ A ◦ τ ~ /µ n ◦ · · · ◦ τ ~ /µ ∈ s but we still did notdefine by this an honest action of R \ s : Explicit we just have by 3.1 τ ~ µ ◦ τ ~ µ ◦ A ◦ τ ~ /µ ◦ τ ~ /µ = τ ~ µ µ ◦ m t ~ µ ◦ A ◦ m t (1+ ~ ) µ ◦ τ ~ µ µ (3.2)= m t µ ~ ) ◦ τ ~ µ µ ◦ A ◦ τ ~ µ µ ◦ m t µ ~ ) and hence again only an action of the multiplicative group R \ ~ = − τ ~ also admits n-th roots, for example ∀ k ∈ Z the n-th power of the maps e i π (1+2 k ) /n − e i π (1+2 k ) /n +1 + τ ~ are proportional to τ ~ , in particular the maps (i ± τ ~ ) / √± i2 square to τ ~ .10he operators (id ± τ ~ ) satisfy (id + τ ~ ) ◦ (id − τ ~ ) = 0, (id ± τ ~ ) m = 2 m (id ± τ ~ ) and (id ± τ ~ ) ◦ τ ~ = ± (id ± τ ~ ). We can split f ∈ H = H + ~ ⊕ H − ~ in f = f + + f − where we project f → f ± := ( f ± τ ~ f ) / ∈ H ± ~ (3.3)onto the two eigenspaces of τ ~ , hence τ ~ f ± = ± f ± is valid and h f + , f − i ~ = 0.The operator m ln( t ) satisfies m ln( t ) ◦ τ ~ = − τ ~ ◦ m ln( t ) and hence defines maps H + ~ ↔ H − ~ ,but admits no inverse because m / ln( t ) could produce a singularity at 1 and is not in anycase well-defined, m ln( t ) maps H − ~ to functions in H + ~ that in addition vanish at 1. We have[ t∂ t , m ln( t ) ] − = id.Let us refer to section 6.1 for a more detailed discussion of the ~ -dependence and just mentionhere that the two projections in 3.3 transform according to the rule id ± τ ~ ′ = Con m t ~ − ~ ′ (id ± τ ~ ).With help of h τ µ ~ f , f i ~ = h f , ( τ ~ µ ) ∗ f i ~ by elementary manipulations we have the formula M (cid:2) τ µ ~ f (cid:3) ( s ) = 1 | µ | M [ f ] (cid:18) ~ + sµ (cid:19) (3.4)Let b again denote the usual Fourier-transform. On H the regularized Poisson summation2.2 can be rewritten with help of τ in the shape H ◦ Z = τ − ◦ Z ◦ b ◦ H and we have H λ ◦ Z λ = τ /λ − ◦ H ◦ Z ◦ τ λ − . Proof I, the four standard symmetrizations of H λ ◦ Z λ Definition
Consider the pre-Hilbert space H ∞ , of at ∞ or at rapidly decaying smoothfunctions equipped with the inner product h f , f i ~ := R ∞ d t t ~ f ( t ) f ( t ) for ~ ∈ R ± respectively. In consideration of τ ~ there is some tension, we have τ ~ H ∞ ⊂ H but a function in τ ~ H could have a t − (1+ ~ ) singularity at zero, and we want to consult this by 3.1.1.Adjunction ∗ = ∗ ( ~ ) is an involution and we invoke a natural projector p = p( ~ ) and squarezero map ∂ = ∂ ( ~ ) = p ◦ m i where we suppress in our notation the ~ dependence: A ∈ g → p( A ) := ( A + A ∗ ) / ∈ s and A ∈ g → ∂ ( A ) := i( A − A ∗ ) / ∈ s (3.5)We can translate between h· , ·i ~ and M by h f , m t i y f i ~ = M [ f · f ]( ~ + 1 + i y ) and have * f , ( p ∂ m t i y f + ~ = ( ℜℑ (cid:16) M [ f · f ]( ~ + 1 + i y ) (cid:17) (3.6)Notice id = p − i ∂ and ∗ = p + i ∂ and the restriction identities p | s = id, ∂ | s = 0 hold, hencep = p, ∂ = ∂ ◦ p = 0 and p ◦ ∂ = ∂ . To call ∂ a differential is misleading, we just have ∂ [ A, B ] − = i2 [p A, p B ] − − i2 [ ∂A, ∂B ] − and p[ A, B ] + = [p A, p B ] + − [ ∂A, ∂B ] + . and clearlyker ∂/ im ∂ = 0. Theorem
Consider the pre-Hilbert space H equipped with the inner product h· , ·i ~ , see3.0.3. For < λ ∈ R the operators Z λ ± : H → H defined by Z λ ± := ( p (cid:0) H λ ◦ Z λ (cid:1) = (cid:0) H λ ◦ Z λ + τ ~ ◦ H λ ◦ Z λ ◦ τ ~ (cid:1) / ∂ (cid:0) H λ ◦ Z λ (cid:1) = i (cid:0) H λ ◦ Z λ − τ ~ ◦ H λ ◦ Z λ ◦ τ ~ (cid:1) / are self-adjoint. We have [ Z λ ± , τ ~ ] ∓ = 0 and hence restrictions Z λ ± : H ± ′ ~ → H ± ′ ± ~ with respect tothe two eigenspaces of τ ~ . Restricted to H we also have hermitian operators by the formulas b Z λ ± := ( H λ ◦ Z λ ◦ τ ~ : τ ~ H ∞ → H ∞ τ ~ ◦ H λ ◦ Z λ : H ∞ → τ ~ H ∞ (3.8)11 roof. For H α the adjoint with respect to the inner product 3.0.3 is given by H ∗ α = (cid:0) − α (1 + ~ ) (cid:1) H αα (1+ ~ ) − (3.9)Hence for ~ = − h· , ·i ~ the operator H ~ /
2i is self-adjointand for the special value ~ = − t∂ t is self-adjoint resolving the for ~ = − H ~ /
2i consistently. It is standard that for A ∈ g with adjoint A ∗ ∈ g wehave A ∗ ◦ A, A ◦ A ∗ ∈ s and also real polynomials of a self-adjoint operator A ∈ s are self-adjoint.Hence for ~ = − ~ ∈ s and for the special value ~ = − t∂ t ) ∈ s .With respect to 3.0.3 we have for d β the adjoint operator by the enlightening rewriting d ∗ β = 1 β ~ d /β = τ ~ ◦ d β ◦ τ ~ (3.10)The compositions of operators τ µ ~ and d β transform by τ µ ~ ◦ d β = β ~ d β /µ ◦ τ ~ µ and we find τ µ ~ ◦ Z α = (cid:0)P ∞ n =1 1 n α (1+ ~ ) d n α/µ (cid:1) ◦ τ µ ~ . Observe Z α is not hermitian: We have h Z α f , f i ~ = h f , ( Z α ) ∗ f i ~ with the twist ( Z α ) ∗ = ∞ X n =1 n d n − α = τ ~ ◦ Z α ◦ τ ~ (3.11)Between the operators τ µ ~ and H α by calculation the commutation relations H α ◦ τ µ ~ = (cid:0) αµ (1 + ~ ) (cid:1) τ µ ~ ◦ H αµ αµ (1+ ~ ) (3.12) τ µ ~ ◦ H α = (cid:0) − α (1 + ~ ) (cid:1) H αµ (1 − α (1+ ~ )) ◦ τ µ ~ are valid. The formulas 3.12 imply that we have τ − /µ − ~ ~ ◦ H ~ = ~ − ~ +1 H ~ ◦ τ − µ (1+ ~ ) ~ , H ~ ◦ τ µ ~ =(1 + 2 µ ) τ µ ~ ◦ H µ (1+ ~ )(1+2 µ ) and H m µ − µ (1+ ~ ) ◦ ( τ µ ~ ) n = µ m · n ( τ µ ~ ) n ◦ H m µ − µ (1+ ~ ) where we suppose µ = 0, ~ = − H / (1+ ~ ) ◦ τ ~ = − τ ~ ◦ H / (1+ ~ ) and hence the rewriting ∆ α = (i τ − αα ± H α ) .With the preparation formulas (cid:0) τ µ ~ (cid:1) ∗ = τ /µ ~ / | µ | , 3.10, 3.9 and 3.12 we have for λ ∈ R (cid:0) ( H λ ◦ Z λ ) n (cid:1) ∗ = (cid:0) ( H λ ◦ Z λ ) ∗ (cid:1) n = τ ~ ◦ ( H λ ◦ Z λ ) n ◦ τ ~ (3.13)by induction where we essentially only use ( A ◦ B ) ∗ = B ∗ ◦ A ∗ , 3.10, τ ~ ∗ = τ ~ , τ ~ ◦ τ ~ = id andid(1 + ~ ) id − ◦ id(1 + ~ )id − < λ ∈ R to the formulas 3.7 where it is manifest that the two operators Z λ ± arewell-defined maps H → H , compare the argumentation in the proof of 2.0.4. It is clear thatby 3.13 the standard symmetrizations p and ∂ are compatible with the dynamics generated by H λ ◦ Z λ , i.e. the iterations ( H λ ◦ Z λ ) n behave similar under the symmetrizations p and ∂ , thesame holds for the other two symmetrization procedures 3.8: The fact that b Z λ ± are hermitian canbe justified in two ways, the first method is just a direct check by calculation. We have( τ ~ ◦ ( H λ ◦ Z λ ) n ) ∗ = (cid:0) τ ~ ◦ ( Z λ ◦ H λ ) n ◦ τ ~ (cid:1) ◦ τ ~ = τ ~ ◦ ( H λ ◦ Z λ ) n because of 3.13 and τ ~ ∗ = τ ~ . The adjoint conjugation Con ∗ B ( A ) = B ◦ A ◦ B ∗ is also self-adjointif A ∈ s , hence H λ ◦ Z λ ◦ τ ~ = Con τ ~ ( τ ~ ◦ H λ ◦ Z λ ) ∈ s if τ ~ ◦ H λ ◦ Z λ ∈ s because τ ~ ◦ τ ~ = id.The second somehow slightly more conceptional proof will be discussed in section 6.12 strategy developed in [7], [12] is to consider the dual of the quotient of a certain largerfunction space by the range of Z λ and the transposed operators acting on the dual space. Thisprocedure yields a spectral realization of the Riemann zeros, in analogy we will consider thequotients by the image of the operators 3.1.2 in the following section 7.0.1. Because in 3.1.1,3.1.2we do not consider the mentioned quotient and the transposition of t∂ t we have still an innerproduct but also corrections to the usual spectral realization. Eigenstates of Z λ ± and some sums of ζ functions In the section 2.0.4 we showed M [ H λ ◦ Z λ f ] (cid:0) sλ (cid:1) = (1 − id) ζ ( s ) M [ f ] (cid:0) sλ (cid:1) and 3.4 applied twiceyields M [ τ ~ ◦ H λ ◦ Z λ ◦ τ ~ f ] (cid:0) sλ (cid:1) = (1 − id) ζ ((1 + ~ ) λ − s ) M [ f ] (cid:0) sλ (cid:1) , hence by linear combinationthe following Mellin transform interpretation: Proposition
