aa r X i v : . [ m a t h . N T ] F e b Designed pseudo-Laplacians
E. Bombieri P. GarrettFebruary 20, 2020
Abstract
We elaborate and make rigorous various speculations about the implica-tions of spectral properties of self-adjoint operators on spaces of automorphicforms for location of zeros of L -functions. Some of these ideas arose in workof Colin de Verdi`ere, Lax-Phillips, and Hejhal, from the late 1970s and early1980s, not to mention semi-apocryphal attributions to P ´olya and Hilbert. Forexample, given a complex quadratic extension k of Q , we give a natural self-adjoint extension of a restriction of the invariant Laplacian on the modularcurve whose discrete spectrum, if any, consists of values s ( s − for zeros s of ζ k ( s ) . Unfortunately, there seems to be no reason for this discrete spec-trum to be large. In fact, Montgomery’s pair correlation, and the behavior of ζ (1 + it ) , imply that at most of zeros of ζ ( s ) can appear in this discretespectrum. Less naively, some preliminary positive results about the dynamicsof zeros do follow from these considerations.
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Friedrichs extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Friedrichs self-adjoint extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Friedrichs extensions of restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103. Eisenstein-Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1 Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Heegner points and Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Spectral decomposition and spectral synthesis . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Eisenstein-Sobolev spaces E r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Automorphic Dirac delta distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6 Eisenstein-Heegner distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.7 Solving ( − ∆ − λ w ) u = θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.8 Constant-term distributions η a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.7 Solving ( − ∆ − λ w ) u = η a η a ( v w,a ) for a > and ℜ ( w ) > / . . . . . . . . . . . . . . . . . . . . . . 385.3 Computing θ d ( v w,a ) for a ≫ θ and ℜ ( w ) > / . . . . . . . . . . . . . . . . . . . . 385.4 Computing η a ( u θ,w ) for a ≫ θ and ℜ ( w ) > / . . . . . . . . . . . . . . . . . . . . 405.5 Computing θ ( u θ,w ) for a > and ℜ ( w ) > / . . . . . . . . . . . . . . . . . . . . . . 415.6 Rewriting the determinant condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.7 Meromorphic continuation and location of zeros . . . . . . . . . . . . . . . . . . . . . . 425.8 An important remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.9 Computing θE s in a special case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466. Meromorphic continuations of spectral expansions . . . . . . . . . . . . . . . . . . . . . . . 476.1 Vector-valued integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.2 Holomorphic vector-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.3 Spaces M of moderate-growth functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.4 Pre-trace formula and E ε ⊂ M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.5 Meromorphic continuation of spectral integrals . . . . . . . . . . . . . . . . . . . . . . . 536.6 Spectral corollaries of meromorphic continuation . . . . . . . . . . . . . . . . . . . . . 596.7 Remarks on appealing but incorrect arguments . . . . . . . . . . . . . . . . . . . . . . . 627. Spacing of spectral parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627.1 Exotic eigenfunction expansions of distributions . . . . . . . . . . . . . . . . . . . . . . 627.2 Exotic eigenfunction expressions for solutions of differential equations . 637.3 Interleaving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667.4 Spacing of spectral parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.5 Juxtaposition with pair-correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728. Spacing of zeros of ζ k ( s ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728.1 Exotic eigenfunction expansions and interleaving . . . . . . . . . . . . . . . . . . . . . 738.2 Dependence of eigenvalues on cut-off height . . . . . . . . . . . . . . . . . . . . . . . . . 768.3 Sample unconditional results on spacing of zeros . . . . . . . . . . . . . . . . . . . . . 78References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 First, a simple example of the content of theorem 10: there is a Hilbert space E (described just below) of automorphic forms on SL ( Z ) \ H such that, for everycomplex quadratic field extension k of Q , there is a self-adjoint extension e S k ofa certain restriction S k of − ∆ , with the invariant Laplace-Beltrami operator ∆ = ombieri and Garrett 3 y ( ∂ /∂x + ∂ /∂y ) , such that the eigenvalues λ s = s (1 − s ) > / of e S k on E , if any , occur only for s a zero of the Dedekind zeta function ζ k ( s ) . Something likethis is suggested by parts of [5], although perhaps often misunderstood as heuristicsrather than as potentially rigorous arguments. In any case, this is not what onemight hope: there is no assertion of existence of any eigenvalues of e S k , and thereis no assertion that every zero of ζ k ( s ) gives an eigenvalue. Corollary 69 showsthat, assuming Montgomery’s pair correlation conjecture [21], at most of thezeros s of ζ ( s ) can occur as parameters for eigenvalues λ s of any particular e S k ,suggesting that perhaps none do.Eigenvalues of self-adjoint operators are real . Thus, when parametrized as λ s = s (1 − s ) ∈ R , either ℜ ( s ) = or s ∈ R . Thus, as apocryphally suggested by G.Polya and by D. Hilbert, one might imagine proving the Riemann Hypothesis byfinding a self-adjoint operator such that, for every non-trivial zero s of ζ ( s ) , λ s isan eigenvalue. (See [24].)In [20] an argument is sketched for discrete decomposition (i.e., pure point spec-trum) of L a (Γ \ H ) , the latter space defined to be L automorphic forms with con-stant term vanishing at height y ≥ a > , with respect to the Friedrichs self-adjointextension e S Θ of the restriction S Θ of − ∆ to C ∞ c (Γ \ H ) ∩ L a (Γ \ H ) . The firstsurprise is that part of the continuous spectrum of the original invariant Laplace-Beltrami operator lying inside L a (Γ \ H ) , consisting of suitable wave-packets ofEisenstein series E s , becomes discrete. Even more surprising, the new eigen-functions with eigenvalues λ > / are certain truncated Eisenstein series ∧ a E s ,namely, for s such that a s + c s a − s = 0 , where a s + c s a − s is the constant termof E s evaluated at height a , seemingly contradicting elliptic regularity. Evidentlyeigenfunctions of the Friedrichs extension of a restriction of an elliptic operator canfail to be smooth . Thus, a s + c s a − s = 0 if and only if λ s = s (1 − s ) is an eigen-value of e S Θ . By self-adjointness, a s + c s a − s = 0 only for s ∈ + i R (or in [0 , ).This idea appears in [20], pages 204-6.In that context, the document [14] was provocative: attempting numerical deter-mination of eigenvalues for − ∆ on SL ( Z ) \ H , the list of spectral parameters s for eigenvalues λ s = s (1 − s ) was observed by H. Stark and D. Hejhal to includelow-lying zeros of ζ ( s ) and of L ( s, χ − ) with the Dirichlet L -function of conduc-tor . [16] reported attempted reproduction of the numerical results, with the zerosof zeta and the L -function notably missing from the list of spectral parameters, andobserving that the spurious appearance of these values in the list of [14] was due to amis-application of the Henrici collocation method [8]. Hejhal further observed thatthe solution procedure of [14] in fact allowed as solution a value u ( z ) = G λ s ( z, ω ) of an automorphic Green’s function G λ s ( z, z o ) as a solution, with ω = e πi/ . Au- Designed pseudo-Laplacianstomorphic Green’s functions and related meromorphic families had been studied in[22], [23], and [7].That is, the values λ s = s (1 − s ) obtained by [14] not belonging to genuine cusp-forms were values λ s fitting into an equation ( − ∆ − λ s ) u = δ a fcω for Γ -invariantDirac δ a fcω on H , supported on images of ω = e πi/ . Since λ s appearing in suchan equation are not necessarily eigenvalues of a self-adjoint operator, they need notbe real . A claim that δ a fcω can be disregarded on the grounds that it has support ofmeasure zero is incorrect. Nevertheless, there were precedents in [20], [4], and [5]for legitimate reinterpretation of certain inhomogeneous equations as homogeneousequations. For example, in [20] the exotic eigenfunctions for λ > / (certain trun-cated Eisenstein series) are solutions of an inhomogeneous distributional equation ( − ∆ − λ w ) u = η a , where η a evaluates the constant term of an automorphic format height y = a . This was a precedent for the suggestion at the end of [5] that theinhomogeneous equation ( − ∆ − λ s ) u = δ a fcω on Γ \ H could be converted to a ho-mogeneous equation involving a self-adjoint operator, so that the values λ s wouldbecome (genuine) eigenvalues, and thus be real .A complication observed in [5] for an equation ( − ∆ − λ s ) u = θ on Γ \ H , withdistribution θ on Γ \ H , is that the Friedrichs extension e S θ of the restriction S of − ∆ to the kernel of θ on C ∞ c (Γ \ H ) converts this inhomogeneous equation to ahomogeneous one with an auxiliary condition, only for θ in a suitable (global au-tomorphic) Sobolev space H − (Γ \ H ) . This is the Hilbert-space dual of H (Γ \ H ) ,which is the completion of C ∞ c (Γ \ H ) with respect to the Sobolev H norm givenby | f | H = (cid:10) (1 − ∆) f, f (cid:11) L for f ∈ C ∞ c (Γ \ H ) , discussed in subsection 3.4.To solve equations ( − ∆ − λ ) u = θ with distributions θ , it is advantageous touse a spectral characterization of (global automorphic) Sobolev spaces. Recall thespectral decomposition of L (Γ \ H ) with Γ = SL ( Z ) , for example from [6] or[18]: first, for f ∈ C ∞ c (Γ \ H ) , f = X F h f, F i · F + h f, i · h , i + 14 πi Z + i ∞ − i ∞ h f, E s i · E s ds where F runs over an orthonormal basis of L cuspforms, with the L hermi-tian product h , i notation abused to denote the corresponding integral for f ∈ C ∞ c (Γ \ H ) : h f, E s i = Z Γ \ H f ( z ) · E s ( z ) dx dyy ombieri and Garrett 5Then one proves a Plancherel theorem for test functions f : | f | = X F |h f, F i| + |h f, i| h , i + 14 πi Z + i ∞ − i ∞ |h f, E s i| ds Further, f → ( s → h f, E s i ) has image dense in the space of L functions g on + i R such that g (1 − s ) = c s · g ( s ) . As with the Plancherel-Fourier transform on L ( R ) , the spectral expansion extends to a Plancherel theorem for L (Γ \ H ) , withthe pairings h f, E s i necessarily interpreted as isometric extensions. The spectralsynthesis integrals R ( ) h f, E s i · E s ds converge only in an L sense, certainly notnecessarily pointwise. Then, for r ∈ R we can define (global automorphic) Sobolevspaces H r (Γ \ H ) to be the completion of C ∞ c (Γ \ H ) with respect to the H r norms | f | H r = X F |h f, F i| · (1 + λ s F ) r + |h f, i| · (1 + λ ) r h , i + 14 πi Z + i ∞ − i ∞ |h f, E s i| · (1 + λ s ) r ds where s F ∈ C gives the eigenvalue λ s F of cuspform F with respect to ∆ , firstdefined for f ∈ C ∞ c (Γ \ H ) . See subsection 3.4.An obstacle appears: subsection 3.5 observes that an automorphic Dirac δ isnot in H − (Γ \ H ) , but only in H − − ε (Γ \ H ) for every ε > . A possible wayaround this obstacle appears briefly at the end of [5], to consider the restriction θ of δ a fcω to a smaller Hilbert space of automorphic forms. In [5], this smaller space issuggested to be the orthogonal complement to the discrete spectrum, but it turns outto be necessary to retain the constants, which appear as square-integrable residuesof Eisenstein series at s = 1 . For precision, a different description of the restriction θ is appropriate, as given in detail in section 3.4. First, the suitable analogue of testfunctions here is the space E ∞ c of pseudo-Eisenstein series Ψ ϕ ( z ) = X Γ ∞ \ Γ ϕ ( ℑ ( γz )) with test-function data ϕ ∈ C ∞ c (0 , ∞ ) . The r -th Eisenstein-Sobolev space E r isthe completion of E ∞ c with respect to the H r -norm. Let S be the restriction of − ∆ to E ∞ c . Initially, let θ be the restriction of δ a fcz o to E ∞ c . As we see just below,this restriction θ has finite H − -norm, so extends continuously to an element of Designed pseudo-Laplacians E − . Then take S θ to be the further restriction of S to have domain E ∞ c ∩ ker θ ,which is still dense in E , since θ E . Friedrichs’ construction applies to theunbounded operator S θ on the Hilbert space E . In present terms, for the analogousrestriction θ of a finite real-linear combination of automorphic δ such that θE s =( √ d k / s · ζ k ( s ) /ζ (2 s ) for d k the (absolute value of) the discriminant of k , since θ is a restriction of a distribution of compact support on Γ \ H , θ is in E −∞ = S r E r .Its H r -norm would be | θ | H r = |h θ, i| · (1 + λ ) r h , i + 14 πi Z ( ) | ( p d k / s ζ k ( s ) /ζ (2 s ) | · (1 + λ s ) r ds By the second-moment bound of [15] and Landau’s bounds on the behavior of ζ ( s ) on the edge of the critical strip, the integral is finite for r < − / . Thus, θ ∈ E − − ε ⊂ E − , and has a spectral expansion converging in E − : θ = h θ, i · h , i + 14 πi Z ( ) a − sk ζ k (1 − s ) ζ (2 s ) · E s ds Certainly this does not converge pointwise, but does converge in E − . Such expan-sions allow rigorous solution of differential equations ( − ∆ − λ w ) u = θ by division :for ℜ ( w ) > and w R , a solution in E is u θ,w = h θ, i · λ − λ w ) · h , i + 14 πi Z ( ) a − sk ζ k ( s ) ζ (2 s ) · ( λ s − λ w ) · E s ds Since θ ∈ E − , the Friedrichs extension e S θ of the restriction S θ of − ∆ to E ∞ c ∩ ker θ behaves as desired: for u ∈ V , ( e S θ − λ ) u = 0 if and only if ( − ∆ − λ ) u = c · θ for some constant c , and θu = 0 . That is, the inhomogeneous distributionalequation ( − ∆ − λ ) u = c · θ is equivalent to a homogeneous equation ( e S θ − λ ) u = 0 together with the auxiliary condition θu = 0 .At this point, we can make precise sense of one speculation from [5]: with θ ∈ E − the restriction of δ a fcω to E (after extending by continuity), theorem 10 showsthat the discrete spectrum λ s > / of e S θ , if any , is λ s = s (1 − s ) with s a zeroof ζ ( s ) · L ( s, χ − ) . There is no assurance of existence of any discrete spectrum.Spectral theory can be applied in less naive ways: systematic construction ofnatural self-adjoint operators S ≥ exhibits meromorphic functions whose zerosombieri and Garrett 7 s are on the critical line + i R , and perhaps also on [0 , . The arguably next-simplest continuation is to consider two elements η, θ ∈ E − , such that ( C · η + C · θ ) ∩ E = { } with η c = η and θ c = θ , the restriction S η,θ of − ∆ to domain e S η,θ . Let u η,w and u θ,w be solutions in E (Γ \ H ) of ( − ∆ − λ w ) u = η and ( − ∆ − λ w ) u = θ ,respectively. The off-line non-vanishing argument (see subsection 4.4) shows thatfor ℜ ( w ) > and w R , h η, u η,w i · h θ, u θ,w i − h η, u θ,w i · h θ, u η,w i 6 = 0 where the pairings are on E − × E . Taking η to be the constant-term evaluation η a f = Z f ( x + ia ) dx with a ≫ θ gives (see subsection 4.4 and the prior calculations in section 5) h η a , u θ,w i = θE w w − h θ, u η a ,w i For complex quadratic k over Q with absolute value of discriminant d k , let a k = √ d k / , take θ ∈ E − such that θE w = a wk ζ k ( w ) /ζ (2 w ) . Then the non-vanishingassertion becomes explicit: in ℜ ( w ) > and w ( , , for a ≥ a k , after somenatural simplifications, a − wk · ( a w − + c w ) · (cid:16) h k /a k λ − λ w + 14 πi Z ( ) (cid:12)(cid:12)(cid:12) ζ k ( s ) ζ (2 s ) (cid:12)(cid:12)(cid:12) dsλ s − λ w (cid:17) − w − · ζ k ( w ) ζ (2 w ) = 0 where h k is the class number of k . In fact, meromorphic continuation and func-tional equation Theorem 29 and Corollary 30 show that the latter expression haszeros only on ℜ ( w ) = and [0 , .Returning to the simplest situation θ ∈ E − , with θE w = a wk ζ k ( w ) /ζ (2 w ) ,definitive proof of presence or absence of discrete spectrum for the operator e S θ seems difficult. Apart from numerical tests, which suggest that there is no discretespectrum, some clarification can be achieved by a subtler application of operator Designed pseudo-Laplacianstheory and distribution theory, as follows. By direct computation, the constant termof a solution u w to the distributional equation ( − ∆ − λ w ) u = θ vanishes at height y ≫ θ . Thus, such u w lies inside the corresponding Lax-Phillips space L a (Γ \ H ) (as above, and as in section 4.2), and can be expanded in terms of the (exotic) eigen-functions for e S Θ (mostly certain truncated Eisenstein series). Expanding (the imageof) θ in terms of those eigenfunctions, we find (see subsection 7.3) an interleavingproperty : there is at most one spectral parameter w for an eigenvalue of e S θ betweenany two adjacent zeros s of a s + c s a − s . Arguing as in [1], the average verticalspacing of zeros of ζ k ( s ) (on the critical line or not) at height T is π/ log T , thesame as that of the spacing of zeros of a s + c s a − s , which bodes well. However,from [27] (5.17.4) page 112 (in an earlier edition, page 98), for log log T large, theargument of ζ ( s ) on the edge of the critical strip is relatively regular, so eventuallythe spacing of the zeros of a s + c s a − s is similarly regular. That is, given ε > , for log log T sufficiently large, the space between consecutive zeros of a s + c s a − s isbetween (1 − ε ) π/ log T and (1 + ε ) π/ log T . Adjusting a slightly, the interleavingproperty shows that, given ε > , for log log T sufficiently large the space betweenconsecutive zeros of ζ Q ( ω ) ( s ) on the critical line is at least (1 − ε ) π/ log T . Thiswould be in conflict with the pair correlation conjecture [21]: for example, underthe Riemann Hypothesis and assuming pair correlation, such a lower bound wouldallow at most 94% of zeros s of ζ ( s ) to appear as parameters for eigenvalues λ s (see corollary 69).Corollary 76 proves an illustrative positive result about spacing of on-the-linezeros of ζ k ( w ) , without any assumptions about point spectrum of self-adjoint op-erators. Namely, let t < t ′ be large, and such that + it and + it ′ are adjacenton-line zeros of ζ k ( w ) . Take θ ∈ E − such that θE w = a wk ζ k ( w ) /ζ (2 w ) . Supposethat neither + it nor + it ′ is a zero of J θ,w = h k ( λ − λ w ) · h , i + 14 πi Z ( ) | θE s | − | θE w | λ s − λ w d s Suppose there is a unique zero + iτ o of J θ, + iτ between + it and + it ′ , and ∂∂τ J θ, + iτ > . Then | t ′ − t | ≥ π log t · (1 + O ( t )) . That is, in this configuration,the distance between consecutive zeros of ζ k ( w ) must be at least the average.Analogous discussions with similar proofs apply to a broad class of self-adjointoperators on spaces of automorphic forms.ombieri and Garrett 9 This section deals with the construction and properties of self-adjoint Friedrichsextensions of operators on a complex Hilbert space.
