Simplest automorphic Schrödinger operators

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1 (August, 3) Smplest automorphc Schrödger operators Paul Garrett garrett/. Compact resolvet of + q for q ε. Mell trasform fuctoals B 3. Hlbert-Schmdt resolvet of + q for q. Evaluatos of stadard L-fuctos, f Λ(f, s) = s f() d (for fxed s C, cuspform f) for cuspforms for Γ = SL (Z), are ot cotuous fuctoals o L (Γ\H). [] formg global automorphc Lev-Sobolev spaces We ca tr to remed ths b H (Γ\H) = Hlbert-space completo of C c (Γ\H) uder f B ( +) = ( + ) f, f L (Γ\H) wth the usual = ( ) x + Compactl-supported automorphc dstrbutos do le H (Γ\H) = colm H (Γ\H) (H (Γ\H) the Hlbert-space dual of H (Γ\H)) However, f Λ(f, s) does ot le H (Γ\H). A related problem s that the cotuous spectrum of etals that (the Fredrchs exteso of) + caot have compact resolvet, so the jectos H (Γ\H) H k (Γ\H) are ot compact, ad H + (Γ\H) = lm H (Γ\H) s ot a uclear Fréchet space. [] These ad other ssues are addressed b cosderg perturbatos + q of b potetals q o Γ\H wth growth at ft, ad correspodg geeralzed Lev-Sobolev spaces [3] B = Hlbert-space completo of C c (Γ\H) uder f B ( +q) = ( + q) f, f L (Γ\H) The perturbato + q has compact resolvet for q(x + ) ε for ε >. The L-fucto evaluato fuctoals f Λ(f, s) are the Hlbert-space dual B of B o vertcal strps σ < α for q(x + ) α. For q(x + ), the resolvet of + q s ot merel compact, but s Hlbert-Schmdt, so B = lm B s uclear Fréchet. [] I effect, [Good 986] solves ( + λ)u = µ s o H, wth µ s(f) = s f() d/ for f C c (H), forms a Pocaré seres F s,w from the free-space soluto u, ad meromorphcall cotues to obta a automorphc form F s such that Γ\H f Fs = Λ(f, s). Aaltc behavor of such Pocaré seres s o-trval. [] Projectve lmts of Hlbert spaces wth Hlbert-Schmdt trasto maps are the most mportat class of uclear Fréchet spaces, b a defto, so we take ths to be the defto of uclear Fréchet. As usual, a chef applcato s exstece of geue tesor products of uclear Fréchet spaces, from whch a Schwartz kerel theorem follows almost mmedatel. E.g., see [Garrett ]. [3] Ths abstracted form of Lev-Sobolev spaces was cosdered at latest b the 96s. For example, see [Petsch 966].

2 Paul Garrett: Smplest automorphc Schrödger operators (August, 3). Compact resolvet of + q for q ε [..] Theorem: For a potetal q wth q(x + ) ε for some ε >, the Fredrchs exteso S of the Schrödger operator S = + q has compact resolvet ( S λ), so has dscrete spectrum. Proof: The argumet s a easer varat of the compactess argumet [Lax-Phllps 976], p. 6. Let B = B ( +q). B costructo, the verse S of the Fredrchs exteso S of S maps cotuousl to L (Γ\H) B, the latter topolog fer tha that of L (Γ\H). Compactess of S : L (Γ\H) L (Γ\H) would follow from compactess of the cluso B L (Γ\H). Stadard perturbato theor would prove that ( S λ) exsts (as bouded operator) for λ off a set wth accumulato pot at most, ad s a compact operator there, ad the spectrum of S s verses of o-zero elemets of the spectrum of S. The total boudedess crtero for relatve compactess requres that, gve ε >, the mage of the ut ball B B L (Γ\H) ca be covered b ftel-ma balls of radus ε. The usual Rellch-Kodrachev compactess lemma, assertg compactess of jectos H s (T ) H t (T ) for s > t of stadard Lev-Sobolev spaces o products of crcles, wll reduce the ssue to a estmate o the tal of Γ\H, whch wll follow from the B codto. Gve c, cover the mage Y o of 3 c+ Γ\H b small coordate patches U, ad oe large ope U coverg the mage Y of c. Ivoke compactess of Y o to obta a fte sub-cover of Y o. Choose a smooth partto of ut {ϕ } subordate to the fte subcover alog wth U, lettg ϕ be a smooth fucto that s detcall for c +. A fucto f B o Y o s a fte sum of fuctos ϕ f. The latter ca be vewed as havg compact support o small opes R, thus detfed wth fuctos o products T of crcles, ad lg H (T ), sce ( + q)ϕ f, ϕ f ( E + )ϕ f, ϕ f (wth usual Eucldea Laplaca E ) The Rellch-Kodrachev lemma apples to each cop of the cluso map H (T ) L (T ), so ϕ B s totall bouded L (Γ\H). Thus, to prove compactess of the global cluso, t suffces to prove that, gve δ >, the cut-off c ca be made suffcetl large so that ϕ B les a sgle ball of radus δ sde L (Γ\H). Sce ϕ, t suffces to show dx d lm f(z) c >c (uforml for f B ) We have dx d f(z) >c c ε f(z) ε dx d >c c ε f(z) ( + ε dx d ) >c c ε (as c +, for f B ) gvg compactess. ///