For γ ± ∈ R we have M h (cid:0) γ + Z λ + + γ − Z λ − (cid:1) f i (cid:18) ~ sλ (cid:19), M [ f ] (cid:18) ~ sλ (cid:19) (3.14)= " γ + + i γ − − id) · ζ (cid:18) (1 + ~ ) λ s (cid:19) + γ + − i γ − − id) · ζ (cid:18) (1 + ~ ) λ − s (cid:19) We have the implications (1 − z ) ζ ( z ) = 0 ⇒ M (cid:2) Z λ ± (cid:0) − λλ (cid:1) f (cid:3) ( z/λ ) = 0 ∀ f ∈ H and areformulation of section 2.1 with help of the convolution ∗ , see definition 4.2: Lemma
The function on the r.h.s of M (cid:2) Z λ + f ∗ Z λ − f (cid:3)(cid:0) ~ + sλ (cid:1) i M [ f ∗ f ] (cid:0) ~ + sλ (cid:1) = (cid:0) (1 − id) · ζ (cid:1) (cid:18) (1 + ~ )2 /λ + s (cid:19) − (cid:0) (1 − id) · ζ (cid:1) (cid:18) (1 + ~ )2 /λ − s (cid:19) admits only strictly complex zeros ∈ i R if the inequality (1+ ~ ) λ ≥ is satisfied The r.h.s. of 3.1.4 is by Hadamard factorization determined by its zeros while the operators Z λ ± depend on their eigenvalues and eigenstates it seems plausible that there is a connection betweenthe two relevant sets of real numbers, but without this missing link this is just a coincidence.Notice that there would be a serious convergence problem when Z λ ± gets applied to t ν , thisfunctions are not elements of H , the continuous spectrum of H λ ◦ Z λ does not carry over to thesymmetrizations. Eigenstates ψ E of Z λ ± would be a bit bizarre, we give a vague sketch how tounravel the eigenvalue equation: Either this would imply the not convincing functional equation(1 − id) · ζ (cid:18) (1 + ~ )2 /λ + s (cid:19) ± (1 − id) · ζ (cid:18) (1 + ~ )2 /λ − s (cid:19) !? = ( E ∈ R E/ i ∈ i R (3.15)or the function M [ ψ E ] (cid:0) ~ + sλ (cid:1) should be concentrated on the isolated points in C that aresolutions of this equation 3.15. Let us informally express the previous concentration statementin a distributional sense with help of Dirac delta functionals δ and assume M [ ψ E ] (cid:18) (1 + ~ )2 + sλ (cid:19) = X a z δ ( s − z ) (3.16)where the sum ranges over the isolated set (cid:26) z (cid:12)(cid:12) (1 − id) ζ (cid:18) (1 + ~ )2 /λ + z (cid:19) ± (1 − id) ζ (cid:18) (1 + ~ )2 /λ − z (cid:19) = (cid:18) E E/ i (cid:19)(cid:27) f ( t ) = 12 π Z r+i ∞ r − i ∞ d s M [ f ]( s ) t − s (3.17)The assumptions for this inversion are that M [ f ]( s ) is analytic for a < ℜ ( s ) < b and convergesabsolutely in this strip, lim t →∞ M [ f ](r + i t ) converges uniformly to zero for a < r < b andsatisfies at any jump discontinuities the equation f ( x ) = [lim t → x + f ( t ) + lim t → x − f ( t )] (cid:14) δ (cid:0) g ( α ) (cid:1) = δ (cid:0) α − z (cid:1) / | g ′ ( z ) | at simple real zeros z of g immediate converts the concentration 3.16 into the statementthat the eigenstates ψ E should be of the shape ψ E ( t ) = X n z (cid:12)(cid:12) ((1 − id) ζ ) ( (1+ ~ )2 /λ + z ) ± ((1 − id) ζ ) ( (1+ ~ )2 /λ − z ) = ( E E/ i ) o a z | λ | t − ( zλ + ~ ) (3.18)Here we used in the computation of the eigenstates that the two dimensional δ distribution δ ( z )can be represented by the product δ ( x ) δ ( y ) of two one dimensional δ functions and in order toavoid integration singularities in the inverse Mellin transform it seems convenient if we blur theintegration line r + i R to the integration strip [r − ǫ, r + ǫ ] + i R by averaging the integration.This blur argumentation also seems to resolve that the inverse Mellin transform is invariantunder slight shifts of the integration line r + i R . However our consideration with δ distributionsis incomplete and conjectural: It is not clear that ψ E as described in 3.18 is a non-trivial elementof H without specifying an infinite series a z = 0, further non-trivial convergence arguments andshowing that the listed conditions for the inverse Mellin transform are not violated. Eigenstates of b Z λ ± and some products of ζ functions With the universal continuation formula 2.0.4 we have the vanishing implications (1 − z ) ζ ( z ) =0 ⇒ M [ b Z λ + ( ~ ) f ]( z/λ ) = M [ b Z λ − ( ~ ) f ] (1 + ~ − z/λ ) = 0 ∀ f ∈ H where we substituted for thefirst equals τ ~ = id and for the second equals used 3.4.In the previous discussion the quasi eigenvalue E = 0 and the “phase transition” inequality (1+ ~ ) λ ≥ E = 0 corresponds to zeros. In this context it may be worth tomention that we have a reformulation of 3.1.4 for Z λ ± : Lemma
For the sum b Z λ + ± b Z λ + and f ∈ H the implication M h(cid:16) b Z λ + ± b Z λ − (cid:17) f i(cid:0) ~ + sλ (cid:1) M [ τ ~ f ] (cid:0) ~ + sλ (cid:1) = (1 − id) · ζ (cid:18) (1 + ~ )2 /λ + s (cid:19) ± (1 − id) · ζ (cid:18) (1 + ~ )2 /λ − s (cid:19) = 0 ⇒ s ∈ i R is true if (1+ ~ ) λ ≥ . As we have seen in 3.14 the eigenstates of Z λ ± in some sense refer to some special sums, but aswe will see eigenstates of b Z λ + refer to some special products, the calculations are quite analogous:For example for eigenstates ψ E ( λ ) of b Z λ + we find M [ ψ E ] (cid:16) sλ (cid:17) = (1 − id) · ζ ( s ) E M [ τ ~ ψ E ] (cid:16) sλ (cid:17) (3.19) ! = ((1 − id) · ζ ) ( s )(1 − id) · ζ (cid:0) (1 + ~ ) λ − s (cid:1) E M [ ψ E ] (cid:16) sλ (cid:17) δ distributions or alternative with 3.17 and the reversedformula of the convolution theorem 4.3, namely M − [ f · f ]( t ) = (cid:0) M − [ f ] ∗ M − [ f ] (cid:1) ( t ).If ℜ ( ν ) < − /λ then t ν is a simultaneous eigenfunction of the commuting operators Z λ and H λ with the eigenvalues ζ ( − λν ) and 1 + λν respectively. With this informal interpretation the ζ zeros should correspond to quasi eigenfunctions of H λ ◦ Z λ , i.e. eigenfunctions to the eigenvalue0, but however for ℜ ( ν ) ≥ − /λ we do not have convergence of H λ ◦ Z λ t ν . If we suppose aconvergent series f ( t ) = P ∞ n =1 a n t − n we symbolically find, evaluating the operator H λ ◦ Z λ oneach term of the expansion, the formula( H λ ◦ Z λ f )( t ) = ∞ X n =1 (1 − λn ) ζ ( λn ) a n t − n (3.20)Another curiosity for b Z λ ± is that t − ( zλ + ~ ) is a eigenfunction of H λ ◦ Z λ if convergence holds butgets mapped by τ ~ to t − ( − zλ + ~ ), hence is not respected by the compositions b Z λ + = H λ ◦ Z λ ◦ τ ~ and b Z λ − = τ ~ ◦ H λ ◦ Z λ and application of 3.20 yields for example b Z λ + ψ E = X n z (cid:12)(cid:12) (1 − id) · ζ ( (1+ ~ )2 /λ + z ) · (1 − id) · ζ ( (1+ ~ )2 /λ − z ) = E o (1 − id) · ζ (cid:18) ~ /λ + z (cid:19) a z | λ | t − ( − zλ + ~ ) (3.21)= X n z (cid:12)(cid:12) (1 − id) · ζ ( (1+ ~ )2 /λ + z ) · (1 − id) · ζ ( (1+ ~ )2 /λ − z ) = E o E a z | λ | t − ( zλ + ~ ) (3.22)This implies by comparison of the two expressions for b Z λ + ψ E the equation E a − z a z = (1 − id) · ζ (cid:16) ~ /λ + z (cid:17) , hence for the coefficients a z the relation ± a − z a z = r (1 − id) · ζ ( ~ /λ + z ) (1 − id) · ζ ( ~ /λ − z ) .As will be alluded in section 4 analogous statements also hold for the eigenstates of the fourstandard symmetrizations of an arbitrary convolution operator obtained by lemma 4.0.9.Compared with the general E → − E eigenvalue symmetry construction 5.2 with help of τ ~ we have a different E → − E observation for b Z λ ± because H α commutes with Z λ and 3.12: Proposition