Consider complex Hilbert spaces V with inner product h , i and required to have acomplex-conjugate linear conjugation map v → v , with expected properties: v = v, α · v = α · v ( α ∈ C ) , h v, w i = h w, v i . Spaces of L -complex-valued functions on measure spaces, for example, have nat-ural conjugations given simply by pointwise conjugation of functions.Let S be an unbounded, symmetric, densely-defined operator on V with domain D dense in V . Assume S is semi-bounded , specifically, that inf x ∈ D, h x,x i =1 h Sx, x i > c > . Suppose D is stabilized by conjugation and that S commutes with the conjugation v → v . For x, y ∈ D let us write h x, y i = h Sx, y i and let i : D −→ V denotethe completion of D with respect to the new inner product h x, y i . The space V has a canonical continuous linear map j : V → V extending by continuitythe identity map D → D , because h x, x i > c h x, x i for x ∈ D . In fact, j isinjective: h w, iv i = h jw, T v i for w ∈ V and v ∈ D , so jw = 0 implies that w is orthogonal to the image of D in V , which is dense. Whenever possible wesuppress the inclusions i and j from the notation. We write inc = j ◦ i , in acommutative diagram V VD ji inc Now assume S is genuinely unbounded, so that V = V . Recall that the Friedrichs extension ( S, D ) e of the pair ( S, D ) is a new self-adjoint operator e S : V −→ V with a new domain e D −→ V , extending S in the sense that there0 Designed pseudo-Laplaciansis a diagram for the composition inc = j ◦ i , in which (only) the outer curvilineartriangle commutes: e D V VD ( S,D ) e inc j( S,D ) i For simplicity, usually we shall write only e S without mention of the domain e D ,but the domain e D is part of the description of the Friedrichs extension.The Friedrichs extension is characterized by its inverse e S − being an everywheredefined, continuous, self-adjoint operator e S − : V → V , with the h , i topologyon V , with the property h x, e S − y i = h jx, y i ( x ∈ V , y ∈ V ) with j the embedding j : V → V defined before. Thus e S is self-adjoint, e S > S ,and inf x ∈ D, h x,x i =1 h Sx, x i inf x ∈ e D, h x,x i =1 h e Sx, x i . When e S = S the spectra of e S and S can be different. If the spectrum of S isdiscrete and ( λ ν ) is the associate sequence of eigenvalues, and similarly for e S , wehave λ ν e λ ν for all ν . Let V − be the complex-linear dual of V , with norm | µ | − := sup x ∈ V , h x,x i | µ ( x ) | . Since V is a Hilbert space, the norm | · | − gives an inner product h , i − bypolarization, and V − is a Hilbert space. Using the conjugation map on V , let Λ : V → V ∗ be the complex-linear isomorphism of V with its complex-linear dualby means of Λ( x )( y ) = h y, x i = h x, y i .ombieri and Garrett 11The inclusion j : V → V dualizes to j ∗ : V ∗ → V ∗ = V − by means of ( j ∗ µ )( x ) = µ ( jx ) for µ ∈ V ∗ and x ∈ V . Thus we have V V V ∗ V − j Λ ee j ∗ Conjugation acts on V − by λ ( x ) = λ ( x ) .Define a continuous linear S : V −→ V − , with h , i and h , i − topologies,respectively, by S ( x )( y ) = h x, y i ( x, y ∈ V ) . By the Riesz-Fr´echet theorem, S is a topological isomorphism. Proposition 1.
The restriction of S to the domain of e S is j ∗ ◦ Λ ◦ e S . The domainof e S is e D = { x ∈ V : S x ∈ ( j ∗ ◦ Λ) V } .Proof. By construction of the Friedrichs extension, its domain is e D = e S − V .Thus, for x = e S − x ′ with x ′ ∈ V and all y ∈ V we have ( S x )( y ) = ( S e S − x ′ )( y ) = h e S − x ′ , y i − = h x, y i = (( j ∗ ◦ Λ) x ′ )( y ) = (( j ∗ ◦ Λ ◦ e S ) x )( y ) . This shows that S e D = ( j ∗ ◦ Λ ◦ e S ) e D .On the other hand, for S x = ( j ∗ ◦ Λ) y with y ∈ V we have, for all z ∈ V : h z, x i = ( S x )( z ) = (( j ∗ ◦ Λ) y )( z ) = (Λ y )( jz ) = ( jz, y ) = h z, e S − y i . Therefore, x = e S − y , proving the second statement of the proposition. Q.E.D.
Let Θ be a finite-dimensional subspace of V − with Θ ∩ ( j ∗ ◦ Λ) V = { } . (2.1)Since Θ consists of linear functionals on V continuous in the h , i -topology, thesimultaneous kernel ker Θ is a closed subspace of V . Lemma 2. D ∩ ker Θ is dense in V . Proof.
This follows from the general fact that for a continuous inclusion of Hilbertspaces j : X −→ Y , for D ⊂ X dense in Y , and for a finite-dimensional subspace Θ ⊂ X ∗ such that j ∗ ( Y ∗ ) ∩ Θ = { } , we have that D ∩ ker Θ ⊂ X is dense in Y .For completeness, we recall the simple proof. Consider first Θ of dimension , spanned by θ . Since θ / ∈ j ∗ ( Y ∗ ) , θ cannot be continuous in the Y -topologyon dense D . This provides for each ε > an element x ε ∈ D with | x ε | Y < ε and θ ( x ε ) = 1 . Given y ∈ Y , density of D in Y yields a sequence z n in D approaching y in the Y -topology. If θ ( z n ) = 0 for infinitely many n , there isnothing to prove. Otherwise, the sequence z ′ n = z n − θ ( z n ) · x − n /θ ( z n ) is in ker Θ and still z ′ n → y in the Y -topology, because | θ ( z n ) · x − n /θ ( z n ) | Y < | θ ( z n ) | · − n | θ ( z n ) | = 2 − n → . Induction on dimension completes the proof.
Q.E.D.
Let S Θ be S restricted to the smaller domain D Θ := D ∩ ker Θ and let ( S Θ , D Θ ) e be the Friedrichs extension associated to ( S Θ , D Θ ) , with domain e D Θ , which is in-deed dense, by the preceding Lemma 2. Our next goal is the analogue of Lemma 1for the domain e D Θ . In order to do this, we need some preparatory observations.The extension ( S Θ ) : V ∩ ker Θ −→ ( V ∩ ker Θ) ∗ is defined in the same way as S , by ( S Θ ) ( x )( y ) = h x, y i ( x, y ∈ V ∩ ker Θ) . Let t Θ : V ∩ ker Θ −→ V , t ∗ Θ : ( V ) ∗ −→ ( V ∩ ker Θ) ∗ be the inclusion and its adjoint. Assume that Θ is stable under the extension of theconjugation map to V − . Lemma 3. ( S Θ ) = t ∗ Θ ◦ S ◦ t Θ . Proof.
Lemma 2 shows that D Θ is dense in V in the V -topology, so formationof the Friedrichs extension as an unbounded self-adjoint operator (densely defined)on V is legitimate. For x, y ∈ V ∩ ker Θ we have ( t ∗ Θ ◦ S ◦ t Θ )( x )( y ) = S ( x )( y ) = h t Θ x, t Θ y i = h x, y i = ( S Θ ) ( x )( y ) , ombieri and Garrett 13which is the statement of the lemma. Q.E.D.
The following re-characterization of the Friedrichs extension of the restriction isstraightforward, but essential.
Theorem 4.
The domain e D Θ of e S Θ is e D Θ = { x ∈ V ∩ ker Θ : ( S ◦ t Θ ) x ∈ ( j ∗ ◦ Λ) V + Θ } . = { x ∈ V ∩ ker Θ : S x ∈ ( t ∗ Θ ◦ j ∗ ◦ Λ) V } . We have e S Θ x = y , with x ∈ V ∩ ker Θ and y ∈ V , if and only if ( S ◦ t Θ ) x = ( j ∗ ◦ Λ) y + θ for some θ ∈ Θ .Proof. The Friedrichs extension e S Θ is characterized by h z, ( f S Θ ) − y i = h z, y i ( z ∈ D Θ , y ∈ V ) . Given S x = ( j ∗ ◦ Λ) y + θ with x ∈ V ∩ ker Θ , y ∈ V , and θ ∈ Θ , take z ∈ D Θ and compute h x, z i = ( S x )( z ) = (( j ∗ ◦ Λ) y + θ )( z ) = ( j ∗ y )( z ) + θ ( z ) = h z, y i + 0= h y, f S Θ − S z i = h f S Θ − y, S z i = h f S Θ − y, z i , thus showing that e S − x = y . On the other hand, by Lemma 3, ( S Θ ) x = y if andonly if ( S ◦ t Θ ) x = y + θ for some θ ∈ ker t ∗ Θ , and ker t ∗ Θ is the closure of Θ in V − . Since Θ is finite-dimensional, this closure is Θ itself.The second description of the domain of the Friedrichs extension is immediatefrom the previous lemma and from the fact that Θ is the kernel of t ∗ Θ . Q.E.D.
Corollary 5.
With the hypotheses of the theorem, for x ∈ V with ( S − λ ) x = θ with θ ∈ Θ , if x ∈ ker Θ then x is a λ -eigenfunction for the self-adjoint operator e S Θ > S > c > . In that case, λ is real and λ > c > . If S is merelynon-negative the same conclusion holds, except for the weaker inequality λ > .Proof. The first part of the corollary is immediate. For the second part, the condition S > c > imposed on S at the beginning of Subsection 2.1 can be removed,replacing it by non-negativity, by applying the first conclusion of the corollary tothe operator S + c and noting that the new eigenvalues are obtained by making ashift by c . Q.E.D. S withdense domain D on Hilbert space V with conjugation v → v , and h Sv, v i ≥ h v, v i for all v ∈ D . Let Θ be an S -stable, conjugation-stable, not necessarily finite-dimensional subspace of D . Let V Θ be the orthogonal complement of Θ in V ,with respect to the hermitian inner product on V . Let D Θ = D ∩ V Θ . Let Θ − bethe closure of Θ in V − in the V − topology. The relevance of the S -stability of Θ is as expected, namely, that S restricted to D Θ really does map to V Θ : Lemma 6. S ( D Θ ) ⊂ V Θ .Proof. For v ∈ D Θ and θ ∈ Θ ⊂ D , h Sv, θ i = h v, Sθ i ∈ {h v, θ ′ i : θ ′ ∈ Θ } = { } giving the indicated inclusion. Q.E.D.
Unlike the previous situation, we must assume that D Θ is dense in V Θ , anddense in V ∩ V Θ in the V topology. We prove the requisite density for cases ofinterest to us in Lemma 20. Let ( V ) Θ be the closure of D Θ in V with respect tothe V topology. Let S Θ be the restriction of S to D Θ , and S : V → V − by ( S v )( w ) = h v, w i . Let t Θ : V Θ1 → V be the inclusion, and t ∗ Θ : V − → ( V Θ1 ) ∗ the adjoint. Let e S Θ be the Friedrichs extension of S Θ . Let ( S Θ ) : V Θ1 → ( V Θ1 ) ∗ by (( S Θ ) v )( w ) = h v, w i for v, w ∈ V Θ1 . The present analogue of Lemma 3 is Lemma 7. ( S Θ ) = t ∗ Θ ◦ S ◦ t Θ .Proof. Identical to that of Lemma 3.
Q.E.D.
The analogue of Theorem 4 has a nearly identical form:
Theorem 8.
The domain of e S Θ is e D Θ = { v ∈ V Θ1 : ( S ◦ t Θ ) v ∈ ( j ∗ ◦ Λ) V Θ + Θ − } = { v ∈ V Θ1 : ( S Θ ) v ∈ ( t ∗ Θ ◦ j ∗ ◦ Λ) V Θ } We have e S Θ x = y for x ∈ e D Θ and y ∈ ( j ∗ ◦ Λ) V Θ if and only if ( S ◦ t Θ ) x = y + θ for some θ ∈ Θ − .Proof. The argument is essentially identical to that of Theorem 4. While Θ gives(continuous) functionals on V , via the hermitian inner product on V , the closureombieri and Garrett 15 Θ − in V − gives (continuous) linear functionals on V , via duality. Because thepairing V × V − → C extends the restriction to D × D of the V × V → C pairing, ker Θ − = V ∩ V Θ . By assumption, V Θ = V ∩ V Θ .As usual, the extension e S Θ is characterized by h z, ( f S Θ ) − y i = h z, y i for z ∈ D Θ and y ∈ V Θ . Given S x = y + θ with x ∈ V Θ1 , y ∈ V Θ , and θ ∈ Θ − , take z ∈ D Θ and compute h x, z i = ( S x )( z ) = (( j ∗ ◦ Λ) y + θ )( z ) = ( j ∗ y )( z ) + θ ( z ) = h z, y i + 0= h y, f S Θ − S z i = h f S Θ − y, S z i = h f S Θ − y, z i , Thus, e S − x = y . On the other hand, by Lemma 3, ( S Θ ) x = y if and only if ( S ◦ t Θ ) x = y + θ for some θ ∈ ker t ∗ Θ , and ker t ∗ Θ is the closure Θ − of Θ in V − . The second description of the domain of the Friedrichs extension is immediatefrom the previous lemma and from the fact that Θ is the kernel of t ∗ Θ . Q.E.D.
Let ∆ be the hyperbolic Laplacian ∆ = y (cid:16) ∂ ∂x + ∂ ∂y (cid:17) (3.1)on the upper half-plane H , Let Γ = SL ( Z ) . The standard inner product is h f, g i = Z Γ \ H f ( z ) g ( z ) d ω (3.2)(the Petersson inner product) with respect to the hyperbolic area element d ω z = y − d x d y . The hyperbolic area of a fundamental domain of Γ \ H is h , i = π/ .Let S be − ∆ restricted to have domain E ∞ c = n Ψ ϕ ( z ) = X γ ∈ Γ ∞ \ Γ ϕ ( ℑ ( γz )) : ϕ ∈ C ∞ c (0 , ∞ ) o . (3.3)These Ψ ϕ are pseudo-Eisenstein series (the incomplete theta series of other au-thors) with test-function data ϕ . The conjugation map f → f on E ∞ c is the ex-pected pointwise conjugation. It commutes with ∆ and S , and E ∞ c is stable by6 Designed pseudo-Laplaciansconjugation. The ambient Hilbert space is the L (Γ \ H ) completion E of E ∞ c .The operator S is a non-negative operator, so the previous discussion of Friedrichsextensions applies to S , for example, and thereby to S itself, as unboundedoperator on E , with domain E ∞ c . Let Γ ∞ be the stabilizer of i ∞ . The Eisenstein series associated to the cusp at i ∞ and z ∈ Γ \ H is explicitly given when ℜ ( s ) > by E s ( z ) := X γ ∈ Γ ∞ \ Γ ℑ ( γz ) s = 12 X ( c,d )=1 ′ y s | cz + d | s (3.4)and by analytic continuation for general s , where the sum is over coprime integerpairs c, d , and the pair , is also excluded.The Eisenstein series are functions of the two complex variables z and s , au-tomorphic in z , while s is the spectral parameter . The family of functions z → E s ( z ) of z form a meromorphic automorphic-function-valued function of s , witha simple pole at s = 1 and infinitely many poles for ℜ ( s ) < . For s not a pole, z → E s ( z ) is a real-analytic function of the variable z ∈ Γ \ H .By the SL ( R ) -invariance of the Laplacian, ( − ∆ − λ s ) E s = 0 , where λ s = s (1 − s ) . The Eisenstein series satisfy the functional equation E s = c s E − s with c s = √ π Γ( s − ) ζ (2 s − s ) ζ (2 s ) = ξ (2 − s ) ξ (2 s ) , (3.5)where ξ ( s ) is the completed Riemann zeta function and we have used the functionalequation of the zeta function. This also yields c s c − s = 1 .Lastly, the residue of the simple pole of the Eisenstein series E s at s = 1 is theconstant / h , i = 3 /π . It is a well known yet remarkable fact that, for Γ an arithmetic group, values ofcertain Eisenstein series for Γ \ H have arithmetical significance. Recall that a fun-damental discriminant is a product of relatively prime factors of the form − , , − , ( − ( p − / p, ombieri and Garrett 17where p is an odd prime. Associated to a fundamental discriminant d there is areal, primitive character χ d ( n ) = (cid:18) dn (cid:19) where ( d/n ) is the Kronecker symbol, which enjoys the multiplicativity χ d ( n ) χ d ′ ( n ) = χ dd ′ ( n ) for d and d ′ coprime . The absolute value | d | is the modulus of the character. The fundamental discrimi-nants are all numbers, positive or negative, of the form N with N square-free and N ≡ or of the form N with N ≡ or .From now on d will denote a negative fundamental discriminant. The integral,positive-definite, Lagrange-reduced quadratic forms of discriminant d are Q ( x, y ) := Ax + Bxy + Cy with d = B − AC, | B | ≤ A ≤ C, ( and when A = | B | = 1 then B = − ) . The root z Q = − B + i p | d | A ∈ Γ \ H of the equation Az + Bz + C = 0 is the Heegner point associated to that reducedquadratic form. We have y − Q · | mz Q + n | = (cid:18) p | d | (cid:19) − Q ( n, − m ) . From this, one shows that for discriminants d < − the value E s ( z Q ) of the Eisen-stein series is E s ( z Q ) = (cid:18) p | d | (cid:19) s ζ (2 s ) − ζ ( s, z Q ) (3.6) In this case the two quadratic forms x ± xy + | d | +14 y are equivalent by ( x, y ) → ( x, x ∓ y ) and here we choose the one with B = − as a representative, the so-called ambiguous form . z Q is the ideal class of the fractional ideal [1 , z Q ] of the imaginary quadraticfield Q ( √ d ) . Therefore, summing over the h ( d ) ideal classes we have h ( d ) X i =1 E s ( z Q i ) = (cid:18) p | d | (cid:19) s ζ (2 s ) − ζ ( s, Q ( √ d )) = (cid:18) p | d | (cid:19) s ζ ( s ) ζ (2 s ) L ( s, χ d ) (3.7)where z Q i runs over the h ( d ) Heegner points for the fundamental discriminant d and where χ d is the quadratic character associated to Q ( √ d ) . The spectral theory of the Laplacian on L (Γ \ H ) , where Γ is a discrete sub-group of SL ( R ) acting on the upper half-plane H and of finite covolume, iswell understood: see for example [6], or Iwaniec’ monograph [18]. This decom-poses L (Γ \ H ) as the direct orthogonal sum of the L cuspidal discrete spec-trum, constants ( L residues of Eisenstein series), and eigen-packets associatedto a continuous spectrum generated by the Eisenstein series. Here we only consider Γ = SL ( Z ) , which has just one family of Eisenstein series giving the continu-ous spectrum, attached to the single cusp i ∞ . For f ∈ C ∞ c (Γ \ H ) , the spectralexpansion is f ( z ) = X F h f, F i · F ( z ) + h f, i · h , i + 14 πi Z ( ) h f, E s i · E s ( z ) ds where F runs over an orthonormal basis for L cuspforms. As usual, h f, E s i cannot be the L pairing, because E s is not