3 Paul Garrett: Smplest automorphc Schrödger operators (August, 3). Mell trasform fuctoals B Wth potetal q(x + ) α as +, a certa rage of Mell trasform maps are B ( + q): [..] Theorem: For Re(s) < α, the Mell dstrbuto µ s (f) = Λ(f, s) = s f() d (for f C c (Γ\H)) s the Hlbert-space dual B ( + q) of B + ( + q). Proof: Frst, a estmate o f Cc (Γ\H) terms of ts B + -orm s obtaed from Placherel appled to the Fourer expaso of f(x + ) as a perodc fucto of x: > f B = dx d dx d ( + q)f f Γ\H ( + q)f f Z\R q ( Z\R x + α )f f dx d α ( + ) c () d ( + α ) c () d Meawhle, for f Cc (Γ\H) a boud o µ s (f) has a smlar expresso, as follows. B the fuctoal equato s s, take σ = Re(s). Use f( /z) = f(z): µ s (f) = s f() d = ( s + s ) f() d σ f() d For a δ >, b Cauch-Schwarz-Buakowsk, ad at the ed rememberg the earler estmate, µ s (f) σ f() d σ c () d δ = ( δ ( + ) δ + σ +δ + c () d ) ( σ +δ ( + ) c () d ( σ +δ ( + ) c () d ) d ) δ f B (for σ + δ α ) Whe σ < α, the codto σ + δ α holds for some δ >. The estmate o µ s(f) holds for f C c (Γ\H) ad the b cotut for f B +. /// 3

4 Paul Garrett: Smplest automorphc Schrödger operators (August, 3) 3. Hlbert-Schmdt resolvet of + q for q Whe + q has Hlbert-Schmdt resolvet, all trasto maps B ( + q) B ( + q) the projectve lmt are Hlbert-Schmdt: for orthoormal bass {u } of egefuctos for L (Γ\H), wth egevalues λ >, the vectors u /λ / form a orthoormal bass for B = B ( + q). Wth respect to these orthoormal bases, the clusos are smpl multplcato maps c u λ Such a map s Hlbert-Schmdt f ad ol f (λ c λ ) < The resolvet of the Fredrchs exteso of +q has egevalues λ, ad the Hlbert-Schmdt propert s the same equalt. I ths stuato B + = lm B s uclear Fréchet, gvg a Schwartz kerel theorem. u λ [3..] Theorem: + q has Hlbert-Schmdt resolvet for q(x + ) as +. Proof: As the proof of compactess of the resolvet, the fact that H s (T ) H s (T ) s Hlbert-Schmdt reduces dscusso to cosderato of the geometrcall smpler o-compact part of Γ\H. Specfcall, t suffces to cosder the restrcto S of + q to test fuctos o the taperg clder X = T [, ), dx d wth measure, ad to take q(x + ) =. Thus, the doma of S cludes test fuctos o X vashg to fte order o the boudar X = T {}. Let S be the Fredrchs self-adjot exteso of S. O ths o-compact but geometrcall smpler fragmet of Γ\H, the crcle group T acts, ad commutes wth S ad S. Thus, L (X) decomposes orthogoall to compoets dexed b characters ψ (x) = e x of T = R/πZ. O the th compoet, the dfferetal equato for that compoet of a fudametal soluto u a at a s ( ) δ a = ( + q) e x u a () smplfg, coveetl, to a costat-coeffcet equato = ( + )u a() = ( u a + ( + )u a ) a δ a = u a + ( + )u a (wth a > ) We ca follow the usual prescrpto for pecg together u a from solutos e ±c to the correspodg homogeeous equato u + ( + )u =, lettg c = +. That s, u a () must have moderateeough growth as + so that t s L (X) wth measure dx d/, ad go to zero as +, addto to beg cotuous but o-smooth eough at = a to produce the requred multple of δ a. Thus, u a must be of the form { Aa e u a () = c + B a e c (for < < a) C a e c (for a < ) for some costats A a, B a, C a, sce e c grows too rapdl as +. The codtos are A a e c + B a e c = (vashg at + ) A a e ca + B a e ca = C a e ca (cotut at = a) c C a e ca (ca a e ca cb a e ca ) = a (chage of slope b a at = a) 4