The following implications are true b Z λ ± ψ E = Eψ E ⇒ b Z λ ± (cid:16) H ~ ψ E (cid:17) = − E (cid:16) H ~ ψ E (cid:17) ⇒ b Z λ ± (cid:16) ∆ ~ ψ E (cid:17) = E (cid:16) ∆ ~ ψ E (cid:17) It is therefore immediate to consider the quotient of the eigenspace of the eigenvalue E bythe equivalence relation ψ E ∼ ψ ′ E ⇔ ∃ n : ψ E = ∆ n ~ ψ ′ E ∨ ψ ′ E = ∆ n ~ ψ E With 3.20 H ~ acts on the coefficients a z by multiplication with − zλ (1+ ~ ) while the secondorder differential operator ∆ ~ acts by multiplication with (cid:0)(cid:0) zλ (1+ ~ ) (cid:1) − (cid:1) .The case (1 + ~ ) λ = 1 is special: Here we have that the eigenstate ψ has to be of the shape P z | ζ (1 / z )=0 t − ( zλ + ~ ) and Weil showed [21] that RH is equivalent to the remarkable positivitycriteria that for Mellin invertible test functions f X M (cid:2) τ f ∗ f (cid:3) ( z ) ≥ z of Ξ and ∗ denotes the multiplicative convolution, see definition4.2. We refer the reader for proofs and a detailed discussion of Weil’s positivity criteria 3.23 andhis so-called explicit formula of number theory to [3].15 The analogy with convolution operators
The algebra of multiplication operators 3.1 became singular because mild multiplied singularitiesat 0 or ∞ get absorbed by the nice behaved test functions in H . For the multiplication operators m V it seems not so easy to give a general Mellin transform interpretation we only have byintegration by parts the symmetric formula s M [ V · f ] ( s ) = s M [ m V f ] ( s ) = s M [ m f V ] ( s ) = −M [ m t∂ t V f ] ( s ) − M [ m t∂ t f V ] ( s ) (4.1)From this interpretation point of view there is a more well-behaved algebra acting on H byintegration: Let as usual the multiplicative convolution ∗ be defined by( f ∗ f )( t ) := Z ∞ d yy f (cid:18) ty (cid:19) f ( y ) (4.2)The bilinear operation ∗ is associative, commutative and we have the famous convolution theorem M [ f ∗ f ]( s ) = M [ f ]( s ) · M [ f ]( s ) (4.3)With the usual additive convolution ( f ˜ ∗ f )( t ) := R ∞−∞ d yf ( t − y ) f ( y ) we recover ∗ by a so-called equivalence formula that uses invertibility to make a trivial associativity connection bythe identity ( f ˜ ∗ g ) ( t ) = (cid:0) ( f ◦ ln) ∗ ( g ◦ ln) (cid:1) (exp( t )).For the norm || · || p defined by || f || p := (cid:0) R ∞−∞ d x | f ( x ) | p (cid:1) /p the Young inequality || f ˜ ∗ g || r ≤|| f || p || g || q is valid for p + q = 1 + r where 1 ≤ p, q, r ≤ ∞ .It is also well-known that functions of the shape f ∗ f are dense in H . Definition
Let c V : H → H with fixed V ∈ H act by f ( t ) → ( c V f ) ( t ) := ( V ∗ f )( t ) . We have for example c δ (1 − t ) = id where δ denotes as usual the Dirac delta function, butnotice δ / ∈ H and δ can only be approximated elements of H . Another interesting example of anidentity for the convolution operator c e − ρ ln2( t ) t s is the interpretation (cid:0) c e − ρ ln2( x ) x s f (cid:1) ( t ) = e − ρ ln ( t ) t s M h f e − ρ ln ( · ) i(cid:16) − s + 2 ρ ln( t ) (cid:17) (4.4)The ring ( H , + , ∗ ) is clearly not noetherian, for instance the spaces m e − ρ ln2( t ) H where 0 ≤ ρ are ∗ -ideals and the inclusion m e − ρ ′ ln2( t ) H ⊂ m e − ρ ln2( t ) H holds if ρ < ρ ′ . Proposition
With respect to the inner product 3.0.3 the adjoint of c V is given by c ∗ V = Con τ ~ c V = τ ~ ◦ c V ◦ τ ~ = c τ ~ V By the convolution theorem 4.3 we have for f, V ∈ H the Mellin transform interpretation M [ c V f ] ( s ) = M [ c f V ] ( s ) = M [ V ] ( s ) · M [ f ] ( s ), especially M [ c ∆ Ψ f ] ( s ) = 4 ξ ( s ) M [ f ] ( s )hence 4.0.7 allows to encode ζ alternative to 3.1.2 as a convolution Hamiltonian: Lemma
For every V ∈ H we have c ( V + τ ~ V ) / ∈ s and c i( V − τ ~ V ) / ∈ s . If V ( t ) = V ( t ) we have also two self-adjoint operators given by the compositions τ ~ ◦ c V ∈ s and c V ◦ τ ~ ∈ s .Proof. We use 4.7, the fact that ∗ is associative and the compatibility τ µ ~ ( f ∗ f ) = | µ | ( τ µ ~ f ∗ τ µ ~ f ) (4.5)16his shows in particular that the multiplication operators m t s are morphisms of ∗ , i.e. m t s ◦ ∗ = ∗ ◦ ( m t s ⊗ m t s ) and more general τ µ ~ / | µ | are morphisms of ∗ . We have the commutation relation τ µ ~ ◦ c V = | µ | c τ µ ~ V ◦ τ µ ~ (4.6)The first two Hamiltonians are now obtained by the two standard symmetrization procedures pand ∂ and the third and fourth by the standard procedures described in 6.Notice that there is literally in some sense an analogy of the following four symmetrizationsin the lemma 4.0.9 with the four more discrete formulas 3.1.2 defined by summation.With τ ~ , d t and h· , ·i we can rewrite the convolution ∗ in the shape( τ ~ f ∗ f )( t ) = h τ ~ ◦ d t ◦ τ ~ f , f i ~ = h f , d t f i ~ (4.7)With the reformulation 4.7 and the compatibilities 4.5, 4.9 and τ ~ ◦ d t = m t − (1+ ~ ) d t ◦ τ ~ we geta reincarnation of theorem 3.1.2 in a simple, basic form concerning convolution operators: Lemma
We have the compatibility H λ ◦ Z λ ( f ∗ f ) = ( H λ ◦ Z λ f ) ∗ f = f ∗ ( H λ ◦ Z λ f ) We have Z λ ± ◦ c V ∈ s if τ ~ ( V ) = V and b Z λ ± ◦ c V ∈ s if V is real.Proof. By 4.5 and the standard identities H α ( f ∗ f ) = H α f ∗ f = ( H α f ∗ f + f ∗ H α f ) / d β ( f ∗ f ) = d β f ∗ f = ( d β f ∗ f + f ∗ d β f ) / Z λ ± ◦ c V is self-adjoint if τ ~ ( V ) = V because we have the identity Z λ ± ◦ c V = c V ◦ Z λ ± (4.10)By 4.6 if Z λ ± ψ E = Eψ E we have the eigenvalue equation Z λ ± ( c V ψ E ) = E ( c V ψ E ), the eigenstatesof Z λ ± are stable under convolution and the convolution of eigenstates with different eigenvaluesvanishes. The eigenvalues of b Z λ ± are not stable under convolution in general: b Z λ ± ◦ c V = c τ ~ V ◦ b Z λ ± (4.11)holds, hence if b Z λ ± ψ E = Eψ E we find b Z λ ± ( c V ψ E ) = E ( c τ ~ V ψ E ) and this implies for potential V ± ∈ H ± ~ in that b Z λ ± ( c V ± ψ E ) = ± E ( c V ± ψ E ). This shows that b Z λ ± ◦ c V ∈ s if V is real.The equation M (cid:2) m ln( t ) f (cid:3) ( s/λ ) = λ∂ s M [ f ] ( s/λ ) holds, hence we have the identity M [ · · · [ |{z} n Z λ , m ln( t ) ] − · · · , m ln( t ) ] − f ( s/λ ) = λ n ( ∂ ns ζ ( s )) M [ f ] ( s/λ )Also a similar compatibility is satisfied for the commutators with convolution operators: The factthat m ln( t ) is a derivation of ∗ , i.e. ∗ ◦ (id ⊗ m ln( t ) + m ln( t ) ⊗ id) − m ln( t ) ◦ ∗ : H × → H , is agadget in the proof of the explicit formulas of number theory contained in [12]. This derivationproperty and the commutativity of ∗ allows to define a Jacobi structure: Lemma