in L , but by standard abuse of no-tation refers to the integral R Γ \ H f ( z ) E − s ( z ) d ω z , which converges absolutely forautomorphic test functions f . For test functions f , the right-hand side convergesuniformly pointwise in z . We have Plancherel for test functions: | f | L = X F |h f, F i| + |h f, i| h , i + 14 πi Z ( ) |h f, E s i| ds If the discriminant is − or − the associated quadratic forms Q ( x, y ) are the ambiguousform x − xy + y and x + y . Besides the obvious automorphism ( x, y ) → ( − x, − y ) arisingfrom − I ∈ Γ ∞ they have the automorphisms ( x, y ) → ( x − y, x ) and ( x, y ) → ( y, − x ) of order and , so equations (3.6) and (3.7) must be corrected by factors and in the right-hand side. ombieri and Garrett 19Extend the spectral expansion to f ∈ L (Γ \ H ) by isometry, and write E f forthe extension of f → ( s → h f, E s i ) . Pseudo-Eisenstein series f ∈ E ∞ c withtest function data are orthogonal to cuspforms, so for such automorphic forms thespectral expansion and Plancherel become f = h f, i · h , i + 14 πi Z ( ) E f ( s ) · E s ds and | f | L = |h f, i| h , i + 14 πi Z ( ) |E f ( s ) | ds The operator S , − ∆ restricted to E ∞ c , is symmetric because the compact supportof pseudo-Eisenstein series with test-function data allows integration by parts. For f ∈ E ∞ c we have the spectral relation , intertwining of S and multiplication by λ s , E ( Sf )( s ) = Z Γ \ H ( − ∆) f ( z ) · E − s ( z ) d ω z = Z Γ \ H f ( z ) · ( − ∆) E − s ( z ) d ω z = Z Γ \ H f ( z ) · λ s E − s ( z ) d ω z = λ s · E f ( s ) . Thus, for all ℓ ∈ Z , E ( S ℓ f )( s ) = λ ℓs · E f ( s ) ( for f ∈ D ) . (3.8) E r For r ∈ R , the r -th global automorphic Sobolev space H r (Γ \ H ) is the completionof C ∞ c (Γ \ H ) with respect to the r -th Sobolev norm, defined on automorphic testfunctions f by | f | H r = X F |h f, F i| · (1 + λ s F ) r + |h f, i| · (1 + λ ) r h , i + 14 πi Z ( ) |h f, E s i| · (1 + λ s ) r ds r -th Eisenstein-Sobolev space is E r = completion of E ∞ c with respect to | · | H r Let X r = C ⊕ X ro where X ro is the weighted L -space of measurable functions g on + i R such that Z ∞−∞ | g ( 12 + it ) | · ( 14 + t ) r dt < + ∞ For all r ∈ R , Plancherel restricts (for r ≥ ) or extends (for r ≤ ) to an isometry E r → X r , by f → h f, ih , i ⊕ E f . Let E ∞ = \ r E r = lim r E r and E −∞ = [ r E r = colim r E r in the category of locally convex topological spaces. There is the expected hermi-tian pairing, h f, θ i E r × E − r = h f, i · h θ, ih , i + 14 πi Z ( ) E f ( s ) · E θ ( s ) d s For all r ∈ R the map S : E ∞ c → E ∞ c is continuous when the domain is giventhe E r topology and the range is given the E r − topology. Extending by continuitydefines L -differentiation E r → E r − . The pre-trace formula (as in [18] and elsewhere) is X F : | λ ( F ) | T | F ( z ) | + |h F, i| h , i + 14 πi Z ( ) | E s ( z ) | d s ≪ C T for z in a fixed compact subset C ⊂ Γ \ H , where F runs over an orthonormalbasis for the cuspidal spectrum and SF = λ ( F ) F . In particular, dropping thecuspidal part we have |h F, i| h , i + 14 πi Z ( ) | E s ( z ) | d s ≪ C T ombieri and Garrett 21Integrating by parts, the function s → E s ( z ) is in the weighted L space X − − ε ,so E s ( z ) = E θ ( s ) for some θ ∈ E − − ε , for all ε > , and θ = h θ, i · h , i + 14 πi Z ( ) E − s ( z ) · E s d s (as an element of E − − ε )Define the non-cuspidal Dirac δ distribution, or Eisenstein-Dirac δ distribution, by δ nc z := 1 h , i + 14 πi Z ( ) E − s ( z ) · E s d s (as an element of E − − ε ) (3.9)That is, δ n cz o ( s ) = E − s ( z o ) . By design, its action on f ∈ E ε is δ nc z f = f ( z ) .From evaluating the hermitian pairing on E ε × E − − ε : h f, δ nc z o i E ε × E − − ε = h f, i · h δ nc z o , ih , i + 14 πi Z ( ) E f ( s ) · E δ nc z o ( s ) d s = h f, i · h , i + 14 πi Z ( ) E f ( s ) · E s ( z o ) d s At least for f a test-function pseudo-Eisenstein series, this is f ( z o ) . The estimateis uniform for z in compact subsets of Γ \ H , so the map z → δ nc z is a continuous E − − ε -valued function of z . Thus, for test-function pseudo-Eisenstein series f ,by the Cauchy-Schwarz-Bunyakowsky inequality, sup z o ∈ C | f ( z o ) | = sup z o ∈ C |h f, δ nc z o i| ≤ sup z o ∈ C | f | E ε · | δ nc z o | E − − ε = | f | E ε · sup z o ∈ C | δ nc z o | E − − ε ≪ C,ε | f | E ε That is, the seminorms obtained by taking suprema on compacta are dominated bythe E ε norm. This proves the Sobolev embedding E ε ⊂ C (Γ \ H ) , with thelatter topologized by suprema on compact subsets. Thus, δ nc z o ( f ) = f ( z o ) for all f ∈ E ε .As a corollary of the above argument, we again see the expected E δ nc z o ( s ) = E − s ( z o ) . For a fundamental discriminant d < − , let H d be the set of Heegner points in Γ \ H representing the ideal classes of the ring of integers Q ( √ d ) , and let θ d be the2 Designed pseudo-Laplaciansfunctional θ d = X z ∈ H d δ nc z ∈ E − − ε (3.10)for all ε > . We call this the Eisenstein-Heegner distribution attached to thefundamental discriminant d . The cardinality h ( d ) = | H d | is the class number of thequadratic field Q ( √ d ) . Let χ d be the quadratic character attached to Q ( √ d ) / Q .This is a primitive character because d is a fundamental discriminant. For d < − we have E θ d ( s ) = (cid:18) p | d | (cid:19) s ζ ( s ) ζ (2 s ) L ( s, χ d ) . (3.11)The Eisenstein-Heegner distributions θ d belong to the space E − − ε for any ε > , from the Landau bound /ζ (1 + it ) = O (log T ) and the deeper second momentbound for ζ ( s ) L ( s, χ ) on the critical line ℜ ( s ) = . The E − − ε spectral expansionis (for d = − or − ): θ d = θ d (1) · h , i + 14 πi Z ( ) θ d E − s · E s d s = h ( d ) h , i + 14 πi Z ( ) (cid:18) p | d | (cid:19) − s ζ (1 − s ) L (1 − s, χ d ) ζ (2 − s ) · E s d s, ( − ∆ − λ w ) u = θ Let θ be a finite real-linear combination of Eisenstein-Heegner distributions θ d .For ℜ ( w ) > , the equation ( − ∆ − λ w ) u = θ has a unique solution u θ,w in E − ε for every ε > , with spectral expansion obtained directly from that of θ via (3.8): u θ,w = θ (1) · h , i · ( λ − λ w ) + 14 πi Z ( ) θE − s · E s d sλ s − λ w . (3.12) η a Let η a denote the constant term distribution at height a > on f ∈ E ∞ c , namely: η a f = Z f ( x + ia ) d x. (3.13)ombieri and Garrett 23This functional is a compactly-supported, real-valued, regular, Borel measure on Γ \ H , so is a continuous functional on C (Γ \ H ) . By the remark at the end of 3.5,there is a continuous injection E ε ⊂ C (Γ \ H ) for every ε > , so η a restricts toa continuous functional on E ε , still denoted η a . Thus, η a ∈ E − − ε for all ε > .As with automorphic Dirac δ and Eisenstein-Dirac δ , we remove a potentialambiguity about correct determination of spectral coefficients E η a . We could againuse a variant pre-trace formula, but, instead, we give an argument relevant to sub-sequent developments: Proposition 9.
A continuous functional µ on the Fr´echet space C (Γ \ H ) given bya compactly-supported real-valued regular Borel measure on Γ \ H , restricted to afunctional on E ∞ , is in E − − ε for every ε > , and E µ ( s ) = µ ( E − s ) .Proof. Fix ε > . We have a Sobolev imbedding H ε (Γ \ H ) ⊂ C o (Γ \ H ) forevery ε > . The continuous dual of C o (Γ \ H ) is exactly compactly-supportedregular Borel measures µ . Thus, µ has a natural image in E − − ε = ( E ε ) ∗ , since E ε ⊂ H ε (Γ \ H ) . Thus, there is a spectral expansion in E − − ε : µ = h µ, i · h , i + 14 πi Z ( ) E µ ( s ) · E s d s For any u ∈ E ε , on one hand, using µ = µ , µ ( u ) = h u, µ i E ε × E − − ε = h u, i · h µ, ih , i + 14 πi Z ( ) E u ( s ) · E µ ( s ) d s On the other hand, the spectral integral for u converges in E ε , and is the limit ofcompactly supported integrals of C (Γ \ H ) -valued functions, so µ ( u ) = µ (cid:16) h u, i · h , i + 14 πi Z ( ) E u ( s ) · E s d s (cid:17) = µ (cid:16) lim T →∞ h u, i · h , i + 14 πi Z |ℑ ( s ) |≤ T E u ( s ) · E s d s (cid:17) = lim T →∞ h u, i · µ (1) h , i + 14 πi Z |ℑ ( s ) |≤ T E u ( s ) · µE s d s = h u, i · µ (1) h , i + 14 πi Z ( ) E u ( s ) · µE s d s µ passes inside the compactly-supported C (Γ \ H ) -valued integral of the continuous C (Γ \ H ) -valued integrand s → E u ( s ) · E s . (Suchstandard properties of Gelfand-Pettis vector-valued integrals are recalled in section6.1.) Since µ = µ , we have µ (1) = h µ, i , and µE s = µE − s for ℜ ( s ) = . Thetwo expressions for u agree for all u ∈ E ε , giving the proposition. Q.E.D.
In fact, η a ∈ E − − ε for all ε > , because the | · | − − ε norm is | η a | − − ε = 1 h , i + 14 π Z ∞−∞ | a s + c s a − s | (1 + 4 t ) + ε d t ≪ a Z ∞−∞ d t (1 + 4 t ) + ε < ∞ . Thus, the E − − ε spectral expansion convergent in E − − ε is η a = 1 h , i + 14 πi Z ( ) ( a − s + c − s a s ) E s d s (3.14) ( − ∆ − λ w ) u = η a For ℜ ( w ) > the equation ( − ∆ − λ w ) u = η a has an unique solution v w,a ∈ E − ε for all ε > , with spectral expansion obtained directly from that of η a via (3.8) v w,a = 1 h , i + 14 πi Z ( ) ( a − s + c − s a s ) · E s d sλ s − λ w . (3.15) For any θ ∈ E − with θ = θ and θ E , let S θ be − ∆ restricted to the domain E ∞ c ∩ ker θ . Symmetry of S θ is inherited from − ∆ . The pseudo-Laplacian e S θ isthe Friedrichs extension of S θ .As above, for θ ∈ E − , the distributional equation ( − ∆ − λ ) u = θ has a uniquesolution u ∈ E for all λ not in { } ∪ [ , + ∞ ) , via spectral expansions and (3.8).ombieri and Garrett 25 Theorem 10.
For real λ w > , if the equation ( − ∆ − λ w ) u = θ has a solutionin E then E θ ( w ) = 0 . More precisely, existence of a solution implies that, for all ε > , Z ℑ ( w )+ ε ℑ ( w ) − ε (cid:12)(cid:12)(cid:12) E θ (cid:16)
12 + it (cid:17)(cid:12)(cid:12)(cid:12) d t ≪ w,ε ε Proof.
The solution u has spectral expansion u = h u, i · h , i + 14 πi Z ( ) E u ( s ) · E s d s ( in E ) and ( − ∆ − λ w ) u = ( λ − λ w ) h u, i · h , i + 14 πi Z ( ) E u ( s ) · ( λ s − λ w ) · E s d s ( in E − ) Since θ itself has a spectral expansion in E − , by (3.8) necessarily E θ ( s ) = E u ( s ) · ( λ s − λ w ) ( at least as locally- L functions on + i R ) Further, from the Cauchy-Schwarz-Bunyakowsky inequality, Z v + εv − ε (cid:12)(cid:12)(cid:12) E θ (cid:16)
12 + it (cid:17)(cid:12)(cid:12)(cid:12) d t ≤ Z v + εv − ε (cid:12)(cid:12)(cid:12) E u (cid:16)
12 + it (cid:17)(cid:12)(cid:12)(cid:12) · | t − v | d t ≤ (cid:18)Z v + εv − ε (cid:12)(cid:12)(cid:12) E u (cid:16)
12 + it (cid:17)(cid:12)(cid:12)(cid:12) d t (cid:19) · (cid:18)Z v + εv − ε (cid:12)(cid:12) t − v (cid:12)(cid:12) d t (cid:19) ≪ w,ε kE u k · ε as asserted. Q.E.D.
As a corollary, we have a necessary , but in general not sufficient , constraint onpossible discrete spectrum of e S θ : Corollary 11.
The discrete spectrum λ w > of e S θ , if any , is of the form w (1 − w ) for w ∈ + i R such that E θ ( w ) = 0 , in the sense of the previous theorem.Proof. From Theorem 4, any solution u ∈ e D θ to ( e S θ − λ w ) u = 0 is a solutionto a distributional equation ( − ∆ − λ w ) u = c · θ for some constant c . For u notidentically and λ w = 0 , the constant c cannot be , since ( − ∆ − λ w ) u = 0 has no non-zero solution in E for λ w = 0 . Thus, without loss of generality, take c = 1 , and apply the theorem. Q.E.D.
Remark 12.
For θ ∈ E − ε for some ε > , theorem 48 will show that on ℜ ( w ) = with w = , E θ ( w ) = 0 is also sufficient for existence of a solution u ∈ E to the distributional equation ( − ∆ − λ w ) u = θ . However, such u is not in thedomain of the self-adjoint operator e S θ unless also θu = 0 , which does not followfrom E θ ( w ) = 0 in general. Remark 13.
The corollary gives a definite relation between the spectrum of a nat-ural self-adjoint operator and the zeros of E θ ( s ) , which, as in section 3.6, in manyinteresting cases an L -function or a finite linear combination of such. However,there appears to be no general assurance of existence of any discrete spectrumwhatsoever. Remark 14.
The larger point of our discussion of self-adjoint operators is to provethat various quantities do not vanish in ℜ ( w ) > (off the real line). Unsurpris-ingly, some non-vanishings are more trivial than might be anticipated. For example,for any θ ∈ E − , with u θ,w ∈ E a solution of ( − ∆ − λ w ) u = θ , by spectral theory θu θ,w = h u, i · h θ, i λ − λ w + 14 πi Z ( ) E θ ( s ) · E s λ s − λ w d s For ℜ ( w ) > and w off ( , , it is elementary that the imaginary part of thatexpression is non-zero. That is, θu θ,w = 0 off the critical line and the real line. Thatis, although facts about self-adjoint operators do yield these particular conclusions,some of these conclusions are elementary. Here we consider families of restrictions of − ∆ similar to [20], pages 204–206,with attention to details. For fixed a > , let Θ ⊂ L (Γ \ H ) be the space of pseudo-Eisenstein series Ψ ϕ with test function ϕ supported on [ a, ∞ ) . Since ∆Ψ ϕ = Ψ ∆ ϕ the space Θ is stable under ∆ . Let E be the orthogonal complement to Θ in E .Let S Θ be the restriction of − ∆ to E ∞ c ∩ E , and e S Θ its Friedrichs extension. Toavoid potential ambiguities, we should be sure that S Θ is densely defined on E : Lemma 15. E ∞ c ∩ E is dense in E , and E ∞ c ∩ E is dense in E ∩ E with the E topology.Proof. To show that E ∞ c ∩ E is dense in E , given a sequence of pseudo-Eisensteinseries Ψ ϕ i ∈ E ∞ c converging to f ∈ E , we produce a sequence of pseudo-Eisenstein series in E ∞ c ∩ E converging to f . We will do so by cutting off theombieri and Garrett 27constant terms of the Ψ ϕ i at height a . Since the limit f of the Ψ ϕ i has constantterm vanishing above height y = a , that part of the constant terms of the Ψ ϕ i mustbecome small. The explicit details are routine. Q.E.D.
Essentially as in [20] but restricting to the orthogonal complement E to cusp-forms, we have Theorem 16.
The resolvent ( e S Θ − λ w ) − is compact for λ w R , and has a mero-morphic continuation to w ∈ C , giving a compact operator for w off a discretesubset of ( + i R ) ∪ [0 , . In particular, e S Θ has purely discrete spectrum.Proof. Since (1 + e S Θ ) − : E → E is continuous with the (finer) E -topology on E , we see below that it suffices to demonstrate a Rellich-lemma-type compactness,namely, that the inclusion E → E is a compact linear map. The correspondingcompactness for compact Riemannian manifolds, possibly with boundary, is stan-dard.The total boundedness criterion for relative compactness requires that, given ε > , the image of the unit ball B ⊂ E by the inclusion into E can be coveredby finitely-many balls of radius ε . The Rellich lemma on compact Riemannianmanifolds reduces the issue to an estimate on the a -tail of the quotient Γ \ H , thatis, the image in Γ \ H of { z ∈ H : ℑ ( z ) ≥ a } . Then the necessary estimate on the a -tail will follows from the E - boundedness.We prove that, given ε > , there is c sufficiently large so that ϕ ∞ · B lies in asingle ball of radius ε inside L (Γ \ H ) , that is, lim c →∞ Z y>c | f ( z ) | d x d yy −→ uniformly for | f | E ≤ .A precise choice of smooth truncations and understanding of their E norms isneeded. Fix a smooth real-valued function ψ on R with ψ ( y ) = 0 for y ≤ , ≤ ψ ( y ) ≤ for < y < , and ψ ( y ) = 1 for y ≥ . For t > , let ψ t ( y ) = ψ ( yt − , and form a pseudo-Eisenstein series Ψ ψ t , a locally finite sum.Then Ψ ψ t · f ( x + iy ) is a smoothly cut-off tail of f starting gradually at height t : (1 − Ψ ψ t ) · f is identically in the region where y ≥ t , and in all images of thisregion under SL ( Z ) . Lemma 17.
The smooth truncation Ψ ψ t · f has E -norm dominated by that of f itself, with implied constant uniform in t ≥ and in f ∈ E . Proof.
Routine computation.
Q.E.D.