5 Paul Garrett: Smplest automorphc Schrödger operators (August, 3) From the frst equato, B a = e c A a, ad the sstem becomes { Aa (e ca e c e ca ) = C a e ca (cotut at = a) c C a e ca ca a ( e ca + e c e ca ) = a (chage of slope b a at = a) Substtutg C a = A a (e ca e c ), from the frst equato, to the secod, gves smplfg to A a = e ca /ca. The so ( A a c(e ca e c )e ca c(e ca + e c e ca) = a C a u a () = C a e c Sce (c )u = f s solved b = A a (e ca e c ) = ec e ca e ca ca = ec e ca e ca ca e c (for > a) u() = a u a () f(a) da the smmetr of + q wth respect to the measure d/ mples that a u a () s smmetrc, a, ad the tegral kerel for the resolvet s e c e ca e ca e c (for > a) a c u a () = e c e c e c e ca (for < < a) c The resolvet beg Hlbert-Schmdt s equvalet to a u a () da a d < B smmetr, t suffces to tegrate over < a < <, ad <a< a u a () da a d = <a< e 4c e ca e c + e ca e c 4c (e 4c e ca + e c + e ca c da ) e + <a< a Replacg, a b +, a +, the tegral becomes (e ca + + e ca ) e c da d (a + ) ( + ) <a< da (a + ) d ( + ) < <a< d da a d da d (a + ) ( + ) Thus, the th compoet of the tegral kerel has L orm bouded b a uform costat multple of /( + ). The sum over Z s fte, provg that the resolvet s Hlbert-Schmdt. /// 5

6 Bblograph Paul Garrett: Smplest automorphc Schrödger operators (August, 3) [CdV 98,83] Y. Col de Verdère, Pseudo-laplaces, I, II, A. Ist. Fourer (Greoble) 3 (98) o. 3, 75-86, 33 (983) o., [Fredrchs ] K.O. Fredrchs, Spektraltheore halbbeschräkter Operatore, Math. A. 9 (934), , , (935), [Garrett a] P. Garrett, Col de Verdère s meromorphc cotuato of Eseste seres, garrett/m/v/cdv es.pdf [Garrett b] P. Garrett, Pseudo-cuspforms, pseudo-laplacas, garrett/m/v/pseudo-cuspforms.pdf [Garrett ] P. Garrett, Hlbert-Schmdt operators, tesor products, Schwartz kerels: uclear spaces I, garrett/m/fu/otes -3/6d uclear spaces I.pdf [Gelfad-Vlek 964] I.M. Gelfad, N. Ya. Vlek, Geeralzed Fuctos, IV: applcatos of harmoc aalss, Academc Press, NY, 964. [Good 986] A. Good, The covoluto method for Drchlet seres, Selberg Trace Formula ad Related Topcs (Bruswck, Mae, 984), Cotemp. Math. 53, AMS, Provdece, 986, 7 4. [Grothedeck 955] A. Grothedeck, Produts tesorels topologques et espaces ucléares, Mem. AMS 6, 955. [Kato 966] T. Kato, Perturbato theor for lear operators, Sprger, 966, secod edto, 976, reprted 995. [Lax-Phllps 976] P. Lax, R. Phllps, Scatterg theor for automorphc fuctos, Aals of Math. Studes, Prceto, 976. [Petsch 966] A. Petsch, Über de Erzeugug vo F -Räume durch selbstadjuderte Operatore, Math. A. 64 (966), 9-4. [Vekov 99] A.B. Vekov, Selberg s trace formula for a automorphc Schrödger operator, Fu. A. ad Applcatos 5 ssue (99), -. 6