Let ( A • , ⋆ ) be a graded super-algebra and ∂ : A • → A • +2 m with m ∈ Z be aneven super-derivation i.e. ⋆ is associative, | a ⋆ b | = | a | + | b | and ∂ ( a ⋆ b ) = ( ∂a ) ⋆ b + a ⋆ ( ∂b ) .Suppose A is super-commutative i.e. a ⋆ b = ( − | a || b | b ⋆ a or that ∂ is a differential i.e. ∂ = 0 :By [ x, y ] ∂ := ( ∂x ) ⋆ y − ( − | x || y | ( ∂y ) ⋆ x we have a even super Lie-bracket [ · , · ] ∂ : A ⊗ → A ofdegree m , i.e. ( − | x | + | z | [ x, [ y, z ] ∂ ] ∂ + ( − | y | + | x | [ y, [ z, x ] ∂ ] ∂ + ( − | z | + | y | [ z, [ x, y ] ∂ ] ∂ = 0 . For the usual derivative we have the interpretation of [ x, y ] ∂ = y ∂ ( x/y ).17 Substitution Hamiltonians and uniqueness of τ ~ Definition
Suppose g : [0 , ∞ ] → [0 , ∞ ] is invertible, differentiable and respects rapiddecay at and ∞ in the sense that if f decays rapidly at this two points then also f ◦ g ( − havethis property and let V be as specified in 3.0.4. ( S V,g f ) ( t ) := (cid:0) ( m V f ) ◦ g (cid:17) ( t ) = V ( g ( t )) f ( g ( t )) (5.1)Definition 3.0.6 is an example of a substitution operator, i.e. we have τ µ ~ = S t ~ ,t µ and alsothe dilation can be written as d β = S ,βt . Clearly 5.0.12 also unifies with 3.0.4 for g = id. Forthe composition of two substitution operators we have the semi-direct composition rule S V ,g ◦ S V ,g = S ( V ◦ g − ) · V ,g ◦ g (5.2)The following proposition at least allows us by the mentioned procedures S V,g → p S V,g ∈ s and S V,g → ∂ S V,g ∈ s etc. to produce some self-adjoint substitution operators or use theconjugation A → Con ∗S V,g ( A ) = S V,g ◦ A ◦ S ∗ V,g to transform A ∈ s . Lemma
The adjoint of the operator S V,g with respect to 3.0.3 is given by (cid:0) S ∗ V,g f (cid:1) ( t ) = ± S t ~ · ( V ◦ g ) g ~ · ∂tg ,g − f ! ( t ) = ± ∂ t g − ( t ) V ( t ) (cid:18) g − ( t ) t (cid:19) ~ f (cid:0) g − ( t ) (cid:1) (5.3) with the + sign in the case g (0) = 0 and the − sign in the case g (0) = ∞ . We have the following corollary of 5.3 that not surprisingly just identifies two times the samesubstitution operator on both sides of h· , ·i ~ with a multiplication operator: Corollary
Let g (0) = 0 . We have for f , f ∈ H the adjunction equation hS V,g f , S V,g f i ~ = ± * f , | V ( t ) | ∂ t g − ( t ) (cid:18) g − ( t ) t (cid:19) ~ f + ~ Hence we have (cid:10) S V,g f , S V,g f (cid:11) ~ = h f , m h f i ~ with some real positive h ∈ H if we set V ( t ) = e i ϕ ( t ) r(cid:16) tg − ( t ) (cid:17) ~ h ( t ) ∂ t g − ( t ) with some real argument ϕ : R + → R . Clearly we also have compatibility of the Lie-algebra structure and the
Chevalley-Eilenbergcomplex with adjoint conjugation by Con ∗S V,g as specified in 5.0.14.If we scale τ µ ~ with p | µ | we could omit the proportionality factor 1 / | µ | in the adjunction (cid:0) τ µ ~ (cid:1) ∗ = τ /µ ~ / | µ | , i.e. ( p | µ | τ µ ~ ) ∗ = p | /µ | τ ~ /µ and the previous observation is a solution of aquestion that we only answer halfway in the following proposition 5.0.15:We ask for the symmetry that 5.3 is 1 /µ proportional to the operator defined in 5.1 but with g replaced by its inverse g − , i.e. we search for real pairs ( V, g ) with √ µ S ∗ V,g = p /µ S V,g − . Proposition If g satisfies µ µ − ∂ t g ( g ( t )) = (cid:0) ∂ t g − ( t ) (cid:1) − µ = (cid:0) ∂ t g (cid:0) g − ( t ) (cid:1)(cid:1) µ (5.4) then we have √ µ S ∗ V,g = p /µ S V,g − if V ( t ) is real proportional to µ − µ t ~ (cid:18) ∂ t g ( t ) ∂ t g ( g ( t )) (cid:19) − µ (5.5)18 roof. In concrete formulas we ask for t ~ V ( t ) = µ (cid:0) g − ( t ) (cid:1) ~ V ( g − ( t )) ∂ t g − ( t ) ∀ t ∈ [0 , ∞ ] (5.6)For some V a g solving 5.6 is unique by the Picard-Lindel¨of theorem. Iteration of 5.6 and theAnsatz V ( t ) = t ~ v ( t ) decouples the parameter ~ and is now more symmetric in g and g − : µ ∂ t g − ( t ) v ( g − ( t )) = µ ∂ t ln (cid:0) v (cid:0) g − ( t ) (cid:1)(cid:1) = ∂ t ln (cid:0) v ( g ( t )) (cid:1) = ∂ t g ( t ) v ( g ( t )) (5.7)By integration and g (0) = g − (0) we yield v ( g ( t )) = v − µ (cid:0) g (0) (cid:1) v µ (cid:0) g − ( t ) (cid:1) . Self-consistencywith 5.7 and the more exquisite equation 5.6 implies if µ = 1 the constraint 5.4. If g satisfies5.4 we have √ µ S ∗ V,g = p /µ S V,g − if V ( t ) is real proportional to 5.5. Lemma
Let g diverge at like some power t − ρ with ρ ∈ R + . If the graph of g is reflectionsymmetric with respect to the diagonal we have an involutive operator S V,g where V = q vv ◦ g withsome v : R + → R + with no zeros and so that V is as specified in 3.0.4.Proof. If we suppose S V,g is an involution we obtain g ◦ g = id and V ( t ) · V ( g ( t )) = 1. Notice g ◦ g = id i.e. g = g − implies if g (0) = 0 readily g = id but else is locally not very restricting,for instance it is well-known that the inverse of g is just the graph obtained by reflection at thedashed diagonal in the picture below:Hence there are plenty involutive substitution operators, but self-adjoint ones are quite rareand the two symmetries of the hero τ ~ of the previous section 3.1 in some sense are unique: Proposition
If we suppose a substitution operator S V,g as specified in 5.0.12 satisfiesthe two identities S V,g ◦ S
V,g = id and √ µ S ∗ V,g = p /µ S V,g − then S V,g = id or S V,g = τ ~ .Proof. We find S V,g = id ∨ τ ~ by substituting g ◦ g = id and V ( t ) · V ( g ( t )) = 1 in 5.5.If B ∗ = B − the adjoint conjugation Con ∗ B ( A ) = B ◦ A ◦ B ∗ obviously coincides with the usualconjugation Con B ( A ) := B ◦ A ◦ B − but Con ∗ B restricts to s ∀ B ∈ g because Con ∗ B commuteswith ∗ while Con B is only defined if B is invertible in the sense B ∈ i i := { A ∈ g |∃ ! A − ∈ g : A ◦ A − = A − ◦ A = id } . Let us mention that we have a map Q : i → i ∩ s ( ~ ) by Q ( A ) := A ◦ A ∗ ( ~ ) and because ( A − ) ∗ = ( A ∗ ) − we have Q (cid:0) ( A ∗ ) − (cid:1) = ( Q ( A )) − . The map Con B restricts to s if B = B − ∈ s and clearly we have Con A ◦ Con B = Con A ◦ B as well as Con ∗ A ◦ Con ∗ B = Con ∗ A ◦ B . Proposition
For the set of smooth involutions of R + the quotient by the equivalencerelation ∼ defined by g ∼ g ⇔ ∃ f : Con f g = g , where f is a smooth function on R + , justconsists of the maps id and .Proof. Let g be an involution with g (0) = ∞ and h : R + → R + with h (0) = 0 and h ′ ( t ) > f ( t ) := h ( t ) /h ( g ( t )). The function f : R + → R + is bijective because f (0) = 0, f ( ∞ ) = ∞ and f ′ ( t ) >
0. We also clearly have 1 /f ( t ) = f ( g ( t )) or equivalent Con f g = .19 .1 Basic compatibilities with uncertainty
Definition
The variance σ f ( A ) of A ∈ s is defined by σ f ( A ) := rD f, (cid:0) A − (cid:10) f, Af (cid:11) ~ (cid:1) f E ~ where f ∈ H is a normalized state, i.e f is on the unit sphere defined by the condition h f, f i ~ = 1 . Only the quotient space g / ∼ ~ with respect to the equivalence relation A ∼ ~ B ⇔ ∃S V,g ∈ g : S V,g ◦ S ∗ ( ~ ) V,g = id ∧ A = Con ∗ ( ~ ) S V,g B is relevant for h· , ·i ~ restricted to the unit sphere. Corollary If V satisfies | V ( t ) | = (cid:0) t/g − ( t ) (cid:1) ~ (cid:14) ∂ t g − ( t ) then S ∗ V,g ◦ S
V,g = id . Thisoperators form a subgroup of g and Con ∗S V,g is compatible with the natural Lie-algebra structure.