Returning to the proof of Theorem 16, given c > , we can assume that f hasbeen smoothly truncated so that in the fundamental domain it is supported insidethe region where y ≥ c , and increasing its E norm by at most a uniform factor. Letthe Fourier coefficients of f ( x + iy ) in x be b f ( n ) , functions of y . Take y ≥ c > a ,so the b f (0) vanishes. Using Plancherel for the Fourier expansion in x , integratingover the part of Y ∞ above y = c , letting F denote Fourier transform in x , there isa direct computation Z Z y>c | f | d x d yy ≤ c Z Z y>c | f | d x d y = 1 c X n =0 Z y>c | b f ( n ) | d y ≤ c X n =0 (2 πn ) Z y>c | b f ( n ) | d y = 1 c X n =0 Z y>c (cid:12)(cid:12)(cid:12) F ∂f∂x ( n ) (cid:12)(cid:12)(cid:12) d y = 1 c Z Z y>c (cid:12)(cid:12)(cid:12) ∂f∂x (cid:12)(cid:12)(cid:12) d x d y = 1 c Z Z y>c − ∂ f∂x · f ( x ) d x d y ≤ c Z Z y>c (cid:18) − ∂ f∂x · f ( x ) − ∂ f∂y · f ( x ) (cid:19) d x d y = 1 c Z Z y>c − ∆ f · f d x d yy ≤ c Z Z Γ \ H − ∆ f · f d x d yy = 1 c | f | E ≤ c This uniform bound completes the proof that the image of the unit ball of E in E is totally bounded. Thus, the inclusion is a compact map.As earlier, Friedrichs’ construction shows that ( e S Θ − λ ) − : E → E is contin-uous even with the stronger topology of E . Thus, the composition E → E ⊂ E by f −→ ( e S Θ − λ ) − f → ( e S Θ − λ ) − f is the composition of a continuous operator with a compact operator, so is compact.Thus, ( e S Θ − λ ) − : E → E is a compact operator. Thus, for λ off a discrete setof points in C , e S Θ has compact resolvent ( e S Θ − λ ) − , and the parametrized familyof compact operators ( e S Θ − λ ) − : E −→ E is meromorphic in λ ∈ C .We recall the standard argument (see [19], page 187 and preceding, for example)for the fact that that, for a (not necessarily bounded) normal operator T , if T − exists and is compact, then ( T − λ ) − exists and is a compact operator for λ off adiscrete set in C , and is meromorphic in λ . First, from the spectral theory of normalombieri and Garrett 29compact operators, the non-zero spectrum of compact T − is all point spectrum.We claim that the spectrum of T and non-zero spectrum of T − are in the obviousbijection λ ↔ λ − . From the algebraic identities T − − λ − = T − ( λ − T ) λ − and T − λ = T ( λ − − T − ) λ , failure of either T − λ or T − − λ − to be injective forces the failure of the other, so the point spectra are identical. For (non-zero) λ − not an eigenvalue of compact T − , T − − λ − is injective and has a continuous,everywhere-defined inverse. That S − λ is surjective for compact normal S and λ = 0 not an eigenvalue is an easy part of Fredholm theory. For such λ , invertingthe relation T − λ = T ( λ − − T − ) λ gives ( T − λ ) − = λ − ( λ − − T − ) − T − from which ( T − λ ) − is continuous and everywhere-defined. That is, λ is not inthe spectrum of T . Finally, λ = 0 is not in the spectrum of T , because T − existsand is continuous. This establishes the bijection. Q.E.D.
Essentially as in [20], identification of the eigenfunctions for e S Θ depends on thecut-off height a > and reduction theory. The truncation ∧ a E s of an Eisensteinseries at height ℑ ( z ) = a is as follows. With y = ℑ ( z ) , let τ s ( z ) be y s + c s y − s for y ≥ a and for < y < a , and form a pseudo-Eisenstein series Ψ s,a ( z ) = X γ ∈ Γ ∞ \ Γ τ s ( γz ) Even though τ s is not a test function, by reduction theory this is a locally finitesum, so converges uniformly absolutely pointwise. For a > , inside the standardfundamental domain for SL ( Z ) , by reduction theory, Ψ s,a ( z ) is unless y ≥ a ,and is y s + c s y − s for y ≥ a . The truncated Eisenstein series is ∧ a E s = E s − Ψ s,a By design, for a > , inside the standard fundamental domain for SL ( Z ) , thistruncation makes the constant term vanish above height y = a . Theorem 18.
For a > , spectral parameters w for eigenvalues λ w > of e S Θ are exactly the zeros of the constant term a w + c w a − w of the Eisenstein series E w with ℜ ( w ) = . The corresponding eigenvalues λ w are simple, the correspondingeigenfunctions are solutions u ∈ E of the equation ( − ∆ − λ ) u = η a , and up toconstants these are truncated Eisenstein series ∧ a E w . Specifically, ( − ∆ − λ w ) ∧ a E w = 2(1 − w ) a w +1 · η a Remark 19.
In particular, all eigenfunctions with eigenvalues λ > fail to besmooth, since their constant terms will be continuous, but have discontinuous firstderivative (in y = ℑ ( z ) ) at height a . In this regard, such eigenfunctions are exotic.Proof. Because the homogeneous equation ( − ∆ − λ ) u = 0 has no non-zero solu-tion in E , it suffices to identify the possible θ in the E − closure Θ − of Θ thatcould fit into an equation ( − ∆ − λ ) u = θ with u in the E closure of E ∞ c ∩ E . Onone hand, because a > , Θ − consists of distributions which, on the standard fun-damental domain, have support inside the Siegel set S a = { x + iy ∈ H : y ≥ a } .Further, on C a = Γ ∞ \ S a , the circle S = Z \ R acts by translations, descending tothe quotient from H . By reduction theory, the restrictions to C a of every pseudo-Eisenstein series Ψ ϕ with ϕ ∈ C ∞ c [ a, ∞ ) are invariant under S , so anything inthe E − closure is likewise translation-invariant.On the other hand, E ∞ c ∩ E consists of functions with constant term vanishingin y ≥ a , and taking E completion preserves this property. Since Θ − is S -invariant and the Laplacian commutes with S , it suffices to look at S -integralaverages of v restricted to some cylinder C b with a > b > . Such an integral is arestriction of the constant term c P v to C b , and vanishes in y > a .Thus, in the standard fundamental domain, the support of distributions θ ∈ Θ − fitting into an equation ( − ∆ − λ ) u = θ , with u in the E closure of E ∞ c ∩ E , isinside the image of the set where y = a , and θ is S -invariant. By the classificationof distributions supported on submanifolds, such θ is a derivative normal to the cir-cle Γ ∞ \{ x + iy : y = a } followed by application of an S -invariant distribution onthe circle. The latter must be integration over the circle, by classification of invari-ant distributions. By standard Sobolev theory, there can be no actual derivatives,for the resulting distribution to lie in the − Sobolev space. Thus, up to a constantmultiple, θ is the evaluation of constant term at height a functional η a , namely, η a f = c P f ( ia ) .Since | a s + c s a − s | ≤ √ a for ℜ ( s ) = , in fact η a ∈ E − − ε for every ε > .As in Theorem 10 and its proof, from the spectral relation (3.8), for v ∈ E with ( − ∆ − λ w ) u = η a , the spectral expansions u = h u, i · h , i + 14 πi Z ( ) E u ( s ) · E s d s (4.1) η a = h η a , i · h , i + 14 πi Z ( ) E η a ( s ) · E s d s (4.2)converging in E and E − , respectively, have E η a ( s ) = η a ( E − s ) = a − s + c − s a s ,ombieri and Garrett 31and the expansions are related by ( λ s − λ w ) · E u ( s ) = E η a ( s ) , so E η a ( w ) = 0 in astrong sense. Since ℜ ( w ) = , by c w c − w = 1 , we have a w + c w a − w = 0 .To finish the proof, it suffices to show that ( − ∆ − λ w ) ∧ a E w is a scalar multipleof η a when a w + c w a − w = 0 . Indeed, with a > , in a fundamental domain,away from y = a we have ( − ∆ − λ w ) ∧ a E w = 0 locally. Further, in y > ,the differential operator annihilates all Fourier components of E w but the constantterm, and in both < y < a and y > a does also annihilate the constant term. Tocompute near y = a , let H be the Heaviside function H ( y ) = 0 for y < and H ( y ) = 1 for y > . Thus, near y = a , as functions of y independent of x , ( − ∆ − λ w ) ∧ a E w = ( − ∆ − λ w ) (cid:16) H ( a − y ) · ( y w + c w y − w ) (cid:17) = ( − y ∂ ∂y − w (1 − w )) (cid:16) H ( a − y ) · ( y w + c w y − w ) (cid:17) = − y (cid:16) H ′′ ( a − y )( y w + c w y − w ) + 2 H ′ ( a − y )( y w + c w y − w ) ′ + H ( a − y )( y w + c w y − w ) ′′ (cid:17) − w (1 − w ) H ( a − y )( y w + c w y − w )= − y (cid:16) δ ′ a · ( y w + c w y − w ) − δ a · ( wy w − + (1 − w ) c w y − w ) (cid:17) Since a w + c w a − w = 0 , the term with δ ′ a vanishes, and the rest simplifies to ( − ∆ − λ w ) ∧ a E w = − aδ a · ( wa w + (1 − w ) c w a − w )= − δ a · (2 w − a w +1 on functions of y independent of x . Thus, this is − w ) a w +1 · η a . Q.E.D.
Keep a > fixed, and, as above, Θ the collection of pseudo-Eisenstein seriesformed from test function data ϕ supported in [ a, + ∞ ) . Any u ∈ E with η b u = 0 for all b ≥ a > lies in E , so admits a spectral expansion in terms of theeigenfunctions for e S Θ , converging in the topology of E . However, we want toapply functionals in E − (termwise) to such an spectral expansion, which requiresthat the expansion converge in E . Thus, we need the following stronger analogueof Lemma 15:2 Designed pseudo-Laplacians Theorem 20.
With a > , E ∞ c ∩ E is dense in E ∩ E with the E topology.Proof. Given a sequence of pseudo-Eisenstein series Ψ ϕ i ∈ E ∞ c converging to f ∈ E in the topology of E , we produce a sequence of pseudo-Eisenstein seriesin E ∞ c ∩ E converging to f in the topology of E . We will do so by smoothcut-offs of the constant terms of the Ψ ϕ i . Since the limit f of the Ψ ϕ i has constantterm vanishing above height y = a and is in E , that part of the constant terms ofthe Ψ ϕ i becomes small. More precisely, we proceed as follows.Let g be a smooth real-valued function on R with g ( y ) = 0 for y < − , ≤ g ( y ) ≤ for − ≤ y ≤ , and g ( y ) = 1 for y ≥ . For ε > , let g ε ( y ) = g (( y − a ) /ε ) . Fix real b with a > b > . Given Ψ ϕ i → f ∈ E , the b - tail of the constant term of Ψ ϕ i is τ i ( y ) = c p Ψ ϕ i ( y ) for y ≥ b , and τ i ( y ) = 0 for < y ≤ b . By design, Ψ ϕ i − Ψ g ε · τ i ∈ E ∞ c ∩ E for small ε . We will show that,as i → + ∞ , for ε i sufficiently small depending on i , the E -norms of Ψ g εi · τ i goto , so Ψ ϕ i − Ψ g εi · τ i → f in the E topology.For b > , let S b = { x + iy ∈ H : y ≥ b, | x | ≤ } . By reduction theory, thecylinder C b = Γ ∞ \ (Γ ∞ · S b ) maps homeomorphically to its image in Γ \ H . For f ∈ C ∞ c (Γ \ H ) , let | f | H ( C b ) = Z C b | f ( z ) | − ∆ f · f d x d yy ≤ Z Γ \ H | f ( z ) | − ∆ f · f d x d yy For each b > , let H ( C b ) be the completion of C ∞ c (Γ \ H ) with respect to thesemi-norm | · | H ( C b ) , allowing for collapsing. The cylinders C b admit natural ac-tions of the circle group S = Z \ R , by translation, inherited from the translationof the real part of x + iy ∈ H . As usual, this induces a continuous action of S on H ( C b ) . Thus, the map F → c P F gives continuous maps of the spaces H ( C b ) to themselves. Thus, c P Ψ ϕ i goes to c P f in H ( C b ) , and c P Ψ ϕ i → c P f = 0 in H ( C a ) .To have a useful Leibniz rule for differentiation, it is convenient to rewrite thenorms: for f ∈ C ∞ c (Γ \ H ) , put | f | H = | f | L (Γ \ H ) + | ( |∇ f | s ) | L (Γ \ H ) where ∇ is the left SL ( R ) -invariant, right SO ( R ) -equivariant tangent-space-valued gradient on SL ( R ) , which therefore descends to H and to Γ \ H , and | · | s is a natural SO ( R ) -invariant norm on the tangent space(s). More explicitly, let s be the space of symmetric -by- matrices of trace , identified with the tangentombieri and Garrett 33space at every point of H via left translation of the exponential map: for β ∈ s , asusual the associated left SL ( R ) -invariant differential operator X β is ( X β f )( g ) = ∂∂t (cid:12)(cid:12)(cid:12) t =0 f ( ge t · β ) It is easy to describe ∇ in coordinates, even though it is provably independent ofcoordinates: let h = (cid:18) − (cid:19) and σ = (cid:18) (cid:19) , and put ∇ F ( g ) = X h F ( g ) · h + X σ F ( g ) · σ ∈ s ⊗ R C Up to a scalar, the SO ( R ) -invariant hermitian inner product h , i s on the complex-ified s , makes h, σ an orthonormal basis for s . Let | · | s be the associated norm.The essential property is the integration by parts identity Z Γ \ H h∇ F , ∇ F i s = Z Γ \ H − ∆ F · F for F , F ∈ C ∞ c (Γ \ H ) . The advantage of this formulation is that, extending ∇ bycontinuity in the H topology, ∇ F exists (in an L sense) for F ∈ H ( C b ) . Thus,we can say that | F | H ( C b ) = | F | L (Γ \ H ) + | |∇ F | s | L (Γ \ H ) Then | Ψ g ε · τ i | E = | Ψ g ε · τ i | H ( C a − ε ) = | g ε · τ i | H ( C a − ε ) ≤ | ( g ε − · τ i | H ( C a − ε ) + | τ i − c P f | H ( C a − ε ) + | c P f | H ( C a − ε ) The middle summand goes to : | τ i − c P f | H ( C a − ε ) ≤ | c P Ψ ϕ i − c P f | E ≤ | Ψ ϕ i − f | E −→ The first and third summands require somewhat more care. Estimate | ( g ε − · τ i | H ( C a − ε ) = Z C a − ε | ( g ε − τ i | + |∇ ( g t − τ i | s ≤ Z C a − ε | g ε − | · ( | τ i | + |∇ τ i | s ) + Z C a − ε |∇ g ε | s · | τ i | + Z C a − ε | g ε | · |∇ g ε | s · | τ i | · |∇ τ i | s as ε → + because g ε − when y ≥ a , and τ i and |∇ τ i | s are continuous.In terms of the coordinates z = x + iy on H , for a smooth function F a standardcomputation gives ∇ F = y ∂F∂x · σ + y ∂F∂y · h so |∇ g ε ( x + iy ) | s = | ε · y g ′ (( y − a ) /ε ) · h | s = 1 ε · | y g ′ (( y − a ) /ε ) | ≪ g ε Similarly, since τ i is a function of y independent of z , ∇ τ i = yτ ′ i ( y ) · h . By thefundamental theorem of calculus and the Cauchy-Schwarz-Bunyakowsky inequal-ity, we recover an easy instance of a Sobolev inequality: | τ i ( a − v ) | = (cid:12)(cid:12)(cid:12) − Z v τ ′ i ( a − v ) d v (cid:12)(cid:12)(cid:12) ≤ (cid:16) Z v | τ ′ i ( a − v ) | d v (cid:17) · (cid:16) Z v d v (cid:17) = o ( √ v ) with Landau’s little- o notation, since τ ′ i is locally L . Thus, Z C a − ε | g ε | · |∇ g ε | s · | τ i | · |∇ τ i | s ≤ ε · o (1) · √ ε · Z ε |∇ τ i | s ≤ ε · o (1) · √ ε · (cid:16) Z ε | τ ′ i | (cid:17) · (cid:16) Z ε (cid:17) ≪ τ i ε · o (1) · √ ε · √ ε = o (1) That is, the summand R C a − ε | g ε | · |∇ g ε | s · | τ i | · |∇ τ i | s goes to . Using the samesubordinate estimates, Z C a − ε |∇ g ε | s · | τ i | ≪ ε Z ε (cid:0) o (1) · √ v (cid:1) d v = 1 ε · o (1) · ε −→ Thus, taking the ε i sufficiently small, the smooth truncations Ψ ϕ i − Ψ g εi · τ i of the Ψ ϕ i are in E ∞ c ∩ E , and still converge to f in E . Q.E.D.
Corollary 21.
An orthogonal basis for E consisting of e S Θ -eigenfunctions is anorthogonal basis for E ∩ E , as well. In particular, for eigenfunction f witheigenvalue λ , we have h f, f i E = λ · h f, f i . ombieri and Garrett 35 Proof.
Since e S − ( E ) ⊃ E ∞ c ∩ E , by the theorem e S − ( E ) is dense in E ∩ E .Since finite linear combinations of the e S Θ -eigenfunctions are dense in E and e S − is continuous, their images in E ∩ E are dense there. From Proposition 1, for two e S Θ -eigenfunctions u, v with (real) eigenvalues λ, µ , h u, v i E = h S u, v i E − × V = h e S Θ u, v i E − × V = h e S Θ u, v i = λ h u, v i Symmetrically, h u, v i E = µ h u, v i . Thus, orthogonality of eigenfunctions in E ⊂ V implies orthogonality in E ∩ E ⊂ E , and for u = v we have h u, u i E = λ · h u, u i . Q.E.D.
Corollary 22.
Let { u k : k = 1 , , . . . } be the eigenfunctions for e S Θ , with eigenval-ues λ k . For f ∈ E ∩ E , f = X k ≥ h f, u k i V · u k h u k , u k i V = X k ≥ h f, u k i E · u k h u k , u k i E and these expansions converge to f not only in E , but also in the finer topologyof E ∩ E .Proof. Again, by the theorem, since f ∈ E ∩ E is in the closure of E ∞ c ∩ E ,and { u k } is an orthogonal basis for E ∩ E , such f has an expansion f = X k ≥ h f, u k i E · u k h u k , u k i E convergent in E ∩ E . As in the previous proof, from Proposition 1, h f, u k i E = h f, S u k i V × E − = h f, e S Θ u k i V × E − = h f, e S Θ u k i = λ k h f, u k i Thus, for every u k , h f, u k i E · u k h u k , u k i E = λ k h f, u k i · u k λ k h u k , u k i = h f, u k i · u k h u k , u k i giving the termwise equality of the expansions. Q.E.D.
Fix a finite real-linear combination θ of Eisenstein-Heegner distributions θ d ∈ E − − ε . Fix a > . Let S θ,a be − ∆ restricted to domain E ∞ c ∩ ker θ ∩ ker η a . Note that S θ,a > on the continuous spectrum, and its domain excludes constants.Symmetry of S θ,a is inherited from S . The pseudo-Laplacian e S θ,a is the Friedrichsextension of S θ,a on E ∩ ker θ ∩ ker η a , with e S θ,a > . Theorem 23.
Given a finite real-linear combination θ of Eisenstein-Heegner dis-tributions θ d , for all a with ℑ z = a for all the Heegner points z involved, theFriedrichs extension e S θ,a ignores θ and η a , in the sense that for u in the domainof e S θ,a the eigenvector condition ( e S θ,a − λ w ) u = 0 is equivalent to the satisfactionof the equation ( S θ,a − λ w ) u = A · θ + B · η a ( for some A, B ∈ C ) . Proof.
The point is to show that ( C · θ ∩ η a ) ∩ ( j ∗ ∩ Λ) E = { } . Then Theorem4 applies.There is an unique highest Heegner point z appearing in θ . In fact, if | d the highest Heegner point in θ d is i p | d | / , while otherwise it is (1 + i p | d | ) / .Then it suffices to take pseudo-Eisenstein series f n = Ψ ϕ n with ϕ n ( ℑ z ) = 1 and ϕ n ( ℑ z ′ ) = 1 for the other Heegner points z ′ appearing in θ . Thus, A · θ + B · η a is not in E for A = 0 . Thus, it suffices to show that η a is not in E , which is evensimpler. Q.E.D.
We continue to keep fixed a finite real-linear combination θ of Eisenstein-Heegnerdistributions θ d . From the preceding Theorem 23, a solution u ∈ E of an equation ( S − λ w ) u = A · θ + B · η a ombieri and Garrett 37is a e S θ,a - eigenfunction precisely when u ∈ ker θ ∩ ker η a . As above, let v w,a be theunique solution in E − ε to ( − ∆ − λ w ) = η a and similarly let u θ,w be the uniquesolution in E ε to ( − ∆ − λ w ) u = θ . The condition for existence of a non-zerosolution ( A, B ) to the homogeneous system ( θ ( Au θ,w + Bv a,w ) = 0 η a ( Au θ,w + Bv a,w ) = 0 (5.1)is the vanishing of the determinant: det θ ( u θ,w ) θ ( v w,a ) η a ( u θ,w ) η a ( v w,a ) = 0 (5.2)We compute the components. η a ( v w,a ) for a > and ℜ ( w ) > By the spectral expansion (3.14), this is η a ( v w,a ) = 1 h , i + 14 πi Z ( ) ( a − s + c − s a s )( a s + c s a − s ) d sλ s − λ w = 1 h , i + 14 πi Z ( ) ( a + c − s a s + c s a − s + a ) d sλ s − λ w (use c s c − s = 1 ) = 1 h , i + 12 πi Z ( ) ( a + c s a − s ) d sλ s − λ w ( s → − s in one term) (5.3)The behavior of c s as a function of s in the half-plane σ > is easily determinedfrom the second formula in (3.5). We find that c s has a simple pole at s = 1 withresidue π = 1 h , i and is of order √ σ ( | t | + 1) + ε for any fixed ε > , uniformlyin σ and t . For a > , | a − s | = a − σ goes exponentially to (uniformly in t ) as σ → + ∞ , so we can compute the integral in (5.3) by moving the line of integrationto σ = + ∞ and see that it tends to as σ → + ∞ . In doing so, we encounterresidues at s = w and s = 1 . The residue at s = 1 cancels the constant term / h , i . Noting that λ s − λ w = − ( s − w )( s − (1 − w )) , and noting the negativeorientation around s = w of the path integral, the final result is8 Designed pseudo-Laplacians Theorem 24.
For a > and ℜ ( w ) > , η a ( v w,a ) = a − w ( a w + c w a − w )2 w − θ d ( v w,a ) for a ≫ θ and ℜ ( w ) > By linearity, it suffices to compute δ nc z ( v w,a ) when z is a Heegner point. Note that v w,a ∈ E − ε for all ε > , so the integral for the pairing E − ε × E − − ε is absolutelyconvergent if ε is sufficiently small.Using c − s E s = E − s and the spectral expansion, we find δ nc z ( v w,a ) = 1 h , i + 14 πi Z ( ) η a E − s ( z ) · E s ( z ) d sλ s − λ w = 1 h , i + 14 πi Z ( ) ( a − s + c − s a s ) · E s ( z ) d sλ s − λ w = 1 h , i + 14 πi Z ( ) ( a − s E s ( z ) + a s E − s ( z )) d sλ s − λ w (use c − s E s = E − s ) = 1 h , i + 12 πi Z ( ) a − s E s ( z ) d sλ s − λ w (5.4)The computation of the integral requires some care, depending on the height of z relative to a . To this end, we proceed as before, moving the line of integrationfrom σ = to σ = C where C > (the actual value of C is immaterial),thereby acquiring the contribution of residues at s = w and also at s = 1 fromthe Eisenstein series. The residue of E s ( z ) at s = 1 is π = 1 h , i , hence itscontribution cancels the constant term and one obtains δ nc z ( v w,a ) = a − w E w ( z )2 w − πi Z ( C ) a − s E s ( z ) d sλ s − λ w (5.5)The series for E s ( z ) with z = x + iy is absolutely convergent for ℜ ( s ) = c > and for y → ∞ it is asymptotic to y s . It is obvious that for σ > we have | E s ( z ) | X m,n ′ y σ | nz + n | σ ombieri and Garrett 39where the dash means that the sum is extended to all pairs ( m, n ) of coprime inte-gers. It follows that if y is bounded away from zero (in our case y > √ / ) then | E s ( z ) | ≪ (max(1 , y σ ) uniformly for σ > c > , with the constant involved in theinequality depending only on c .Therefore, if y/a < one may move the line of integration all the way to + ∞ ,showing that the integral in question vanishes. We have proved that in this case δ nc z ( v w,a ) = a − w E w ( z )2 w − ℑ ( z ) < a ) (5.6)If instead y/a > the analysis is more complicated. In this case, we split the sumfor E s ( z ) into two components: E s ( z ) = 12 X | mz + n | >y/a GCD( m,n )=1 y s | mz + n | s + 12 X | mz + n | y/a GCD( m,n )=1 y s | mz + n | s = Σ + Σ . (5.7)The evaluation of the integral πi Z ( C ) a − s Σ d sλ s − λ w can be done as beforeby letting C → + ∞ , obtaining πi Z ( C ) a − s Σ d sλ s − λ w = 0 . (5.8)To deal with the integral involved in Σ we note that the sum involved is a finitesum. For | cz + d | = y/a we can integrate term-by-term and move the line ofintegration backwards all the way to → −∞ , with the limit of the integral being .In doing this we encounter two residues at s = w and s = 1 − w , and concludethat πi Z ( C ) a − s Σ d sλ s − λ w = − a − w w − X | mz + n | y/a GCD( m,n )=1 y w | mz + n | w + 12 a w w − X | mz + n | y/a GCD( m,n )=1 y − w | mz + n | − w . (5.9)0 Designed pseudo-LaplaciansWe have proved: If ℑ ( z ) < a then δ nc z ( v w,a ) = a − w E w ( z )2 w − . (5.10)while if y = ℑ ( z ) > a then δ nc z ( v w,a ) = a − w E w ( z )2 w − − a − w w − X | mz + n |
Let a > , and let d < be a fundamental discriminant. Then θ d ( v w,a ) = 12 w − (cid:8) a − w θ d E w − R w ( d, a ) (cid:9) (5.13) where R w ( d, a ) = X x + iy ∈ Hdy>a ( a − w y w − a w y − w ) (5.14) η a ( u θ,w ) for a ≫ θ and ℜ ( w ) > Let θ be a finite real-linear combination of Eisenstein-Heegner distributions θ d .ombieri and Garrett 41 Theorem 26. η a ( u θ,w ) = θ ( v w,a ) . (5.15) Proof.
One computes η a ( u θ,w ) = η a (1) θ (1) h , i + 14 πi Z ( ) θE − s · η a E s d sλ s − λ w = θ (1) h , i + 14 πi Z ( ) θE − s · ( a s + c s a − s ) d sλ s − λ w = θ (1) h , i + 14 πi Z ( ) ( θE − s · a s + θE s · a − s ) d sλ s − λ w (because c s E − s = E s ) = θ (1) h , i + 12 πi Z ( ) θE − s · a s d sλ s − λ w . (by changing s → − s in one term)The theorem follows by linearity and equation (5.4). Q.E.D. θ ( u θ,w ) for a > and ℜ ( w ) > The outcome here does not admit much simplification, in contrast to the other cases.That is, from the spectral expansions of subsections 3.6 and 3.7, via the E × E − pairing of 3.4, we obtain Theorem 27.
For θ a finite real-linear combination of Eisenstein-Heegner distri-butions θ d , θ ( u θ,w ) = | θ (1) | h , i · ( λ − λ w ) + 14 πi Z ( ) | θE s | d sλ s − λ w . (5.16) First, by Theorem 5.15 the determinant-vanishing condition (5.2) becomes η ( v w,a ) θ ( u θ,w ) − η a ( u θ,w ) = 0 In view of Theorems 24, 25, 27,2 Designed pseudo-Laplacians
Corollary 28.
Let θ = P d ν d θ d be a finite real-linear combination of Eisenstein-Heegner distributions θ d with d < − . For all a > , all w with ℜ ( w ) > andoff ( , , a − w ( a w + c w a − w ) × (cid:16) | θ (1) | h , i · λ − λ w ) + 14 πi Z ( ) | θE s | d sλ s − λ w (cid:17) = 12 w − (cid:0) a − w θE w − R w ( θ, a ) (cid:1) (5.17) where θE s = X d ν d (cid:18) p | d | (cid:19) s ζ ( s ) ζ (2 s ) L ( s, χ d ) (5.18) and where R w ( θ, a ) = X d ν d X x + iy ∈ Hdy>a ( a − w y w − a w y − w ) . (5.14) Proof. In ℜ ( w ) > , u θ,w and v a,w are in E . The vanishing condition η ( v w,a ) θ ( u θ,w ) − η a ( u θ,w ) = 0 is thus necessary and sufficient for some non-zero linear combinationof u θ,w and v a,w to be an eigenfunction for the self-adjoint semibounded operator e S θ,a , with eigenvalue λ w = w (1 − w ) . Such an eigenvalue must be real and satisfy λ w . Q.E.D.
Here we give a direct, relatively elementary argument for meromorphic continua-tion of the two-by-two determinant above. Theorem 43 in section 6.5 gives severalstronger results by less elementary means, useful in the discussion of unconditionalresults on zero spacing in Section 8. For brevity, write F ( a, w ) := ( a w + c w a − w ) (cid:16) | θ (1) | h , i · ( λ − λ w ) + 14 πi Z ( ) | θE s | d sλ s − λ w (cid:17) − a − w θ ( E w ) w − (5.19)for the determinant above, where for the time being ℜ ( w ) > and a is real with a > .ombieri and Garrett 43The analytic continuation of F ( a, w ) beyond the line ℜ ( w ) = is easily ac-complished. Except for a possible simple pole at w = 1 , the function s → θE s isholomorphic for ℜ ( s ) > − C/ log(2 + | s | ) for some absolute positive constant C . Hence, when < ℜ ( w ) < + C/ log(2 + | w | ) we can evaluate the integral in(5.19) as follows : πi Z ( ) | θE s | d sλ s − λ w = 14 πi Z ( ) (cid:16) θE s · θE − s − θE w · θE − w (cid:17) d sλ s − λ w + θE w · θE − w πi Z ( ) d sλ s − λ w . (5.20)The left-hand side of this equation is a holomorphic function for ℜ ( w ) > . For ℜ ( w ) > the last integral is evaluated by the calculus of residues, moving the lineof integration to ℜ ( s ) → + ∞ : πi Z ( ) d sλ s − λ w = 12(2 w − . (5.21)Therefore, for < ℜ ( w ) < + C/ (2 + | w | ) we have F ( w, a ) = ( a w + c w a − w ) | θ (1) | h , i · ( λ − λ w )+ ( a w + c w a − w ) 14 πi Z ( ) (cid:16) θE s · θE − s − θE w · θE − w (cid:17) d sλ s − λ w + ( a w + c w a − w ) θE w · θE − w w − − a − w θ ( E w ) w − (5.22)Using the functional equation c w θE − w = θE w simplifies this into F ( w, a ) = ( a w + c w a − w ) | θ (1) | h , i · ( λ − λ w )+ ( a w + c w a − w ) 14 πi Z ( ) (cid:16) θE s · θE − s − θE w · θE − w (cid:17) d sλ s − λ w + ( a w − c w a − w ) θE w · θE − w w − (5.23)4 Designed pseudo-LaplaciansThis formula, so far proved for < ℜ ( w ) < + C/ log(2 + | w | ) , extends to anopen neighborhood of the line ℜ ( w ) = in the complex plane w ∈ C , becausenow the integral is well defined there as a continuous function of w . Indeed, thenumerator of the integrand vanishes as w → s with ℜ ( s ) = at least to the firstorder when s = and at at least to the second order when s = , while the growthof θE s · θE − s is of order not more than | s | − δ there for some fixed δ > if theneighborhood is sufficiently small, hence the integral is absolutely convergent. Theorem 29.
With a > , let G ( w, a ) := F ( w, a ) a w + c w a − w (5.24) Then G ( w, a ) is a meromorphic function in the whole complex w -plane and satis-fies the functional equation G ( w, a ) = G (1 − w, a ) . (5.25) Proof.
A simple computation using the expansion (5.7) shows that the stated func-tional equation holds in an open neighborhood of the critical line. Due to the im-portance of this symmetry, we carry out this computation in detail. G ( w, a ) = | θ (1) | h , i · ( λ − λ w )+ 14 πi Z ( ) (cid:16) θE s · θE − s − θE w · θE − w (cid:17) d sλ s − λ w + a w − c w a − w a w + c w a − w θE w · θE − w w − (5.26)The first two summands are indeed invariant under w → − w , as is θE w · θE w inthe third summand. Finally, under w → − w , using c w · c − w = 1 , the part a w − c w a − w a w + c w a − w w − of the third summand becomes a − w − c − w a w a − w + c − w a w − w − c w a − w − a w c w a − w + a w − w − a w − c w a − w a w + c w a − w w − giving the claimed invariance. The conclusion of the theorem follows by analyticcontinuation. Q.E.D. ombieri and Garrett 45
Corollary 30.
The only zeros of the function G ( w, a ) defined in equation 5.24 areon ℜ ( w ) = and [0 , .Proof. Corollary 28 shows that G ( w, a ) cannot vanish in ℜ ( w ) > except pos-sibly on ( , , because otherwise λ w = w (1 − w ) would be an eigenvalue for anon-negative self-adjoint operator. Then the symmetry of 5.25 shows non-vanishingin ℜ ( w ) < except possibly on [0 , ) . Q.E.D.
The preceding considerations also apply to a general real-linear combination θ = P ν b ν · δ nc z ν of Eisenstein-Dirac distributions δ nc z ν . With the simplifying assumption P b ν E s ( z ν ) = 0 , we have a − w ( a w + c w a − w ) × πi Z ( ) (cid:12)(cid:12)(cid:12)X b ν E s ( z ν ) (cid:12)(cid:12)(cid:12) d sλ s − λ w = 11 − w (cid:26)X b ν (cid:20) a − w E w ( z ν ) − p a ℑ ( z ν ) sinh (cid:18)
12 (1 − w ) log + ℑ ( z ν ) a (cid:19)(cid:21)(cid:27) (5.27)where log + x = max(log x, .If a > max ℑ ( ζ ν ) then for almost all a the vanishing of P b ν E w ( z ν ) impliesthe vanishing of the left-hand term, which means either a − w ( a w + c w a − w ) = 0 ,which vanishes only when ℜ ( s ) = , or πi Z ( ) (cid:12)(cid:12)(cid:12)X b ν E s ( z ν ) (cid:12)(cid:12)(cid:12) d sλ s − λ w = 0 . Thus if zeros on the critical line of a function P ν b ν E s ( z ν ) with P b ν = 0 had aspectral interpretation for some a = a > max ℑ ( z ν ) this would be so for all a >a , implying that the corresponding eigenvalues w (1 − w ) would be independent of a . For a → ∞ this is analogous to the condition formulated by Colin de Verdi`erein the special case { ν } = { } , b = 1 , z = + i √ , and, tentatively, suggestedby him as a spectral intepretation of the zeros of ζ Q ( √− ( s ) on the critical line.6 Designed pseudo-Laplacians θE s in a special case The fundamental discriminants d are the odd squarefree numbers d = m with m ≡ or numbers of the type d = 4 m with m squarefree and m ≡ , . To m , one associates the real primitive character χ m given by χ m ( n ) = (cid:16) mn (cid:17) m ≡ (cid:18) mn (cid:19) m ≡ , (5.28)where on the right-hand side we have the Kronecker symbol. Then ζ ( s ) L ( s, χ m ) isthe zeta function of the quadratic field Q ( √ d ) .A precise asymptotic formula for a certain simple linear combination of quadratic L -functions has been obtained in the paper [12] of Goldfeld and Hoffstein. We re-call verbatim their Theorem (1), where their reference (0.6) is our equation (5.28).(Compare also [28].) Theorem (1)
Let ε > be fixed. Let χ m be defined as in (0.6). Then there existanalytic functions c ( ρ ) and c ∗± ( ρ ) with Laurent expansion c ( ρ ) = c / ( ρ − ) + c ′ + O ( ρ − ) , c ∗± ( ρ ) = − c such that X < − m
24 + 16Re( ρ ) if ≦ Re( ρ ) ≦ − √ and all O-constants depend at most on ρ , ε . The authors do not give the dependence on ρ in the proportionality factor in-volved in the symbol O ( ... ) , but there must be a function ω ( x ) slowly increasingto ∞ such that the estimates remain uniform in ρ as long as |ℑ ( ρ ) | < ω ( x ) .What is of interest to us is not the sum P L ( ρ, χ m ) (with m < ) but rather thesum divided by ζ (2 ρ ) . This yields the following result.Let A ( s ) = 34 (1 − − s ) Y p =2 (1 − p − − p − s − + p − s − ) (5.29) Theorem 31.
There is a function ω ( x ) , slowly increasing to ∞ as x → ∞ , suchthat X < − m
In a quasi-complete, local convex topological vector space, the convexhull of a compact set has compact closure.
The latter property ensures existence of certain Gelfand-Pettis integrals:
Theorem 33.
Let X be a locally compact Hausdorff topological space with a fi-nite, positive, regular Borel measure. Let V be a locally convex topological vec-torspace in which the closure of the convex hull of a compact set is compact. Thencontinuous, compactly-supported V -valued functions f on X have Gelfand-Pettisintegrals. Further, Z X f ∈ meas ( X ) · (cid:16) closure of convex hull of f ( X ) (cid:17) is the basic estimate substituting for estimating a Banach-space norm of an integralby the integral of the norm of the integrand. The legitimacy of passing continuous operators inside such integrals is an easycorollary:
Corollary 34.
Let T : V → W be a continuous linear map of locally convextopological vectorspaces, where convex hulls of compact sets in V have compactclosures. Let f be a continuous, compactly-supported V -valued function on a finiteregular measure space X . Then the W -valued function T ◦ f has a Gelfand-Pettisintegral, and T (cid:16) R X f (cid:17) = R X T ◦ f .Proof. (of corollary) To verify that the left-hand side of the asserted equality fulfillsthe requirements of a Gelfand-Pettis integral of T ◦ f , we must show that λ (cid:16) left-hand side (cid:17) = Z X λ ◦ ( T ◦ f ) for all λ ∈ W ∗ . Starting with the left-hand side, λ (cid:16) T (cid:16) Z X f (cid:17)(cid:17) = ( λ ◦ T ) (cid:16) Z X f (cid:17) = Z X ( λ ◦ T ) ◦ f = Z X λ ◦ ( T ◦ f ) proving that T (cid:0) R X f (cid:1) is a weak integral of T ◦ f . Q.E.D.