It is well-known that the Cauchy-Schwarz inequality h f, f i ~ h g, g i ~ ≥ |h f, g i ~ | implies σ f ( A ) σ f ( A ) ≥ (cid:12)(cid:12)(cid:12)D f,
12i [ A , A ] − f E ~ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)D f,
12 [ A , A ] + f E ~ − h f, A f i ~ h f, A f i ~ (cid:12)(cid:12)(cid:12) (5.8)for self-adjoint operators A i ∈ s called Schr¨odinger uncertainty relation and this slightlystronger inequality clearly implies the usual
Heisenberg uncertainty principle . Proposition
Let f = f + + f − with f ± ∈ H ± ~ . For A ∈ s ( ~ ) we have D f,
12 [
A, τ ~ ] ± f E ~ = 2 ℜ (cid:18)D f − ,
12 [
A, τ ~ ] ± f + E ~ (cid:19) + h f ± , Af ± i ~ − h f − , Af − i ~ Proof.
The variance of τ ~ in a normalized eigenstate f ± / p h f ± , f ± i ~ vanishes, hence the l.h.s.of 5.8. For self-adjoint B the expectation value h g, Bg i ~ = h Bg, g i ~ = h g, Bg i ~ is real, hence thesquare greater than zero if the expectation value is = 0. The resulting equations are invariantunder scaling, we can drop the unit sphere normalization and yield (cid:10) f ± , i2 [ A, τ ~ ] − f ± (cid:11) ~ = 0 = (cid:10) f ± , (cid:0) [ A, τ ~ ] + ∓ A (cid:1) f ± (cid:11) ~ . The proposition is now obtained by multiplying out this in f ∈ H quadratic expressions and substitution of h f, Ag i ~ = h Ag, f i ~ = h g, A ∗ f i ~ .It is standard that the direct sum ⊕ i ∈ I H i of an indexed family ( H i , h· , ·i i ) i ∈ I of Hilbert spacesbecomes a Hilbert space with inner product h· , ·i on ⊕ i ∈ I H i specified by h f, g i := P i ∈ I h f i , g i i i and this sum converges because it is by definition finite, the same holds for pre-Hilbert spaces. Proposition
Let f ∈ ( H , h· , ·i ~ ) ~ ∈ ( −∞ , ∞ ) be normalized with support contained in [0 , , [1 , ∞ ] or the union of two intervals [ a , , b , ] with < a ≤ b ≤ ≤ a ≤ b < ∞ and a a ≥ or ≥ b b . For every self-adjoint A : ⊕ ~ ∈ ( −∞ , ∞ ) H → ⊕ ~ ∈ ( −∞ , ∞ ) H we have the inequality σ f ( A ) ≥ (cid:12)(cid:12)(cid:12)D f,
12i [
A, τ ~ ] − f E(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)D f,
12 [
A, τ ~ ] + f E(cid:12)(cid:12)(cid:12) (5.9) In particular 5.8 for A = b Z λ ~ + and A = τ ~ if supp( f ~ ) ⊂ [1 , ∞ ] or for A = b Z λ ~ − with λ ~ > and A = τ ~ if f ∈ H ∞ with supp( f ~ ) ⊂ [0 , respectively corresponds to X ~ ∈ ( −∞ , ∞ ) h b Z λ ~ ± f ~ , b Z λ ~ ± f ~ i ~ ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ~ ∈ ( −∞ , ∞ ) h f ~ , Z λ ~ + f ~ i ~ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ~ ∈ ( −∞ , ∞ ) h f ~ , Z λ ~ − f ~ i ~ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.10) Proof.
We only use properties of τ ~ and that Z λ is a sum of dilations d β with β ≥
1: Thevanishing h f, τ ~ f i ~ = 0 is a straight forward calculation. It is immediate that h f, τ ~ f i ~ = 0implies for the variance σ f ( τ ~ ) = h f, τ ~ ◦ τ ~ f i ~ − h f, τ ~ f i ~ = h f, f i ~ = 1 and 5.8 reduces to 5.9.20n particular supp( f ~ ) ⊂ [1 , ∞ ] or ⊂ [0 ,
1] implies h f, τ ~ f i ~ = 0 and let us now justify thepositivity statement 5.10: It is not difficult to verify i2 [ τ ~ , b Z λ ± ] − = ∓Z λ − and [ τ ~ , b Z λ ± ] + = Z λ + .Because of the implication 2.7 the inequality 5.9 reduces to 5.10 respectively and this equationsare equivalent because of τ ~ ◦ Z λ ± ◦ τ ~ = ±Z λ ± and supp( f ) ⊂ (1 , ∞ ] ⇔ supp( τ ~ f ) ⊂ [0 , h f, τ ~ f i ~ = 0 we find σ f ( Z λ ± ) ≥ (cid:12)(cid:12)(cid:10) f, ( b Z λ + ± b Z λ − ) f (cid:11)(cid:12)(cid:12) .Symmetrization of convolution operators 4.0.9 and the four symmetrizations of the operator H λ ◦ Z λ defined in 3.1.2 seem similar, but the analogy of 5.9 for the corresponding convolutionoperators seems like a non-trivial question because here we don’t have 2.7, it is only immediateto verify the standard inclusion supp( f ∗ f ) ⊂ supp( f ) · supp( f ) := { r · r | r i ∈ supp( f i ) } . Involutive construction of the E → − E eigenvalue symmetry The easiest method to incorporate the eigenvalue symmetry E → ± E for any self-adjointoperator is by commutation or anti-commutation with τ ~ respectively: For A ∈ s we have[ A, τ ~ ] + , i[ A, τ ~ ] − ∈ s and [ A, τ ~ ] ± ψ E = Eψ E ⇔ [ A, τ ~ ] ± τ ~ ψ E = ± Eτ ~ ψ E . Proposition (cid:16) Z λ ∓ + m V ∓ m + V ∓ m + c V ∓ c − V ∓ c (cid:17) ψ E = Eψ E ⇔ (cid:16) Z λ ∓ + m V ∓ m + V ∓ m + c V ∓ c − V ∓ c (cid:17) τ ~ ψ E = ∓ Eτ ~ ψ E Proposition 5.2.1 shows that the Z λ − + m V − m + V − m + c V − c − V − c eigenvalues can only be non-zeroif τ ~ ψ E = ± ψ E , but for the Hamiltonian Z λ + + m V + m + V + m + c V + c − V + c all the speculative eigenstateshave to decompose in two eigenstates ψ + E ∈ H + ~ and ψ − E ∈ H − ~ in the two eigenspaces of τ ~ . Proof II, a closer look at the symmetrization
The second promised proof that b Z λ ± ∈ s is more conceptional, just relies on standard proceduresand we want to emphasise this way because some other interesting maps arise: It is well-knownthat for A, B ∈ s also real linear combinations and i[ A, B ] − as well as [ A, B ] + are self-adjoint,the two binary operations i[ · , · ] − and [ · , · ] + endow ∈ s with the structure of a Lie-algebra and
Jordan-algebra respectively. In words we can deduce by real linear combinations of2i[ Z λ ± , τ ~ ] − = − (1 ∓ H λ ◦ Z λ , τ ~ ] − and 2[ Z λ ± , τ ~ ] + = (1 ± H λ ◦ Z λ , τ ~ ] + (6.1)that b Z λ ± ∈ s . Notice also H λ ◦ Z λ = b Z λ + ◦ τ ~ = τ ~ ◦ b Z λ − and we have in some sense dual to 6.1i[[ H λ ◦ Z λ , τ ~ ] − , τ ~ ] ± = 2(1 ∓ Z λ − and [[ H λ ◦ Z λ , τ ~ ] + , τ ~ ] ± = 2(1 ± Z λ + (6.2)The standard construction scheme of b Z λ ± that lurks in the background is now represented by g (p ,∂ ) / / s × τ ~ , · ] + , i[ τ ~ , · ] − ) / / s × ± pr / / s (6.3)We investigate some computations on the moral of the hidden procedure 6.3:Notice, if we set [ A, B ] ∗± := ( A ◦ B ± B ∗ ◦ A ∗ ) we can obviously define two binary operations[ · , · ] + / ◦ , i[ · , · ] − / ∂ ◦ : g ⊗ → s ( ~ ) (6.4)where the tensor product is over R and ◦ : g ⊗ → s ( ~ ) maps ( A, B ) to the composition A ◦ B .21xplicit the two maps g → s ( ~ ) sending H λ ◦ Z λ to b Z λ ± ( ~ ) described in the sequence 6.3 act by A → ( τ ~ ◦ A + A ∗ ◦ τ ~ ) / [ τ ~ , · ] ∗ + ( A ) and the τ ~ conjugated formula A → ( τ ~ ◦ A ∗ + A ◦ τ ~ ) / s ( ~ ) this maps are just [ τ ~ , · ] + / τ ~ ◦ [ τ ~ , · ] + /
2. Notice that thereis also a negative version of this two maps, namely A → i( τ ~ ◦ A − A ∗ ◦ τ ~ ) / i2 [ τ ~ , · ] ∗− ( A )and minus the τ ~ conjugated formula A → i( τ ~ ◦ A ∗ − A ◦ τ ~ ) /
2, restricted to s ( ~ ) this negativemaps are i[ τ ~ , · ] − / τ ~ conjugate. The negative maps can essentially be recovered fromthe positive maps by substituting A ∈ g → i A ∈ g and vice versa by a small computation. Lemma
Let
I ∈ g be an involution. The following iteration formulas hold: ( / / I , · ] ∗± ! n +1 = ( ± n/ (id ± Con ∗I ) (cid:14) if n is even ( ± n +12 ( / / I , · ] ∗± if n is odd ◦ ( / / I , · ] ∗± Proof.