Now we recall basic facts about holomorphic vector-valued functions, mostly with-out proof, for which we refer to the original source [13], or expositions such as[26] or [10], chapter 14. Existence and properties of vector-valued integrals areingredients in the proofs of the assertions below.A function f on an open set Ω ⊂ C taking values in a quasi-complete, locallyconvex topological vector space V is (strongly) complex-differentiable when lim z → z o z − z o · (cid:0) f ( z ) − f ( z o ) (cid:1) exists (in V ) for all z o ∈ Ω , where z → z o specificially means for complex z approaching z o . The function f is (strongly) analytic when it is locally expressibleas a convergent power series with coefficients in V . The function f is weaklyholomorphic when the C -valued functions λ ◦ f are holomorphic for all λ in V ∗ . Theorem 35.
For V a locally convex quasi-complete topological vector space,weakly holomorphic V -valued functions f are strongly holomorphic, in the fol-lowing senses. First the usual Cauchy-theory integral formulas apply: f ( z ) = 12 πi Z γ f ( ζ ) ζ − z d ζ with γ a closed path around z having winding number +1 . Second, the function f ( z ) is infinitely differentiable, in fact strongly analytic, that is, expressible as aconvergent power series f ( z ) = P n ≥ c n ( z − z o ) n with coefficients c n ∈ V givenby Gelfand-Pettis integrals echoing Cauchy’s formulas: c n = f n ( z o ) n ! = 12 πi Z γ f ( ζ )( ζ − z o ) n +1 d ζ The appropriate vector-valued notion of meromorphy is completely parallel tothe scalar-valued version: a V -valued function f defined on a punctured neighbor-hood N of z o is meromorphic at z o when there is a positive integer n such that ( z − z o ) m · f ( z ) has an extension to a V -valued holomorphic function on N ∪ { z o } .Fix a non-empty open subset Ω of C . Let V be quasi-complete, locally convex,with topology given by seminorms { ν } . The space Hol(Ω , V ) of holomorphic V -valued functions on Ω has the natural topology given by seminorms µ ν,K ( f ) = ombieri and Garrett 51 sup z ∈ K ν ( f ( z )) for compacts K ⊂ Ω , seminorms ν on V . This is the analogueof the sups-on-compacts seminorms on scalar-valued holomorphic functions, andthere is the analogous corollary of the vector-valued Cauchy formulas: Corollary 36.
Hol(Ω , V ) is locally convex, quasi-complete. A V -valued function f ( z, w ) on a non-empty open subset Ω ⊂ C is complexanalytic when it is locally expressible as a convergent power series in z and w ,with coefficients in V . The two-variable version of the above discussion of powerseries with coefficients in V succeeds without incident.Again by the vector-valued form of Cauchy’s integral formulas: Corollary 37.
Let V be quasi-complete, locally convex. Let f ( z, w ) be complex-analytic V -valued in two variables, on a domain Ω × Ω ⊂ C . Then the function w −→ ( z → f ( z, w )) is a holomorphic Hol(Ω , V ) -valued function on Ω . There is a vector-valued version of Abel’s theorem on convergent power seriesin one complex variable, proven in similar fashion:
Proposition 38.
Let c n be a bounded sequence of vectors in a locally convex quasi-complete topological vector space V . Then on | z | < the series f ( z ) = P n c n z n converges and gives a holomorphic V -valued function. That is, the function isinfinitely-many-times complex-differentiable. M of moderate-growth functions The space of moderate-growth continuous functions on Γ \ H is M = { f ∈ C (Γ \ H ) : sup ℑ ( z ) ≥√ / y − r · | f ( x + iy ) | < ∞ , for some r ∈ R } The correct topology is as a strict inductive limit of Banach subspaces M r eachobtained as a completion of C c (Γ \ H ) , with respect to norms | f | M r = sup ℑ ( z ) ≥√ / y − r · | f ( x + iy ) | Thus, lim y →∞ y − r | f ( x + iy ) | = 0 for f ∈ M r . That is, the set M is M = S r M r ,but the topology is perhaps not quite as expected. Nevertheless, M is a strict colimit(in the locally convex category) of Banach spaces, so is an LF-space, so is quasi-complete and locally convex.2 Designed pseudo-Laplacians E ε ⊂ M From the Sobolev imbedding at the end of 3.5, we have E ε ⊂ C (Γ \ H ) for all ε > . Certainly E s ∈ C (Γ \ H ) away from poles, and s → E s is a meromorphic C (Γ \ H ) -valued function. Thus, the Fr´echet space C (Γ \ H ) (with seminormsgiven by suprema on compact subsets) contains both E ε and Eisenstein series.However, the space M is much smaller, contains Eisenstein series, and s → E s isa meromorphic M -valued function. Thus, we want Lemma 39. E ε ⊂ M , for all ε > .Proof. A slightly more refined form of the pre-trace formula from 3.5 is X | s F |≤ T | F ( z ) | + 14 π Z T − T | E + it ( z ) | dt ≪ T + T · ℑ ( z ) as ℑ ( z ) → + ∞ , where F runs over an orthonormal basis for cuspforms, and F has eigenvalue s F (1 − s F ) for the invariant Laplacian. Thus, certainly, π Z T − T | E + it ( z ) | dt ≪ T + T · ℑ ( z ) as ℑ z → ∞ . As earlier, this asserts that the functional δ n cz given by δ n cz f = f ( z ) is in E − − ε , a weaker assertion than what we know to be true. However, theintegration by parts does also yield an estimate for the E − − ε norm depending onheight ℑ ( z ) , namely, | δ n cz | E − − ε ≪ ε y as y → ∞ , for all ε . For u ∈ E ε , by theCauchy-Schwarz-Bunyakowsky inequality extended to the pairing E ε × E − − ε → C , | u θ,w ( z ) | = | δ n cz u θ,w | ≤ | u | E ε · | δ n cz | E − − ε ≪ ε | u | E ε · ℑ ( z ) Thus, E ε ⊂ M ε ⊂ M for every ε > . Q.E.D.
Therefore, the continuous dual M ∗ of M has a natural map to the duals E − − ε of the spaces E ε for all ε > , removing a potential ambiguity: Corollary 40. E θ ( s ) = θE − s for θ = θ ∈ M ∗ .Proof. The proof of Proposition 9 applies here, with the Fr´echet space C (Γ \ H ) replaced by the (quasi-complete, locally convex) LF-space M . Q.E.D.
Using the latter, we haveombieri and Garrett 53
Corollary 41.
The function s → E θ ( s ) = θE s is a meromorphic function of s .Proof. From the previous corollary, indeed E θ ( s ) = θE − s . The function s → E s is a meromorphic M -valued function, so s → θE s is a meromorphic C -valuedfunction. Q.E.D.
Consider vector-valued integrals u θ,w = θ (1) · λ − λ w + 14 πi Z ( ) E θ ( s ) · E s λ s − λ w d s initially defined in ℜ ( w ) > (and w = 1 ), where λ s = s (1 − s ) , and where θ = θ ∈ E − ε for some ε > . In that region, u θ,w solves the differential equation ( − ∆ − λ w ) u = θ , and is in E ε . More generally, let Φ : M → N be a continuouslinear map to a quasi-complete locally convex topological vector space N , andconsider the N -valued integrals u θ,w, Φ = θ (1) · λ − λ w + 14 πi Z ( ) E θ ( s ) · Φ E s λ s − λ w d s Of course, for Φ the identity map M → M , this just gives u θ,w itself. We anticipatethat Φ( u θ,w ) = u θ,w, Φ , but this needs proof.From the previous section, θ ∈ E − ε ⊂ E − − ε extends to an element of thedual M ∗ to the space M of moderate growth continuous functions. Thus, θ can beapplied to Eisenstein series, and, further, E θ ( s ) = θE s by Corollary 40. Thus, theprevious expression can be rewritten somewhat more concretely as u θ,w, Φ = θ (1) · λ − λ w + 14 πi Z ( ) θE − s · Φ E s λ s − λ w d s Unsurprisingly, we have
Lemma 42. Φ( u θ,w ) = u θ,w, Φ in the region ℜ ( w ) > and w = 1 .Proof. Again, in the region ℜ ( w ) > and w = 1 , the hypotheses guarantee,via the extended Plancherel theorem, that the integral for u θ,w is an E ε -valued4 Designed pseudo-Laplaciansholomorphic function of w . In that region, using properties of compactly supported,continuous-integrand Gelfand-Pettis integrals from 6.1, Φ (cid:16) Z ( ) θE − s · E s λ s − λ w d s (cid:17) = Φ (cid:16) lim T → + ∞ Z |ℑ ( s ) |≤ T θE − s · E s λ s − λ w d s (cid:17) = lim T → + ∞ Φ Z |ℑ ( s ) |≤ T θE − s · E s λ s − λ w d s = lim T → + ∞ Z |ℑ ( s ) |≤ T θE − s · Φ E s λ s − λ w d s = Z ( ) θE − s · Φ E s λ s − λ w d s since the limit is approached in E ε ⊂ M . Q.E.D.
Further specific applications to u θ,w will be presented later as corollaries to thefollowing theorem. Theorem 43.
With continuous linear
Φ : M → N , with N quasi-complete andlocally convex, the Φ M -valued function w → u θ,w, Φ has a meromorphic continu-ation as an N -valued function of w . In further detail, the function J θ,w, Φ = θ (1) · Φ(1)( λ − λ w ) · h , i + 14 πi Z ( ) θE − s · Φ E s − θE − w · Φ E w λ s − λ w d s has a meromorphic continuation to an N -valued function with the functional equa-tion J θ, − w, Φ = J θ,w, Φ , and u θ,w, Φ = J θ,w, Φ + θE − w · Φ E w − w ) Remark 44.
The continuation assertion for u θ,w itself is stronger than, for example,the assertion of meromorphic continuation of the numerical, pointwise integrals u θ,w ( z o ) = θ (1) · λ − λ w ) · h , i + 14 πi Z ( ) θE − s · E s ( z o ) λ s − λ w d s for fixed z o ∈ H Proof.
From the lemma, in the region ℜ ( w ) > and w = 1 , the expression for u θ,w, Φ converges as an N -valued integral. The meromorphic continuation will beombieri and Garrett 55obtained through rearrangement of the integral. First, in ℜ ( w ) > and w = 1 , wecan certainly add and subtract to obtain u θ,w, Φ = θ (1) · Φ(1)( λ − λ w ) · h , i + 14 πi Z ( ) θE − s · Φ E s λ s − λ w d s = θ (1) · Φ(1)( λ − λ w ) · h , i + 14 πi Z ( ) θE − s · Φ E s − θE − w · Φ E w λ s − λ w d s + θE − w · Φ E w πi Z ( ) d sλ s − λ w = J θ,w, Φ + θE − w · Φ E w πi Z ( ) d sλ s − λ w By residues, θE − w · Φ E w πi Z ( ) d sλ s − λ w = θE − w · Φ E w (cid:16) − · Res s = w λ s − λ w (cid:17) = θE − w · Φ E w − w ) Since w → E − w is a meromorphic M -valued function and θ ∈ M ∗ , the function w → θE − w is a meromorphic C -valued function. Similarly, w → Φ E w is mero-morphic N -valued. Thus, θE − w · Φ E w is a meromorphic N -valued function, witha meromorphic continuation from the meromorphic continuation of the Eisensteinseries. Although the numerator is invariant under w → − w by the functionalequation of the Eisenstein series, the denominator is skew-symmetric.Now we meromorphically continue the integral J θ,w, Φ . Constrain w to lie ina fixed compact set C , and take T large enough so that T ≥ | w | for all w ∈ C . At first for ℜ ( w ) > , make the obvious attempt to cancel vanishing of the6 Designed pseudo-Laplaciansdenominator when s is close to w , by rearranging J θ,w, Φ − θ (1) · Φ(1)( λ − λ w ) · h , i = 14 πi Z ( ) θE − s · Φ E s − θE − w · Φ E w λ s − λ w d s = 14 πi Z | t |≥ T θE − s · Φ E s λ s − λ w d s + θE − w · Φ E w · πi Z | t |≥ T λ s − λ w d s + 14 πi Z | t |≤ T θE − s · Φ E s − θE − w · Φ E w λ s − λ w d s The meromorphy of the leading integral in the case of u θ,w itself is understood viathe Plancherel theorem on the continuous automorphic spectrum. Ignoring con-stants, the Plancherel theorem for E asserts that, for t → A ( t ) in L ( R ) , thespectral synthesis integral B ( z ) = 14 π Z ∞−∞ A ( t ) · E s ( z ) d s for z ∈ H produces a function B in E , and the map A → B gives an isometry.Since L functions in E certainly need not have moderate pointwise growth at in-finity, to have a continuous inclusion to M it is necessary to use E ε . For E r forgeneral index r , Plancherel becomes the following. Let X r be the measurable func-tions t → A ( t ) (modulo null functions) on R such that R R | A ( t ) | · ( + t ) r dt < ∞ .Then the spectral synthesis integral produces a function B in X r , and A → B gives an isometry E r → X r (ignoring constants). Since θ ∈ E − ε , for w in afixed compact, Z | t |≥ T (cid:12)(cid:12)(cid:12) θE − it λ + it − λ w (cid:12)(cid:12)(cid:12) · (cid:18)
14 + t (cid:19) ε d t < ∞ Indeed, the V ε o -valued function that is w → ( t → θE − it λ
12 + it − λ w ) for | t | ≥ T , and is for | t | < T , is directly seen to be complex differentiable M -valued in w , hence,holomorphic. Composition with the Plancherel isometry shows that w → πi Z | t |≥ T θE − s · E s λ s − λ w d s ombieri and Garrett 57is a meromorphic E ε -valued function in w in the fixed compact. Since | w | ≪ T ,the meromorphic continuation is given by the same integral, the invariance of theintegrand under w → − w remains. This treats the first term for u θ,w .To address the first term for u θ,w, Φ for continuous Φ : M → N , with inclusion E ε ⊂ M , use properties of compactly supported, continuous-integrand Gelfand-Pettis integrals from 6.1: Φ (cid:16) Z |ℑ ( s ) |≥ T θE − s · E s λ s − λ w d s (cid:17) = Φ (cid:16) lim T ′ → + ∞ Z T ′ ≥|ℑ ( s ) |≥ T θE − s · E s λ s − λ w d s (cid:17) = lim T ′ → + ∞ Φ Z T ′ ≥|ℑ ( s ) |≥ T θE − s · E s λ s − λ w d s = lim T ′ → + ∞ Z T ′ ≥|ℑ ( s ) |≥ T θE − s · Φ E s λ s − λ w d s = Z |ℑ ( s ) |≥ T θE − s · Φ E s λ s − λ w d s since the limit is approached in E ε ⊂ M . Thus, the meromorphic continuationin E ε of the first term for u θ,w gives that for u θ,w, Φ .In the second summand θE − w · Φ E w · πi Z | t |≥ T λ s − λ w d s the leading θE w · Φ E w has a meromorphic continuation and is invariant under w → − w . Since | w | ≪ T , the meromorphic continuation of the integrand is given bythe the same integral, and the invariance under w → − w remains.The remaining summand πi Z | t |≤ T θE − s · Φ E s − θE − w · Φ E w λ s − λ w d s is a compactly-supported vector-valued integral. To show that the integral is ameromorphic N -valued function of w , we invoke the Gelfand-Pettis criterion forexistence of a weak integral. Let Hol(Ω , N ) be the topological vector space ofholomorphic N -valued functions on a fixed open Ω avoiding poles of E w , withcompact closure C . It suffices to show that the integrand extends to a continuous Hol(Ω , N ) -valued function of s , where Hol(Ω , N ) has the natural quasi-completelocally convex topology as in section 6.2. Unsurprisingly, to show that the integrand8 Designed pseudo-Laplaciansextends to a holomorphic (hence, continuous) Hol(Ω , N ) -valued function of s , itsuffices to show that the integrand extends to a holomorphic N -valued function ofthe two complex variables s, w .By Cauchy-Goursat theory for vector-valued holomorphic functions, near a point s o , the N -valued function s → Φ E s has a convergent power series expansion Φ E s = A + A ( s − s o ) + A ( s − s o ) + . . . with A i ∈ N , and θE s has a scalar power series expansion θE s = θ ( A ) + θ ( A ) · ( s − s o ) + θ ( A ) · ( s − s o ) + . . . Thus, for suitable coefficients B n ∈ N , θE − s · Φ E s = B + B ( s − s o ) + B ( s − s o ) + . . . Then θE − s · E s − θE − w · Φ E w = B (( s − s o ) − ( w − s o )) + B (( s − s o ) − ( w − s o ) ) + . . . = (( s − s o ) − ( w − s o )) · (convergent power series in s − s o , w − s o ) = ( s − w ) · (convergent power series in s − s o , w − s o )Holomorphy is a local property. Thus, the integrand, initially defined only for s = w , extends to a holomorphic N -valued function F ( s, w ) including the diagonal s = w = + it with | t | ≤ T , as well. Thus, the H ol (Ω , N ) -valued function f ( s ) given by f ( s )( w ) = F ( s, w ) is holomorphic in w . Thus, there is a Gelfand-Pettisintegral R | t |≤ T f ( + it ) dt in Hol(Ω , N ) , as desired. This proves the meromorphiccontinuation. Further, the w → − w symmetry is retained by the extension of theintegrand to the diagonal. Q.E.D.
Corollary 45.
For
Φ : M → N a continuous map from the space M of moderate-growth continuous functions on Γ \ H to a quasi-complete, locally convex topologi-cal vector space N , the meromorphic continuations satisfy Φ( u θ,w ) = u θ,w, Φ Proof. In ℜ ( w ) > , lemma 42 proves the asserted equality. Then the vector-valued version of the identity principle of complex analysis gives equality of themeromorphic continuations. Q.E.D. ombieri and Garrett 59
Corollary 46.
With θ ∈ E − ε for some ε > , for ℜ ( w ) < , u θ,w = u θ, − w + θE w · E − w − w In particular, for ℜ ( w ) < , u θ,w ∈ E ε if and only if θE − w = 0 .Proof. First, θE − w · E w = θE w · E − w , using the functional equations E − w = E w /c w and c w c − w = 1 . Then, from the theorem, with ℜ ( w ) < , u θ,w = J θ,w + θE − w · E w − w ) = J θ, − w − θE w · E − w − − w ))= (cid:16) J θ, − w + θE w · E − w − − w )) (cid:17) − θE w · E − w − − w )) = u θ, − w − θE w · E − w − − w )) Since ℜ (1 − w ) > , u θ, − w ∈ E ε , so u θ,w ∈ E ε if and only if the extrasummand vanishes. Q.E.D.
Again, taking
Φ = θ in the above, let J θ,w = J θ,w,θ = θ (1) · λ − λ w ) · h , i + 14 πi Z ( ) θE − s · θE s − θE − w · θE w λ s − λ w d s Corollary 47.
For ℜ ( w ) = , ℜ θu θ,w = J θ,w and ℑ θu θ,w = − θE − w · θE w / ℑ ( w ) .Proof. In Theorem 43, with
Φ = θ , on ℜ ( w ) = , complex conjugation appliedto (the meromorphic continuation of the) the integral for J θ,w,θ sends w → − w ,under which J θ,w,θ is invariant, since θ = θ . On the other hand, by the functionalequation of E w , the term θE − w · θE w / − w ) is purely imaginary. Q.E.D.
Corollary 48.
With θ ∈ E − ε for some ε > , at points w with ℜ ( w ) ≤ , themeromorphic continuation of u θ,w as M -valued function of w is in E ε if andonly if θE w = 0 .Proof. The theorem 43 with Φ the identity map of M to itself gives the meromor-phic continuation of u θ,w as an M -valued function. For θE w = 0 , the extra term0 Designed pseudo-Laplaciansinvolving Eisenstein series in the expression u θ,w = J θ,w + θE − w · E w − w ) disappears. Further, in the numerator in the integral J θ,w = θ (1) · λ − λ w + 14 πi Z ( ) θE − s · E s − θE − w · E w λ s − λ w d s the term θE w vanishes. Thus, the spectral coefficient s → θE − s / ( λ s − λ w ) in theintegrand is in X ε . By the extended Plancherel theorem, the integral (extendedby continuity) is in E ε . Q.E.D.
Corollary 49.
With θ ∈ E − ε for some ε > , if the Friedrichs extension e S θ has an eigenfunction u with eigenvalue λ w > , then u is a multiple of the mero-morphic continuation of u θ,w to ℜ ( w ) = . Further, θE w = 0 = θE − w and θu θ,w = 0 . Conversely, if θE w = 0 = θE − w and θu θ,w = 0 , then u θ,w is aneigenfunction of e S θ .Proof. From theorem 11, if λ w > is an eigenvalue for e S θ , then θE w = 0 , and ℜ ( w ) = . Then, also, θE − w = 0 by the functional equation of the Eisensteinseries. Thus, u θ,w ∈ E ε , by the previous. Then for some constant c , v = u − c · u θ,w satisfies the homogeneous equation ( − ∆ − λ w ) v = 0 , which hasno non-zero solution in E . For u ∈ E to be in the domain of the Friedrichsextension, it is necessary and sufficient that θu = 0 . Q.E.D.
As a special case of the theorem 43 with
Φ = θ : M → C , whose relevance isclearer in light of the two previous corollaries, we have Corollary 50.
The condition θu θ,w = 0 on the meromorphically continued u θ,w is u θ,w,θ = 0 with the meromorphic continuation of u θ,w,θ as in theorem 43. Remark 51.
If we did not know that any solution u ∈ E ε to the differentialequation were of the form u θ,w , it would be more difficult to appraise the vanishingrequirement θu = 0 . Corollary 52.
A necessary and sufficient condition for λ w > to be an eigenvaluefor e S θ is that θu θ,w = 0 . ombieri and Garrett 61 Proof.
By corollary 49, in that range, the eigenfunctions for e S θ are exactly u θ,w with θE w = 0 and θu θ,w = 0 . As above, θu θ,θ = θ (1) · θ (1)( λ − λ w ) · h , i + 14 πi Z ( ) θE − s · θE s − θE − w · θE w λ s − λ w d s + θE − w · θE w − w ) On ℜ ( w ) = , the first two summands are real, while the third is purely imaginary.Thus, for this to vanish on ℜ ( w ) = requires vanishing of θE w . That is, on ℜ ( w ) = , vanishing of θu θ,w implies that of θE w . Q.E.D.
For example, the self-adjoint operator e S Θ on non-cuspidal pseudo-cuspforms,has purely discrete spectrum, by Theorem 16. Theorem 18 already noted that, for ℜ ( w ) = , λ w = w (1 − w ) is an eigenvalue if and only if a w + c w a − w = 0 .Indeed, λ w = w (1 − w ) is real if and only if ℜ ( w ) = or w ∈ [0 , . However, thisconclusion does not address the possible vanishing of a w + c w a − w off ℜ ( w ) = and [0 , . As an instance of desirable corollaries of spectral theory, we have a resultalready observed in [17], but there for non-spectral reasons: Corollary 53. a w + c w a − w = 0 implies ℜ ( w ) = or w ∈ [0 , .Proof. Let u η a ,w be the meromorphic continuation of the spectral expansion u η a ,w = η a (1) · λ − λ w ) · h , i + 14 πi Z ( ) η a E − s · E s λ s − λ w d s By Corollary 49, u η a ,w gives an eigenfunction of e S Θ if and only if η a E w = 0 and η a u η a ,w = 0 . Of course, η a E w = a w + c w a − w . The computation of Theorem 24gives η a u η a ,w = a − w − w · ( a w + c w a − w ) The leading factor a − w / (1 − w ) does not vanish, and does not have poles exceptat w = . Thus, if a w + c w a − w = 0 , then u η a ,w is an eigenfunction for e S Θ ,and necessarily λ w is real and non-negative. Thus, for w outside the set where ℜ ( w ) = or w ∈ [0 , , the expression a w + c w a − w cannot vanish. Q.E.D.
The invariance of λ w under w ←→ − w might suggest invariance of u θ,w un-der w ←→ − w . However, this is illusory, by corollary 46. Nevertheless, theimplications of (the generally incorrect claim) u θ, − w = u θ,w are striking. That is,if we were to believe that u θ, − w = u θ,w , the the trivial non-vanishing of remark14 would (fairly provocatively, but incorrectly, seem to) show that θu θ,w = 0 off ℜ ( w ) = and [0 , .Corollary 52 (correctly) shows that for λ w > , that is, for ℜ ( w ) = and w = , a necessary and sufficient condition for λ w to be an eigenvalue for the self-adjoint operator e S θ is that θu θ,w = 0 . An argument principle discussion (goingback to [1], see [27] pp. 212-213) shows that the asymptotic count of the zeros w of θu θ,w with imaginary parts ≤ ℑ ( w ) ≤ T is Tπ log T + O ( T ) .Thus, if we (erroneously) believe the confinement of zeros of θu θ,w to ℜ ( w ) = (and [0 , ), we (seem to) find that θu θ,w asymptotically has Tπ log T + O ( T ) zerosfrom height to T on the critical line. Since θu θ,w = 0 on the critical line doestruly entail θE w = 0 , we (seem to) conclude that the asymptotic count of zerosof θE w on is Tπ log T + O ( T ) . For θ the Eisenstein-Heegner distribution attachedto a fundamental discriminant − d < and k = Q ( √− d ) , this would (seem to)imply that ζ k ( s ) has of its zeros on the critical line, the asymptotic versionof the Riemann Hypothesis for ζ k ( s ) . The argument is fundamentally flawed, at thepoint where one fallaciously imagines that u θ, − w = u θ,w : the functional equationof u θ,w involves an extra term, which vanishes exactly for θE w = 0 , thus thwartingthis naively optimistic approach. Fix a > . Let { f j : j = 1 , , . . . } be eigenfunctions for the operator e S Θ ofsection 4.2, with eigenvalues λ s ≤ λ s ≤ . . . , with | f j | V = 1 . As in Corollary 21, | f j | E = p λ s j . Let i Θ : E ∩ E → E be the inclusion, and i ∗ Θ : E − → ( E ∩ E ) ∗ its adjoint. By Theorem 20 , E ∩ E is the E -topology completion of D ∩ E ,where D is the space of pseudo-Eisenstein series with test-function data. Thus, { f j } is an orthonormal basis for E , and { f j / p λ s j } is an orthonormal basis forombieri and Garrett 63 E ∩ E .At first for finite sums, define Sobolev-like norms by (cid:12)(cid:12)(cid:12) X j c j · f j (cid:12)(cid:12)(cid:12) W r = X j | c j | · λ rs j and let W r be the completion of the space of finite linear combinations of the vec-tors f j with respect to this norm. Thus, { f j · λ − r/ s j } is an orthonormal basis for W r . Theorem 54.
The map W − ≈ ( E ∩ E ) ∗ given by (at first for finite sums, andthen extending by continuity) (cid:16) X i a i · f i (cid:17)(cid:16) X j b j · f j (cid:17) = X j a i · b j for P j b j · f j ∈ E ∩ E is an isomorphism.Proof. First, since f j ∈ E by the Friedrichs construction, and f j ∈ E , it makessense to apply such functionals in E − to the eigenfunctions f j . By Plancherel andCorollary 21, W = E and W = E ∩ E . Generally, the natural pairing h X j a j · f j , X j b j · f j i = X j a j b j on W r × W − r puts these two spaces in duality. Thus, we have a natural isomorphism W − → W ∗ = ( E ∩ E ) ∗ respecting the duality pairings. Q.E.D.
Thus, functionals µ ∈ ( E ∩ E ) ∗ admit expansions µ = X j µ ( f j ) · f j convergent in the W − topology, and compatible with evaluation on the correspond-ing expansions of elements in W = E ∩ E . Let { f j : j = 1 , , . . . } be eigenfunctions for the operator e S Θ of section 4.2, witheigenvalues λ s ≤ λ s ≤ . . . , with | f j | V = 1 .4 Designed pseudo-Laplacians Proposition 55.
The operator S of Lemma 7 is a topological isomorphism from W = E ∩ E to W − = ( E ∩ E ) ∗ , expressible as S X j b j · f j = X j b j · S f j = X j λ s j · b j · f j Proof.
Since constants are excluded from E , S Θ is strictly positive, so we arein the situation of section 2.2, and its Friedrichs extension gives the asserted iso-morphism, as observed just prior to Proposition 1. Because S is continuous W → W − , it commutes with limits, giving the formula. Q.E.D.
Theorem 56.
For µ ∈ W − = ( E ∩ E ) ∗ , for λ w [0 , + ∞ ) , the equation ( e S Θ − λ w ) v = µ has a unique solution in W = E ∩ E , given by v = X j µ ( f j ) · f j λ s j − λ w This expansion is holomorphic W -valued off ℜ ( w ) = , and has a W -valuedanalytic continuation to C with the discrete set of spectral parameters s j removed.Proof. First, by the construction of the Friedrichs extension, all eigenfunctions f j are in E , so any θ ∈ E − can be sensibly applied to them via the E × E − pairing.The bound X j | µ ( f k ) | · λ − s j < ∞ implies X j (cid:12)(cid:12)(cid:12) µ ( f j ) λ s j − λ w (cid:12)(cid:12)(cid:12) · λ s j < ∞ so the given expansion for v is indeed in W . Since S : W → W − is continu-ous, it commutes with the implied limits in the infinite sums.To show the holomorphy, from section 6.2 it suffices to prove that for w in a fixedcompact C not meeting R , for every ε > there is i o such that for all i , i ≥ i o sup w ∈ C (cid:12)(cid:12)(cid:12) X i ≤ j ≤ i µ ( f j ) λ s j − λ w (cid:12)(cid:12)(cid:12) W < ε ombieri and Garrett 65This follows from | λ s j − λ w | ≫ C | λ s j | . By Theorem 16, e S Θ has compact resolvent,so the parameters s j have no accumulation point in C . Thus, the same inequalityholds for C compact not meeting the set of spectral parameters s j , giving the ana-lytic continuation. Q.E.D.
Let θ = θ = θ D = P d ν d θ d be a finite real-linear combination of Heegner dis-tributions θ d , as in 3.6, with fundamental discriminants d < . As noted in Section3.6, θ ∈ E − ε for some ε > . Let m ( D ) = max d | d | / be the maximum over d appearing with non-zero coefficient, and increase a if necessary so that a > . Corollary 57.
The equation ( S − λ w ) v = i ∗ Θ θ has unique solution v θ,w = X j θ ( f j ) λ s j − λ s · f j in W = E ∩ E , a meromorphic W -valued function of w away from the spectralparameters s j , and θv θ,w = X j | θf | λ s j − λ s · f j is uniformly absolutely convergent on compacts away from the spectral parameters s j , producing a holomorphic C -valued function there.Proof. The image i ∗ Θ θ is in W − . Q.E.D.
Corollary 58.
A solution u ∈ E to the equation ( − ∆ − λ w ) u = θ is in fact in E ∩ E , so is v θ,w of the previous corollary, so has an expansion u = X j θ ( f j ) λ s j − λ w · f j convergent in E ∩ E , holomorphic in w away from spectral parameters s j , andthere θu = X j | θ ( f j ) | λ s j − λ w Proof.
First, we recall why such a solution u must be of the form u θ,w . Certainly in ℜ ( w ) > (and w = 1 ) that equation has the solution given by a spectral expansion6 Designed pseudo-Laplaciansconverging in E : u θ,w = θ (1) · λ − λ w ) · h , i + 14 πi Z ( ) θE − s · E s λ s − λ w d s On ℜ ( s ) = , Theorem 10 shows that the equation is solvable only if E θ ( w ) = 0 ,and Corollary 40 gives θE − w = E θ ( w ) = 0 . Thus, by Theorem 48, at such w the meromorphic continuation of u θ,w is in E . Since the homogeneous form ofthe equation has no solution in E , it must be that u = u θ,w . From Theorem 26,the condition θE w = 0 gives u θ,w ∈ E ∩ E for a ≥ m ( D ) (and a > ). Theprevious corollary applies to the image µ = i ∗ Θ θ , noting that for f ∈ E ∩ E , ( i ∗ Θ θ )( f ) = θ ( i Θ f ) = θ ( f ) suppressing explicit reference to the inclusion i Θ . Then u θ,w must be the solution v θ,w of the previous corollary. Q.E.D.
Continue to let θ = θ = θ D = P d ν d θ d be a finite real-linear combination ofEisenstein-Heegner distributions θ d , as in 3.6, so that all the previous results apply.Let v θ,w = X j θ ( f j ) λ s j − λ s · f j be as in Corollary 57. Corollary 59.
Let s j = + it j and s j +1 = + it j +1 be two adjacent zerosof a s + c s a − s on the line ℜ ( s ) = , with < t j < t j +1 . For w = + iτ with t j < τ < t j +1 , the function τ → θv θ,w is continuous, real-valued, strictlyincreasing, goes from −∞ as τ → t j to + ∞ as τ → t j +1 .Proof. For τ ∈ R , θv θ, + iτ = X j | θ ( f j ) | t j − τ is real-valued (away from the t j ). Being the restriction of a meromorphic function, τ → θv θ, + iτ is continuous away from the t j . On one hand, for τ ∈ ( t j , t j +1 ) ,as τ → t + j , the summand | θ ( f j ) | / ( t j − τ ) goes to −∞ and the rest of the sumombieri and Garrett 67remains finite. On the other hand, in that interval, as τ → t − j +1 , the summand | θ ( f j +1 ) | / ( t j +1 − τ ) goes to + ∞ and the rest of the the sum remains finite. Awayfrom the poles, the derivative is ∂∂τ θv θ, + iτ = ∂∂τ X j | θ ( f j ) | t j − τ = 2 τ · X j | θ ( f j ) | ( t j − τ ) > so the function is strictly increasing. Q.E.D.
The first corollary directly addresses on-the-line zeros of θE w : Corollary 60.
Let s j = + it j and s j +1 = + it j +1 be adjacent zeros of a s + c s a − s on the line ℜ ( s ) = , with < t j < t j +1 . Let w = + iτ and w = + iτ with t j < τ < τ < t j +1 with θE w j = 0 . Then θu θ,w < θu θ,w .Proof. As above, the condition θE w = 0 implies that the meromorphic continuationof u θ,w is in E , and also that u θ,w ∈ E . By Corollary 57, u θ,w must be v θ,w , and θv θ,w is monotone in intervals ( t j , t j +1 ) . Q.E.D.
This second corollary is a variant that more directly addresses solutions of theequation ( − ∆ − λ w ) u = θ : Corollary 61.
Let s j and s j +1 be adjacent zeros of a s + c s a − s on the line ℜ ( s ) = . On the line ℜ ( w ) = , between s j and s j +1 , there is at most one w such that asolution u ∈ E of the equation ( − ∆ − λ w ) u = θ satisfies θu = 0 .Proof. Again, if a solution u to ( − ∆ − λ w ) u = θ is in E , then θE w = 0 ,by Theorem 10. And, again, the meromorphically-continued u θ,w is in E at thatpoint, and is the unique solution to the equation, since the homogeneous equation ( − ∆ − λ w ) u = 0 has no non-zero solution. Also, again, θE w = 0 implies that u θ,w ∈ E . Thus, by Corollary 58, u θ,w = v θ,v , and then θu θ,w = θv θ,w = X j | θ ( f j ) | λ s j − λ w The monotonicity assertion of Corollary 59 shows that there is exactly one w = + iτ in the given interval such that θv θ,w = 0 . Q.E.D.
Corollary 62.
There is at most one w on ℜ ( w ) = between adjacent zeros s j and s j +1 of a s + c s a − s on ℜ ( s ) = such that λ w is an eigenvalue for e S θ .Proof. Combine the previous corollary with Corollary 52.
Q.E.D.
Continue to let θ = θ = θ D = P d ν d θ d be a finite real-linear combination ofEisenstein-Heegner distributions θ d , as in 3.6, so that the immediately previousresults apply.From the previous section, the positions of the zeros of a s + c s a − s influence thepositions of the parameters w appearing in differential equations ( − ∆ − λ w ) u = θ with condition θu = c ∈ R . The zeros s of a s + c s a − s = 0 allow a degree ofadjustment by choice of the cut-off height a > .To anticipate the point of the present discussion, for log log ℑ ( s ) large, the be-havior of ζ ( s ) on the edge of the critical strip is relatively regular, by [27] (5.17.4)page 112 (in an earlier edition, page 98). This gives an eventual regularity of spac-ing of the zeros of a s + c s a − s , with implications for the exotic eigenfunction ex-pansions above.As originally in [1], or from [27] pages 212–213, by the argument principle andJensen’s inequality, (cid:16) zeros of a s + c s a − s with ≤ ℑ ( s ) ≤ T (cid:17) = Tπ log T + O ( T ) Thus, near height T , the average gap is π/ log T . As recalled in Corollary 53, allthe zeros are on + i R or [0 , , and are in bijection with the discrete spectrum of e S Θ (modulo w ↔ − s ) by s → λ s = s (1 − s ) .First, | c s | = 1 on ℜ ( s ) = , so c + it = e − iψ ( t ) with real-valued ψ ( t ) =arg ξ (1 + 2 it ) . Since ζ ( s ) does not vanish on ℜ ( s ) = 1 , ψ ( t ) is differentiable.Letting s = = it , rearrange a s + c s a − s = √ a · e − iψ ( t ) · (cid:16) e it log a + iψ ( t ) + e − it log a − iψ ( t ) (cid:17) = 2 √ a · e − iψ ( t ) · cos (cid:16) t log a + ψ ( t ) (cid:17) Thus, the on-the-line vanishing condition is cos( t log a + ψ ( t )) = 0 . Proposition 63. ψ ( t ) = arg ξ (1 + 2 it ) satisfies ψ ( t ) = t log t + O ( t log t log log t ) and ψ ′ ( t ) = log t + O ( log t log log t ) ombieri and Garrett 69 Proof.
Of course, ψ ( t ) = arg ξ (1 + 2 it ) = − t log π + arg Γ( 12 + it ) + arg ζ (1 + 2 it ) From log Γ( s ) = ( s − ) log s − s + log 2 π + O ( s − )log Γ( 12 + it ) = it log(1 + it ) − ( 12 + it ) + 12 log 2 π + O (cid:16) it (cid:17) = it (cid:16) log t + O ( 1 t ) + i ( π O ( 1 t )) (cid:17) − ( 12 + it ) + 12 log 2 π + O (cid:16) + it (cid:17) Thus, ℑ log Γ( 12 + it ) = t log t − t + O ( 1 t ) and ψ ( t ) = ( − t log π ) + ( t log t − t ) + arg ζ (1 + 2 it ) + O ( t − ) Similarly, the asymptotic Γ ′ ( s ) / Γ( s ) = log s + O ( s − ) gives ℑ dds log Γ( 12 + it ) = log t + O ( t − ) and ψ ′ ( t ) = ( − log π ) + (log t ) + ddt arg ζ (1 + 2 it ) + O ( t − ) From [27] (5.17.4) page 112 (in an earlier edition, page 98), for u ≥ t , log ζ (1 + iu ) − log ζ (1 + it ) = O (cid:16) log t log log t (cid:17) · ( u − t ) This gives the asymptotics for ψ and ψ ′ . Q.E.D.