For an involution
I ∈ g the equations [ I , · ] ∗ + ◦ i2 [ I , · ] ∗− = id +Con ∗I ◦ ∂ , i2 [ I , · ] ∗− ◦ [ I , · ] ∗ + = id − Con ∗I ◦ ∂ and (id ± Con ∗I ) = 2(id ± Con ∗I ), [ I , · ] ∗± ◦ Con ∗I = ± [ I , · ] ∗± ◦ ∗ = Con ∗I ◦ [ I , · ] ∗± hold. Proposition
Let
I ∈ g be an involution. If I ◦ ( A − A ′ ) = ( A − A ′ ) ∗ ◦ I ∗ then the negativemaps i[ I , · ] ∗− / , Con ∗I ◦ i[ I , · ] ∗− / will send A and A ′ to the same element of s ( ~ ) , i.e. the space C ∗ g ( I ) := { A ∈ g |I ◦ A = A ∗ ◦ I ∗ } = { ( A + Con ∗I ( A ∗ )) / , A ∈ g } ⊂ g called the adjoint centralizer of I is the kernel of the negative maps i2 [ I , · ] ∗− and its I adjointconjugate. Analog iC ∗ g ( I ) is the kernel of the two positive maps [ I , · ] ∗ + and Con ∗I ◦ [ I , · ] ∗ + . We can restate 6.0.3 saying that the two negative maps are well-defined injective operators g / C ∗ g ( I ) → s ( ~ ) and analogous the positive maps descend to well-defined maps g / iC ∗ g ( I ) → s ( ~ ).The projection A → ( A + Con τ ~ ( A ∗ )) / ∈ C ∗ g ( τ ~ ) does not respect i2 [ · , · ] ∗− . C ∗ g ( I ) is notclosed under adjunction ∗ ( ~ ) or composition, but still a sub Lie- or Jordan-algebra with respectto the usual Lie bracket i2 [ · , · ] − or [ · , · ] + .The two inclusions τ ~ ◦ A, A ◦ τ ~ ∈ C ∗ g ( τ ~ ) ∀ A ∈ s ( ~ ) and the complementary inclusions τ ~ ◦ A, A ◦ τ ~ ∈ s ( ~ ) ∀ A ∈ C ∗ g ( τ ~ ) allow to identify the space s ( ~ ) in different ways with C ∗ g ( τ ~ ).It is well-known that the eigenvalues of operators contained in s ( ~ ) are real but this is notunconditionally true for operators contained in C ∗ g ( τ ~ ) but almost: Proposition
For A ∈ C ∗ g ( τ ~ ) the eigenvalue equation Aψ E = Eψ E implies E ∈ R or thevanishing h ψ E , τ ~ ψ E i ~ ⇔ h ψ + E , ψ + E i ~ − h ψ − E , ψ − E i ~ with ψ ± E ∈ H ± ~ according to 3.3.Proof. In fact the proof only assumes h αf, βg i ~ = α · β h f, g i ~ and we do not need that h· , ·i ~ ispositive: Assume Aψ E = Eψ E , Aψ E ′ = E ′ ψ E ′ for A ∈ C ∗ g ( τ ~ ) and we have E h ψ E , τ ~ ψ E ′ i ~ = h Aψ E , τ ~ ψ E ′ i ~ = h τ ~ ◦ A ∗ ( ~ ) ◦ τ ~ ψ E , τ ~ ψ E ′ i ~ = h τ ~ ψ E , Aψ E ′ i ~ = E ′ h ψ E , τ ~ ψ E ′ i ~ Proposition
For an involution I the maps [ I , · ] ∗ + , Con ∗I ◦ [ I , · ] ∗ + = [ I , · ] ∗ + ◦ ∗ and i2 [ I , · ] ∗− , Con ∗I ◦ i2 [ I , · ] ∗− = i2 [ I , · ] ∗− ◦ ∗ commute with adjoint conjugation by elements of the centralizer C g ( I ) := { A ∈ g |I ◦ A = A ◦ I} = { ( A + Con I ( A )) / , A ∈ g } ⊂ g For example
Con ∗ A ◦ [ I , · ] ∗ + = [ I , · ] ∗ + ◦ Con ∗ A and also [Con ∗I ◦ ∗ , Con ∗ A ] − = 0 hold ∀ A ∈ C g ( I ) . g ( I ) is closed under adjunction ∗ ( ~ ) if I ∗ = I and also under compositionand the natural Lie-bracket, we have for A, B ∈ C g ( I ) clearly A ◦ B ◦ I = I ◦ A ◦ B .We have a self-adjoint intersection C ∗ g ( τ ~ ) ∩ C g ( τ ~ ) ⊂ s ( ~ ). The spaces C ∗ g ( τ ~ ) = (cid:0) C ∗ g ( τ ~ ) (cid:1) ∗ and C g ( τ ~ ) = (C g ( τ ~ )) ∗ both contain for example Z λ + and m V ( t ) with real potential conjugatesymmetric under t → /t , hence the friendly Brujin-Newman-P´olya operators m e − ρ ln2( t ) with ρ ≥ H λ ◦ Z λ and i Z λ − are only contained in C ∗ g ( τ ~ ). Proposition
For every V ∈ H the maps c ( V + V ) / are elements of the adjoint centralizer C ∗ g ( τ ~ ) of the hermitian involution τ ~ and c ( V + τ ~ V ) / are elements of the centralizer C g ( τ ~ ) . If I ∈ g is an involution we have (Con ∗I ) = id and we can analogous to 3.3 also split A = A + + A − ∈ g = g + I ⊕ g −I where A ± are in the two eigenspaces g ±I of Con ∗I , for instance weset A ± := ( A ± I ◦ A ◦ I ∗ ) / ∈ g ±I . We have Con ∗ τ ~ b Z λ ± = b Z λ ∓ i.e. b Z λ ± are in the same conjugacyclass while τ ~ ◦ Z λ ± ◦ τ ~ = ±Z λ ± i.e. Z λ ± are in the respective eigenspaces g ± τ ~ of Con ∗ τ ~ .It is also immediate to distinguish two quite familiar operators: We showed the adjunction ∗ ± Con τ ~ ( i H λ ◦ Z λ H λ ◦ Z λ = p + i ∂ ± Con τ ~ ( i H λ ◦ Z λ H λ ◦ Z λ = 0 (6.5)and rewriting Con ∗ τ ~ = Con τ ~ in 6.5 is more practical: Because ∗ is an involution and commuteswith Con ∗ B we have for all B ∈ g the formula (cid:16) ∗ ± Con ∗ B (cid:17) = id +Con ∗ B ◦ B ± ∗ B ◦ ∗ . Lemma
Let the operator
I ∈ g be an involution i.e. I ◦ I = id . We have (cid:18) ∗ ± Con ∗I (cid:19) n = ( ∗ ± Con ∗I if n is odd id ± Con ∗I ◦ ∗ if n is evenand the operators (id ± Con ∗I ◦ ∗ ) / are projectors. By flipping signs we have in fact two projectors (cid:18) id − id ± Con ∗I ◦ ∗ (cid:19) = id ∓ Con ∗I ◦ ∗ g → ker ∗ ± Con ∗I ±
1) + 1 ∓
12 C ∗ g ( I ) onto the solutions S of the respective equations ∗ ± Con ∗I S = 0 .If ∗ − Con ∗I S = 0 then I ◦
S, S ◦ I ∗ ∈ s and if ∗ +Con ∗I S = 0 then we have i I ◦ S, i S ◦ I ∗ ∈ s . The artificial ~ -dependence of the symmetrizations Consider the category with one object H and morphisms the set of adjoinable operators g .The map m t ~ − ~ ′ : H → H defines an isomorphism ∗ ( ~ ) → ∗ ( ~ ′ ) between the two fully faithfulcontravariant functors ∗ ( ~ ) and ∗ ( ~ ) ′ because ∗ ( ~ ′ ) = Con m t ~ − ~ ′ ◦ ∗ ( ~ ) = ∗ ( ~ ) ◦ Con m t ~ ′− ~ .The symmetrization procedures p( ~ ) and ∂ ( ~ ) are not compatible with composition, we have ( p ∂ ( ~ ′ ) = ( (1 − Con m t ~ − ~ ′ ) i2 (1 − Con m t ~ − ~ ′ ) + Con m t ~ − ~ ′ ◦ ( p ∂ ( ~ )hence restrictions (p , ∂ )( ~ ′ ) = Con × m t ~ − ~ ′ ◦ (p , ∂ )( ~ ) : C × g ( m t ~ − ~ ′ ) → s × ( ~ ′ ).The space of multiplication operators m V ( t ) as defined in 3.0.4 is a subspace of C g ( m t ~ − ~ ′ ). Proposition
The involutive contravariant functors