Corollary 64.
Fix a o > . Given ε > (with ε < for definiteness), for suffi-ciently large T o > , for all < a ≤ a o , for consecutive real zeros t j < t j +1 of a + it + c + it a − it with T o ≤ t j < t j +1 , (1 − ε ) · π log t j ≤ t j +1 − t j ≤ (1 + ε ) π log t j Proof.
This is elementary from the previous.
Q.E.D.
Theorem 65.
The real zeros t = t ( a ) of a + it + c + it a − it , which are also the realzeros of cos( t log a + ψ ( t )) , are differentiable functions of cut-off height a , with ∂t∂a = − t/a log a + ψ ′ ( t ) with non-vanishing denominator.Proof. Implicit differentiation of cos( t log a + ψ ( t )) = 0 gives the formula. Non-vanishing of the denominator at points where cos( t log a + ψ ( t ) = 0 follows fromthe Maaß-Selberg relation, as follows. On one hand, for ℑ ( s ) > the higherFourier terms of E s do not vanish identically, so h∧ a E s , ∧ a E s i > . On the otherhand, the Maaß-Selberg relation is h∧ a E s , ∧ a E r i = a s + r − s + r − c s a (1 − s )+ r − (1 − s ) + r − c r a s +(1 − r ) − s + (1 − r ) − c s c r a (1 − s )+(1 − r ) − (1 − s ) + (1 − r ) − For s = + it + ε and r = + it , this is h∧ a E s , ∧ a E r i = a ε ε + c s a − it − ε − it − ε + c r a it + ε it + ε + c s c r a − ε − ε As ε → , the two middle terms go to c + it a − it − it + c − it a it it = e − iψ ( t ) a − it − it + e ψ ( t ) a it it = 1 t · sin(2 log a + 2 ψ ( t )) The vanishing condition is cos( t log a + ψ ( t )) = 0 , so t log a + ψ ( t ) ∈ π Z , so t log a + 2 ψ ( t ) ∈ π Z , and the sine function vanishes there. That is, the sum ofthese two terms is .Modulo O ( ε ) , the sum of the first and last terms is ε ( a ε − c + it + ε c − it a − ε ) = 1 ε (cid:16) (1 + ε log a ) − ( c + it + εc ′ + it ) c − it (1 − ε log a ) (cid:17) = 1 ε (cid:16) ε log a − (1 + εc ′ + it c − it )(1 − ε log a ) (cid:17) = 2 log a − c ′ + it c + it ombieri and Garrett 71since c s c − s = 1 . Since c + it = e − iψ ( t ) and c s is holomorphic, c ′ + it = dd ( it ) c + it = − i ddt c + it = − i ddt e − iψ ( t ) = − ψ ′ ( t ) · e − iψ ( t ) Thus, c ′ + it /c + it = − ψ ′ ( t ) . That is, ε → , these terms go to a + 2 ψ ′ ( t ) .That is, when a s + c s a s = 0 , < h∧ a E s , ∧ a E s i = 2 log a + 2 ψ ′ ( t ) giving the non-vanishing of the denominator. Q.E.D.
Thus, to change a given zero t for cut-off height a by the expected gap amount π/ log t at that height, change a by roughlygapderivative ∼ π/ (log a + log t )( − t/a ) / (log a + log t ) ∼ − πat Corollary 66.
Fix a o > . Given ε > (and ε < ), there is T o sufficiently largesuch that, for every a in the range < a ≤ a o , for every real zero t = t ( a ) of a + it + c + it a − it , (1 − ε ) · t/a log t ≤ − ∂t∂a ≤ (1 + ε ) t/a log t Proof.
Elementary from the above.
Q.E.D.
Combining the previous with Corollary 60:
Theorem 67.
Given ε > , there is T o > sufficiently large such that, for twozeros w = + iτ and w = + iτ of θE w with T o < τ < τ , if θu θ,w ≥ θu θ,w ,then we have the lower bound | τ − τ | ≥ (1 − ε ) π/ log τ .Proof. Elementary from the above.
Q.E.D.
As a very special case, relevant to the discrete spectrum (if any) of e S θ : Corollary 68.
Given ε > , there is T > sufficiently large such that, for twozeros w = + iτ and w = + iτ of θE w with T < τ < τ , if θu θ,w =0 = θu θ,w , then we have the lower bound | τ − τ | ≥ (1 − ε ) π/ log T . Thatis, the discrete spectrum, if any, of e S θ , is λ w j with a lower bound | w j +1 − w j | ≥ (1 − ε ) π/ log T for adjacent w j and w j +1 at height T .Proof. This is a special case of the previous theorem and Corollary 52.
Q.E.D.
The asymptotic lower bound π/ log T on spacing of consecutive spectral param-eters w j at height T for eigenvalues λ w j of e S θ is in conflict with Montgomery’spair-correlation [21]. We continue to take θ to be a real-linear combination ofEisenstein-Heegner distributions, so that all previous results apply.For example, assuming the Riemann Hypothesis and pair correlation : Corollary 69.
At most 94% of the zeros of ζ ( s ) give eigenvalues λ s = s (1 − s ) for e S θ . Remark 70.
This shows that the optimistic simple version of the conjecture at theend of [5] cannot hold. Namely, with θ the Eisenstain-Dirac δ at e πi/ , it cannotbe the case that all zeros ρ j of ζ ( s ) give eigenvalues for e S θ . Remark 71.
Unless we believe that a subset of on-the-line zeros of ζ ( s ) is naturallydistinguished, this might suggest that the discrete spectrum is empty.Proof. Let the imaginary parts of zeros be . . . ≤ γ − < < γ ≤ γ ≤ . . . .Montgomery’s pair correlation conjecture is that, for ≤ α < β , n m < n : 0 ≤ γ m , γ n ≤ T with πα log T ≤ γ n − γ m ≤ πβ log T o ∼ Z βα (cid:16) − (cid:0) sin πuπu (cid:1) (cid:17) d u For example, the asymptotic fraction of pairs of zeros within half the average spac-ing π log T up to height T is Z (cid:16) − (cid:0) sin πuπu (cid:1) (cid:17) d u ≈ . > From the lower bound in the previous section, for at least one of every such pair ( m, n ) the corresponding zero cannot appear among discrete spectrum parameters w for e S θ . Q.E.D. ζ k ( s ) Without assumptions on zeros ζ k ( s ) as spectral parameters for any of the pseudo-Laplacians above, we have non-trivial corollaries on spacing of those zeros.ombieri and Garrett 73 Let θ continue to be a finite real-linear combination of Eisenstein-Heegner distribu-tions. For this section, assume that the cut-off height a is above the highest Heegnerpoint involved in θ , so that the determinant vanishing condition η ( v w,a ) θ ( u θ,w ) − η a ( u θ,w ) = 0 of (5.2), amplified as in the computations leading up to Corollary 28,simplifies to a − w ( a w + c w a − w ) × (cid:16) | θ (1) | h , i · λ − λ w ) + 14 πi Z ( ) | θE s | d sλ s − λ w (cid:17) = 12 w − (cid:0) a − w θE w ) (cid:1) (8.1)where θE s = X d ν d (cid:18) p | d | (cid:19) s ζ ( s ) ζ (2 s ) L ( s, χ d ) (8.2)To apply the meromorphic continuation results of subsections 5.7 or 6.5, let J θ,w = θ (1) · λ − λ w ) · h , i + 14 πi Z ( ) θE − s · E s − θE − w · E w λ s − λ w d s This allows elementary algebraic rearrangement of the determinant-vanishing con-dition to a symmetrized form, as in subsection 5.7: ( a w − + c w a − w ) · J θ,w = ( a w − − c w a − w ) · θE − w · θE w − w ) On the critical line w = + iτ , letting ψ ( τ ) = arg ξ (1 + 2 iτ ) as above, this is cos (cid:16) τ log a + ψ ( τ ) (cid:17) · J θ,w = sin (cid:16) τ log a + ψ ( τ ) (cid:17) · θE − w · θE w − τ Let S Θ ,θ be the restriction of − ∆ to test functions in the Lax-Phillips space L a (Γ \ H ) of pseudo-cuspforms which are also annihilated by θ . Let e S Θ ,θ be its Friedrichs ex-tension. We have a variant of earlier results:4 Designed pseudo-Laplacians Theorem 72.
The parameters w = + iτ for eigenvalues λ w = w (1 − w ) < − / of e S Θ ,θ are the solutions of cos (cid:16) τ log a + ψ ( τ ) (cid:17) · J θ,w = sin (cid:16) τ log a + ψ ( τ ) (cid:17) · θE − w · θE w − τ For such w , the eigenfunction is a linear combination Au θ,w + Bv a,w of the mero-morphic continuations of u θ,w and v a,w , where A, B are not both , such that θ ( Au θ,w + Bv a,w ) = 0 = η a ( Au θ,w + Bv a,w ) Proof.
The equation (∆ − λ ) u = 0 has no solution u ∈ E ∩ E except u = 0 , sothe only possible λ w -eigenvectors are solutions u ∈ E to equations (∆ − λ w ) u = A · θ + B · η a (with not both constants ), with the additional condition θu = 0 = η a u . In termsof spectral expansions of elements of E , as in subsection 4.1, if λ w < − / isan eigenvalue, then ( Aθ + Bη a ) E w = 0 . From the vector-valued meromorphiccontinuation results of subsection 6.5, the meromorphic continuation of Au θ,w + Bv a,w = Aθ + Bη a )(1) · λ − λ w ) · h , i + 14 πi Z ( ) ( Aθ + Bη a ) E − s · E s λ s − λ w ds at w with ℜ ( w ) ≤ is in E when ( Aθ + Bη a ) E w = 0 . Further, η b ( Au θ,w + Bv a,w ) = 0 for all b ≥ a ≫ θ . Thus, when λ w < − / is an eigenvalue, somenon-zero Au θ,w + Bv a,w is the eigenfunction. There exist such A, B , not both , exactly when the relevant two-by-two determinant vanishes, as above. That is, λ w < − / is an eigenvalue exactly when cos (cid:16) τ log a + ψ ( τ ) (cid:17) · J θ,w = sin (cid:16) τ log a + ψ ( τ ) (cid:17) · θE − w · θE w − τ and the eigenfunction is a linear combination Au θ,w + Bv a,w of the meromorphiccontinuations. Q.E.D.
Via the Baire category theorem, given a ≥ and ε > , we can increase thecut-off height a by an amount less than ε so that θE s j = 0 for all parameters s j with η a E s j = 0 , that is, for all eigenfunctions of the form ∧ a E s j for e S Θ . In thesequel, assume that such an adjustment has been made.We have the interleaving result: ombieri and Garrett 75 Theorem 73.
Between two consecutive zeros τ ( a ) j and τ ( a ) j +1 of cos( τ log a + ψ ( τ )) there is a unique τ such that λ + iτ is an eigenvalue of e S Θ ,θ .Proof. Since η a ( Au θ,w + Bv a,w ) = 0 implies that f w = Au θ,w + Bv a,w is in E ∩ E , this f w is expressible in terms of the orthonormal basis of eigenfunctions ϕ j for e S Θ , in an expansion converging in E : f w = X i h f w , ϕ i i · ϕ i Since we have arranged that θE s j = 0 for all s j , necessarily A = 0 , so A =1 without loss of generality. As earlier, let ϕ i have eigenvalue s i (1 − s i ) , with ℑ ( s i ) ≤ ℑ ( s i +1 ) , and all ϕ i real-valued. Let K be the kernel of η a on E ∩ E ,and j : K → E the inclusion, with adjoint j ∗ : E − → K ∗ . Let e S =: K −→ K ∗ be as in subsection 2.2. Then ( e S − λ w ) f w = j ∗ ( θ + Bη a ) . Conveniently, j ∗ ( θ + Bη a ) = j ∗ θ + B · j ∗ η a = j ∗ θ + B · j ∗ θ The image j ∗ ( θ + Bη a ) = j ∗ θ in K ∗ admits an eigenfunction expansion j ∗ ( θ + Bη a ) = j ∗ θ = X i ( j ∗ θ )( ϕ i ) · ϕ i = X i θ ( jϕ i ) · ϕ i and then X i j ∗ ( θ + Bη a )( ϕ i ) · ϕ i = X i ( j ∗ θ )( ϕ i ) · ϕ i = j ∗ θ = ( e S − λ w ) f w = ( e S − λ w ) X i h f w , ϕ i i · ϕ i = X i h f w , ϕ i i · ( e S − λ w ) ϕ i = X i h f w , ϕ i i · ( λ s i − λ w ) ϕ i Thus, h f w , ϕ i i = ( j ∗ θ ) ϕ i / ( λ s i − λ w ) , and we have an expansion convergent in E : f w = X i ( j ∗ θ )( ϕ i ) · ϕ i λ s i − λ w θ ( jf w ) = 0 for f w to be an eigenfunction for e S Θ ,θ is θ ( jf w ) = ( j ∗ θ ) f w = X i | ( j ∗ θ ) ϕ i | λ s i − λ w By the intermediate value theorem, there is at least one w between any two s i and s j +1 . The derivative of θf w never vanishes, and no ( j ∗ θ )( ϕ i ) is , so between s i and s j +1 there is exactly one w on + i R satisfying the equation. Q.E.D.
From the previous section, the necessary and sufficient condition on w = + iτ for λ w to be an eigenvalue of e S a,θ , equivalently, of e S ≥ a,θ is the vanishing condition cos (cid:16) τ log a + ψ ( τ ) (cid:17) · J ( w ) = sin (cid:16) τ log a + ψ ( τ ) (cid:17) · θE − w · θE w τ A simultaneous zero of J θ,w and θE w makes λ w an eigenvalue for e S θ already,hence of e S Θ ,θ and of e S a,θ , but we expect that there are few or no eigenvalues for e S θ . For λ w not a simultaneous zero of J θ,w and θE w , with w = + iτ , thisvanishing condition is equivalent to tan (cid:16) τ log a + ψ ( τ ) (cid:17) = 2 τ · J ( w ) θE w · θE − w At least for < τ ∈ R , let R ( τ log τ ) = 2 τ · J ( + iτ ) θE + iτ · θE − iτ denote the right-hand side. Proposition 74.
For an eigenvalue λ + iτ of e S ≥ a,θ (equivalently, of e S a,θ ), ∂τ∂a = τa ( R ( τ log τ ) + 1)(log τ + 1) · R ′ ( τ log τ ) − (log a + ψ ′ ( τ ))( R ( τ log τ ) + 1) ombieri and Garrett 77 Corollary 75.
For all large τ , R ′ ( τ log τ ) ≤ log a + ψ ′ ( τ )log τ + 1 · ( R ( τ log τ ) + 1) That is, for large τ , the graph of t → R ( t log t ) has slope essentially boundedabove by the slope of t → tan( t log t ) at the same height, since ψ ′ ( τ ) ∼ log τ .Proof. (Of claim) It makes sense to rescale the right-hand side of the eigenvaluecondition, expressing it as a function of something asymptotic to τ log τ , both be-cause of the analogous scaling of the left-hand side, and because of the verticalasymptotics of zeros of θE w . One simple choice is R ( τ log τ ) , although one mightuse R ( ψ ( τ )) instead. Using the simple rescaling, differentiating with respect tocut-off height a , (cid:16) τa + ∂τ∂a (log a + ψ ′ ( τ )) (cid:17) · sec (cid:16) τ log a + ψ ( τ ) (cid:17) = ∂τ∂a · (log τ + 1) · R ′ ( τ log τ ) so ∂τ∂a (cid:16) (log τ +1) · R ′ ( τ log τ ) − (log a + ψ ′ ( τ )) sec ( τ log a + ψ ( τ )) (cid:17) = τa sec ( τ log a + ψ ( τ )) and ∂τ∂a = τa sec ( τ log a + ψ ( τ ))(log τ + 1) · R ′ ( τ log τ ) − (log a + ψ ′ ( τ )) sec ( τ log a + ψ ( τ )) Since sec α = tan α + 1 , by the relation tan( τ log a + ψ ( τ )) = R ( τ log τ ) , atsuch points sec ( τ log a + ψ ( τ )) = R ( τ log τ ) + 1 . Thus, ∂τ∂a = τa ( R ( τ log τ ) + 1)(log τ + 1) · R ′ ( τ log τ ) − (log a + ψ ′ ( τ ))( R ( τ log τ ) + 1) which is the assertion of the claim. Q.E.D.Proof. (Of corollary)
From the min-max principle, when ∂τ /∂a it exists, it satisfies ∂τ /∂a ≤ . We also have the interleaving of parameters τ for eigenvalues λ + iτ of e S a,θ between those of e S Θ . The parameter values τ for e S Θ can be adjusted bychanging cut-off height a : changing a by about πa/τ moves the zeros by about8 Designed pseudo-Laplaciansthe average gap amount π/ log τ . Thus, given large τ , by the intermediate valuetheorem, we need increase a only slightly to make λ + iτ an eigenvalue for e S a,θ .Thus, R ′ ( τ log τ ) ≤ log a + ψ ′ ( τ )log τ + 1 · ( R ( τ log τ ) + 1) holds for all large τ . Q.E.D.
Without any assumption that zeros of θE w do or do not give eigenvalues λ w for aself-adjoint operator, we have illustrative corollaries about spacing of zeros of θE w . Corollary 76.
Let t < t ′ be large such that + it and + it ′ are adjacent on-line zeros of θE w , and neither a zero of J θ,w . Suppose there is a unique zero + iτ o of J θ, + iτ between + it and + it ′ , and ∂∂τ J θ, + iτ > . Then | t ′ − t | ≥ π log t · (1+ O ( t )) . That is, in this configuration, the distance between consecutivezeros must be at least the average. Corollary 77.
Let t < t ′ be large such that + it and + it ′ are adjacent on-linezeros of θE w , and neither a zero of J θ,w . Suppose there is no zero of J θ,w on thecritical line between + it and + it ′ , but there is a pair of off-line zeros ± ε + iτ o of J θ,w with t < τ o < t ′ and ε very small. Then | t ′ − t | ≥ π t · (1 + O ( t )) .That is, in this configuration, the distance between the consecutive zeros must beleast essentially half the average.Proof. (Of both) In the scenario of the first corollary, this bound implies that R ( t log t ) cannot get from −∞ (at + iτ ) to + ∞ (at + iτ ′ ) much faster than tan( t log t ) , which is by a change of π/ log t .In the scenario of the second corollary, R ( t log t ) goes to ±∞ (with the samesign) approaching t from the right and t ′ from the left, and is very near at τ o .Depending on the sign, the bound by comparison to the tangent function impliesthat either from t to τ o , or else from τ o to t ′ , the function R ( t ) cannot changefaster than tan( t log t ) . Q.E.D.
Remark 78.
Similar, somewhat more complicated, and perhaps less interesting,corollaries hold, as well, for off-the-line zeros. ombieri and Garrett 79
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