Con τ ~ ◦ ∗ ( ~ ) = ∗ ( ~ ) ◦ Con τ ~ do notdepend on the deformation parameter ~ . roof. First τ ~ ′ = Con m t ~ − ~ ′ ( τ ~ ) is immediate. Again the two covariant functors Con τ ~ andCon τ ~ ′ are connected by the natural transformation m t ~ − ~ ′ : H → H , we have Con τ ~ ′ =Con m t ~ − ~ ′ ◦ Con τ ~ = Con τ ~ ◦ Con m t ~ ′− ~ . The substitution τ ~ ◦ m t λ = m t − λ ◦ τ ~ implies forthe composed contravariant functors Con τ ~ ′ ◦ ∗ ( ~ ′ ) = Con τ ~ ◦ ∗ ( ~ ) and the commutative diagram H m t ~ − ~ ′ (cid:15) (cid:15) H Con τ ~ A / / A ∗ ( ~ ) o o m t ~ − ~ ′ (cid:15) (cid:15) H m t ~ − ~ ′ (cid:15) (cid:15) H m t ~ ′− ~ O O H m t ~ ′− ~ O O Con τ ~ ′ A / / A ∗ ( ~ ′ ) o o H m t ~ ′− ~ O O (6.6)The following commutative diagram 6.7 summarizes some considerations, the dashed linerepresents the contravariant ~ independent functor of 6.1.1: g Con τ ~ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ ∗ ( ~ ) t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ Con m t ~ − ~ ′ (cid:7) (cid:7) ✏✏✏✏✏✏✏✏✏✏✏✏✏✏ g ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ (cid:30) (cid:30) Con τ ~ ◦ ∗ ( ~ ) = Con τ ~ ′ ◦ ∗ ( ~ ′ ) (cid:30) (cid:30) g i i ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ (cid:0) (cid:0) (cid:0) (cid:0) g Con m t ~ ′− ~ G G ✏✏✏✏✏✏✏✏✏✏✏✏✏✏ Con τ ~ ′ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ∗ ( ~ ′ ) i i ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ (6.7)We can rewrite the ~ -dependence of the respective involutions by τ ~ ′ = m t ~ − ~ ′ ◦ τ ~ = τ ~ ◦ m t ~ ′− ~ and consider the ~ dependence of the other two symmetrization procedures described in 3.1.2,again the natural transformation ∗ ( ~ ) → ∗ ( ~ ′ ) given explicit by m t ~ − ~ ′ : H → H appears, but asa composition acting on the space of adjoinable operators g : Lemma
We have the symmetrization identities ( [ τ ~ ′ , · ] ∗ ( ~ ′ )+i2 [ τ ~ ′ , · ] ∗ ( ~ ′ ) − = m t ~ − ~ ′ ◦ ( [ τ ~ , · ] ∗ ( ~ )+i2 [ τ ~ , · ] ∗ ( ~ ) − and ( [ τ ~ ′ , · ] ∗ ( ~ ′ )+i2 [ τ ~ ′ , · ] ∗ ( ~ ′ ) − ◦ ∗ ( ~ ′ ) = ( [ τ ~ , · ] ∗ ( ~ )+i2 [ τ ~ , · ] ∗ ( ~ ) − ◦ ∗ ( ~ ) ◦ m t ~ ′− ~ and can express this for example in the commutative diagram s ( ~ ) - - ◦ m t ~ ′− ~ (cid:15) (cid:15) s ( ~ ) m t ~ − ~ ′ ◦ (cid:15) (cid:15) Con τ ~ q q g / (cid:0) iC ∗ g ( τ ~ ) ∩ iC ∗ g ( τ ~ ′ ) (cid:1) [ τ ~ , · ] ∗ ( ~ )+ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ [ τ ~ , · ] ∗ ( ~ )+ ◦ ∗ ( ~ ) k k ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ [ τ ~ ′ , · ] ∗ ( ~ ′ )+ + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ [ τ ~ ′ , · ] ∗ ( ~ ′ )+ ◦ ∗ ( ~ ′ ) s s ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ s ( ~ ′ ) ◦ m t ~ − ~ ′ O O s ( ~ ′ ) Con τ ~ ′ m m m t ~ ′− ~ ◦ O O (6.8) By the propositions 6.0.3 and 6.1.1 we have in fact iC ∗ g ( τ ~ ) = iC ∗ g ( τ ~ ′ ) = iC ∗ g ( τ ~ ) ∩ iC ∗ g ( τ ~ ′ ) . Certain quotient spaces and cohomologies
We proceed quite in analogy with the absorption spectrum of Connes but instead of quotient outthe image of the subspace defined by the condition 2.3 under the operator Z λ we quotient outthe images of 3.7: Let H λ ± ( ~ ), b H λ ± ( ~ ) be the quotients of H by the respective image of the fouroperators Z λ ± ( ~ ), b Z λ ± ( ~ ), in formulas H λ ± := H / Z λ ± , b H λ ± := H / b Z λ ± , i.e. elements of this space areequivalence classes ψ λf ( ~ ) with f ∈ H , we have by definition for two representatives ψ λf ( ~ ) = ψ λf ′ ( ~ ) ⇔ f = f ′ + Z λ ± ( ~ ) g and ψ λf ( ~ ) = ψ λf ′ ( ~ ) ⇔ f = f ′ + b Z λ ± ( ~ ) g (7.1)with g ∈ H . There are technical issues but in consideration of the following discussion 7.0.1 itseems desirable to further restrict g , maybe Hurwitz stability could play a role.Because we suppose ~ ∈ R and λ ∈ R + complex conjugation ψ λf ( ~ ) = ψ λf ( ~ ) is a involution.Also τ ~ induces a well-defined involution τ ~ on H λ ± ( ~ ) by the formula τ ~ ψ λf := ψ λτ ~ f . If ~ = − λλ the issue seems more delicate in consideration of ζ (1 − s ) = 2(2 π ) − s cos (cid:0) πs (cid:1) Γ( s ) ζ ( s ), however τ ~ : ψ λf → ψ λτ ~ f is well-defined for the quotient H / h b Z λ ± ( ~ ) i by the linear span of b Z λ ± ( ~ ).In fact the previous defined quotients are not only vector spaces, they are commutative rings:Because of 4.0.10 and 4.5 we have induced by the usual convolution 4.2 a well-defined associativeproduct ∗ : b H λ ± ( ~ ) × b H λ ± ( ~ ) and ∗ : H λ ± ( ~ ) × H λ ± ( ~ ) → H λ ± ( ~ ) naturally by the same formula ψ λf ∗ ψ λf := ψ λf ∗ f (7.2)The computations in 3.1 show that H α descend by H α ψ λf ( ~ ) := ψ λH α f ( ~ ) to well-definedoperators on the four quotient spaces because H α commutes with Z λ and we have 3.12. Let µ ( n ) be the M¨obius function defined by 1 for n = 1, 0 if n is not square free and ( − m if n = p · · · p m with p i = p j . The map Z λ admits in some sense an inverse ( Z λ ) − := P ∞ n =1 µ ( n ) d n λ werefer the reader to [12] for a precise definition of the domain of this operators, but this operatorsare not maps H → H , they could produce a singularity at zero. Although H α : H → H is injective( t − /α is not an element of H hence the kernel of H α : H → H trivial) H α is also not invertibleby the natural candidate geometric series, this operation is highly singular. Hence the existenceof non-trivial solutions of the following eigenvalue equations 7.3 is a priori not justified: H α ψ λf ( ~ ) = Eψ λf ( ~ ) ⇔ ∃ g ∈ H : H α f − Ef = ( Z λ ± ( ~ ) b Z λ ± ( ~ ) g (7.3)We set ~ = (1 − λ ) /λ and α = 2 λ and Mellin transform of 7.3 implies(1 − E − s ) M [ f ] (cid:16) sλ (cid:17) = ( (cid:2) (1 − id) · ζ ( s ) ± (1 − id) · ζ (1 − s ) (cid:3) M [ g ] (cid:16) sλ (cid:17) (7.4)(1 − E − s ) M [ f ] (cid:18) (1 ∓ / ± sλ (cid:19) = (1 − id) · ζ (cid:0) (1 ∓ / ± s (cid:1) M [ g ] (cid:18) (1 ± / ∓ sλ (cid:19) The l.h.s of the equations 7.4 clearly vanishes at z = (1 − E ) / " (1 − id) · ζ (cid:18) − E (cid:19) ± (1 − id) · ζ (cid:18) E (cid:19) M [ g ] (cid:18) − E λ (cid:19) (7.5)0 = (1 − id) · ζ (cid:18) − E (cid:19) M [ g ] (cid:18) ± E λ (cid:19) ζ ( z ) = 0: By the convolution theorem 4.3 and the well-known functional equationΞ( s ) = Ξ(1 − s ) there exist well-defined sesquilinear maps h· , ·i λz : H λ ± (cid:0) − λλ (cid:1) × H λ ± (cid:0) − λλ (cid:1) → C and h· , ·i λz : H / h b Z λ ± i (cid:0) − λλ (cid:1) × H / h b Z λ ± i (cid:0) − λλ (cid:1) → C respectively defined by the same formula D ψ λf , ψ λf E λz := M h τ − λλ f ∗ f i ( z/λ ) (7.6)We have h ψ λf , ψ λf i λ − z = D ψ λτ − λλ f , ψ λτ − λλ f E λz = h ψ λf , ψ λf i λz . The calculations3.12 and 4.11 imply D H λ ψ λf (cid:18) − λλ (cid:19) , ψ λf (cid:18) − λλ (cid:19)E λz = − D ψ λf (cid:18) − λλ (cid:19) , H λ ψ λf (cid:18) − λλ (cid:19)E λz (7.7)Hence if ψ λf and ψ λf are eigenstates of H λ with eigenvalues E and E we have “orthogonality” ( E + E ) D ψ λf (cid:18) − λλ (cid:19) , ψ λf (cid:18) − λλ (cid:19)E λz = 0 (7.8)If H λ ψ λf = Eψ λf with z = − E vanishing of 7.8 implies ℜ ( z ) = or 0 = h ψ λf , ψ λf i λz , but we cannot justify the existence of solutions of 7.3 with g restricted to have only zeros on the criticalline and also ∗ is a priori not well-defined on such a restricted quotient space. We adapt a idea developed in [6], [12]: The dual spaces (cid:0) H λ ± ( ~ ) (cid:1) ∗ , (cid:0) b H λ ± ( ~ ) (cid:1) ∗ are the spaces oflinear functionals ϕ : H → C that annihilate the image of Z λ ± ( ~ ), b Z λ ± ( ~ ) respectively, hencespanned by ϕ lz given by ψ f → M (cid:2) m ln l ( t ) f (cid:3) ( ~ + zλ ) with 0 ≤ l < k z ∈ N + and where z is azero of multiplicity k z of the respective functions (1 − id) ζ (cid:16) (1+ ~ )2 /λ + z (cid:17) ± (1 − id) ζ (cid:16) (1+ ~ )2 /λ − z (cid:17) ,(1 − id) ζ (cid:16) (1+ ~ )2 /λ ± z (cid:17) . The cup product defined by (cid:0) ϕ ∪ ϕ (cid:1) ( ψ λf , ψ λf ) := ϕ ( ψ λf ( ~ )) ϕ ( ψ λf ( ~ ))satisfies the inequality (cid:0) ϕ z ∪ ϕ z (cid:1) ( ψ λf , ψ λf ) ≥
0. The convolution theorem implies (cid:0) ϕ l z ∪ ϕ l z (cid:1) ( ψ λf , ψ λf ) = M h m t z − z λ ln l ( t ) f ∗ m t z − z λ ln l ( t ) f i (cid:18) ~ z + z λ (cid:19) (7.9)Hence the transposed operators t H α acting on the respective dual spaces satisfy t H α ϕ l z ∪ ϕ l z − ϕ l z ∪ t H α ϕ l z = α ( z − z ) λ ϕ l z ∪ ϕ l z + α (cid:0) l ϕ l z ∪ ϕ l − z − l ϕ l − z ∪ ϕ l z (cid:1) (7.10) t H α ϕ lz ( ψ λf ) = M (cid:2) m ln l ( t ) H α f (cid:3) (cid:18) ~ zλ (cid:19) = M (cid:2) H α m ln l ( t ) f (cid:3) (cid:18) ~ zλ (cid:19) − αl M (cid:2) m ln l − ( t ) f (cid:3) (cid:18) ~ zλ (cid:19) (7.11)where we used 7.9 and the Leibniz rule. By integration by parts M (cid:2) H α f (cid:3) ( s ) = (1 − αs ) M (cid:2) f (cid:3) ( s )holds for f ∈ H and 7.11 implies the formula (cid:0) t H α − α (cid:0) ~ + zλ (cid:1)(cid:1) ϕ lz = − αlϕ l − z . Henceif we suppose (1 − id) ζ (cid:16) (1+ ~ )2 /λ + z (cid:17) ± (1 − id) ζ (cid:16) (1+ ~ )2 /λ − z (cid:17) = 0 or 0 = (1 − id) ζ (cid:16) (1+ ~ )2 /λ ± z (cid:17) respectively the functionals ϕ lz with 0 ≤ l < k z are in the kernel of ( t H α − α (cid:0) ~ + zλ (cid:1) ) l +1 .In particular ϕ z is a eigenstate of H α with the eigenvalue 1 − α (cid:0) ~ + zλ (cid:1) and the dimension of ∪ l ∈ N ker (cid:0) t H α − α (cid:0) ~ + zλ (cid:1)(cid:1) l +1 is k z . 26 .2 Some cohomologies In the following lemmas ( R , ∗ , +) denotes a commutative ring, τ : R → R an involutive ringhomomorphism and R ± denote the eigenspaces of τ for the eigenvalues ± Lemma
The formulas ⋆ ± := ∗ ◦ (id ± τ ) ⊗ (id ± τ ) define associative products and f ⋆ ± f ∈R + . We have a Lie-bracket [ · , · ] ∗ := ∗ ◦ (cid:0) τ ⊗ id − id ⊗ τ (cid:1) and [ f , f ] ∗ ∈ R − . τ is a morphism of ⋆ ± and a Lie morphism of [ · , · ] ∗ .Proof. ⋆ ± = ∗ ◦ (id ± τ ) ⊗ also is associative if τ symbolizes a involutive morphism of a non-commutative product ∗ . Observe if ∗ is commutative f ∗ V g := f ∗ g ∗ V is also associative.The following lemma provides two linear maps R / R + → (cid:8) d : R ~ ~ | d linear , d = 0 (cid:9) . Lemma
The maps d + V := [(id − τ ) V, · ] ∗ : R → R are inner derivations of [ · , · ] ∗ and d − V defined by d − V f := V ⋆ − f satisfies d − V ( f ∗ g ) = d − V f ∗ g + τ f ∗ d − V g . The equations d ± V ◦ d ± V = 0 hold ∀ V , V ∈ R and we have the inclusion R ∓ ⊂ ker(d ± V ) . The product rule d ± V ∗ V f = V ∗ d ± V f + τ V ∗ d ± V f holds, hence ker(d ± V ) ∩ ker(d ± V ) ⊂ ker(d ± V ∗ V ) .Proof. More general if V ∈ R − and V ∗ V = − V ∗ τ V we have a square zero map d V ,V bysetting d V ,V f := V ∗ f + V ∗ τ f . In particular we have d (id − τ ) V, ± (id − τ ) V = d ± V .As usual we can induce a Lie-bracket [ · , · ] ∗ on the cohomology H ( R , d + V ) := ker(d + V ) / im(d + V )by [ ⌊ f ⌋ + V , ⌊ f ⌋ + V ] ∗ := ⌊ [ f , f ] ∗ ⌋ + V for representatives ⌊ f i ⌋ + V ∈ H ( R , d + V ). Also ∗ induces a producton H ( R , d − V ) := ker(d − V ) / im(d − V ) by ⌊ f ⌋ − V ∗ ⌊ f ⌋ − V := ⌊ f ∗ f ⌋ − V for ⌊ f i ⌋ − V ∈ H ( R , d − V ). We havea involution on the cohomologies by τ ⌊ f ⌋ ± V := ⌊ τ f ⌋ ± V . Proposition
For R = H λ ± ( ~ ) and R = H / h b Z λ ± i respectively the operator H ~ defines amap H ( R , d ± V ) → H ( R , d ∓ V ) and ∆ ~ ⌊ ψ f ⌋ V := ⌊ ψ ∆ ~ f ⌋ V descends to H ( R , d ± V ) .Proof. We have shown 3.12 that H / (1+ ~ ) anti-commutes with τ ~ , b Z λ ± ( ~ ) and commutes with Z λ ± ( ~ ) and we have the identity ∆ α = H α − α + α ) t∂ t + α t ∂ t .For the usual convolution 4.2 we have the adjunction (d ± V ) ∗ = − d ∓ V with respect to the innerproduct 3.0.3, because d ± V = c V − τV ◦ (id ± τ ) and 4.0.8, 4.5 implies (d ± V ) ∗ = (id ± τ ) ◦ c V − τV = d ∓ V .Observe if V − τ ~ V = 0 we have H ∓ ~ = ker(d ± V ) by Mellin transform and the identity theorem.An interesting case is to consider potentials (1 − τ ~ )∆ Ψ and (1 − τ ~ ) H ∆ Ψ ∈ H − ~ , here wehave a to 7.1 quite analogous spectral realization.The ∗ derivation m ln( t ) is not well-defined for H λ ± ( ~ ) and H / h b Z λ ± i , nevertheless m ln( t ) inducesan operation H ( H , d ± V ) → H ( H , d ∓ V ): Lemma
Assume ∂ is a derivation of ∗ and satisfies ∂τ = − τ ∂ . If R ∓ = ker(d ± V ) wehave a well-defined map ∂ : H ( R , d ± V ) → H ( R , d ∓ V ) by ∂ ⌊ f ⌋ ± V := ⌊ ∂f ⌋ ∓ V .Proof. The assumptions imply the computation d ∓ V ∂f = ∂ d ± V f − ((id + τ ) ∂V )) ∗ ((id ± τ ) f ).Inspired by the strategy proposed in [9] it seems natural to consider for ρ ≥ Brujin-Newman-P´olya operators e − ρ m t ) = P ∞ n =0 − ρ n n ! m ln n ( t ) : H ( H , d ± V ) → H ( H , d ± V ). Lemma
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