Nodal densities of Gaussian random waves satisfying mixed boundary conditions

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1 INSTITUTE OF PHSICSPUBLISHING JOURNAL OFPHSICSA: MATHEMATICAL AND GENERAL J. Phys. A: Mth. Gen. 35 () PII: S35-447()3695-X Nol ensities of Gussin rnom wves stisfying mixe ounry onitions MVBerry n H Ishio HHWills Physis Lortory, Tynll Avenue, Bristol BS8 TL, UK Shool of Mthemtis, University of Bristol, University Wlk, Bristol BS8 TW, UK Reeive 4 April Pulishe July Online t stks.iop.org/jphysa/35/596 Astrt Nol sttistis re erive for ounry-pte Gussin rnom wves ψ = u +iv in theplne, stisfying the Helmholtz eqution n mixe ounry onitions, in whih liner omintion of ψ n its norml erivtive, hrterize y prmeter, vnishesonline. For oth the ensity of nol lines for Re ψ = u n the ensity of nol points for omplex ψ,effets of the ounry persist, surprisingly, infinitely fr from the ounry, n, lso surprisingly, re inepenent of. As inreses from the Dirihlet vlue =, nol line n point strutures migrte from the ounry, n n e esrie nlytilly. PACS numers:.5. r, 3.65.Sq, 5.45.Mt. Introution Awiely-isusse lss of Gussin rnom funtions of two vriles R ={X, }= {kx, ky} onsists of superpositions of plne wves trvelling in ifferent iretions ut with the sme wvelength λ = π/k. These re moels for onfine quntum wves ssoite with lssilly hoti trjetories, for exmple, eigenfuntions in ergoi quntum illirs (Berry 977, O Connor et l 987, Heller 99, Sihev et l, Blum et l ). The wvesre rel if the system hs time-reversl symmetry (T) n omplex otherwise. In reent improvement of the moel (Berry, hereinfter lle I), ounry-pte Gussin rnom wves were onstrute to stisfy Dirihlet or Neumnn ounry onitions long stright line, hosen s =. For these wves, some sttistis of nol lines (with T) n points (without T) were lulte, reveling interesting -epenene. Our im here is to generlize this improvement to ounry-pte Gussin rnom wves stisfying the fmily of mixe onitions ( ψ(x,; ) os + ψ (X, ; ) sin = π< π). () Here n herefter, itli susripts enote erivtives, n is prmeter lelling memers of the fmily; for Dirihlet onitions, =, while for Neumnn = π/. This fmily of //9596+$3. IOP Pulishing Lt Printe in the UK 596

2 596 M V Berry n H Ishio reltions etween the wvefuntion n its norml erivtive is speil se of the Roin ounry onition (Gustvsonn Ae 998). The generliztion hs more thn tehnil interest, sine ssoite with it re two unntiipte phenomen tht we wish to rw ttention to. First, the long-rnge epletion (for nol lines) n enhnement (for nol points) of nol ensities, lrey note in I for Dirihlet n Neumnn onitions, persist (setion ) inepenent of the prmeter. Seon, for smll n positive there is rih nol struture for smll (setion 3), whih n e regre s ghost of the eprte Dirihlet nol line tht exists t = when =, n n e esrie nlytilly. As short lultion onfirms, the Gussin rnom superposition of omplex wves ψ,with rel n imginry prts u n v,stisfying the Helmholtz eqution n the ounry onition (), n e written s ψ(r; ) u(r; ) +iv(r; ) = J J j= [sin( sin θ j ) tn sin θ j os( sin θ j )] + tn sin θ j exp(i(x os θ j + φ j )). () This onsists of J ( ) wves trvelling in iretions θ j,equiistriute on the rnge [, π], with phses φ j,equiistriute on the rnge [, π]. Figure shows three plots of smple funtion ψ of this type, for = π/4, initing the nol lines for the rel prt u of the wve n the nol points of ψ (intersetions of nol lines of u n v).. Nol ensities We will lulte the ensity ( ; ) ofnol lines for rel wves u(r; ), n the ensity ρ P ( ; ) ofnol points for omplex wves ψ(r; ), oth quntities eing sle so tht ρ s.the tehniques re ientil to those in I (see lso Berry n Dennis ), involving ertin verges over the ensemle of rnom wves. The only new feture is tht these verges epen on, n nnot e expresse omptly in terms of Bessel funtions s in the Dirihlet n Neumnn ses. The relevnt verges (the sme for the rel n imginry prts u n v, whihre lso sttistilly inepenent) re esily lulte s B(; ) u = π/ f (θ; ) θ π +tn sin θ D X ( ; ) u X = π/ θ os f (θ; ) θ π +tn sin θ (3) D ( ; ) u = DX ( ; ) B(; ) + K(; ) uu = π/ f (θ; ) θ sin θ π +tn sin θ where f (θ; ) [( tn sin θ)os( sin θ)+tnsin θ sin( sin θ)] (4) f (θ; ) [( tn sin θ)sin( sin θ) tnsin θ os( sin θ)]. For eonomy of nottion, we will frequently not inite the vriles expliitly, n refer simply to B,D X,et. Forrel wves, the nol ensity to e lulte is ( ; ),efiney line length per unit re = k ( ; ) (5)

3 Noes of rnom wves ner ounries λ 5 λ () X 5 5 () X 5 λ () X Figure. Smples of Gussin rnom funtion ψ with J = 87 plne wves, for ounryonition prmeter = π/4. ()Nol lines of u = Re ψ (thik) n v = Im ψ (thin), with nol points of ψ inite y intersetions of the nol lines; () ensity plot of u, with nol lines lk; () ensity plot of ψ, with nol points lk. The wvelength λ is inite. inorporting the previously erive ulk ensity k/( ) for nol lines of isotropi Gussin rnom funtions fr from ounries. Applition of ientil resoning to tht in I les to ( ; ) = π D ( X BD K ) π/ θ ( BDX os θ + ( BD K ) sin θ ) 3/ = ( DX B(B ) + K ) π B E (6) BD X where in the seon equlity (not given previously) E enotes the omplete ellipti integrl (the efinition is tht of Mthemti (Wolfrm 996)). Figure shows nol line ensities for severl vlues of, inexellent greement with numerilly-ompute ensemle verges over smple wves of type (). The pek for = π/4will e isusse in etil in setion 3.

4 5964 M V Berry n H Ishio Figure. Nol line ensities ( ; ), for () = π/4, () =, () = +π/4, () = +π/. Thik urves: theoretil formul (6); otte urves: verges over 5 numerilly ompute Gussin rnom funtions eh me from J = 87 plne wves. The inset is mgnifition of the smll- pek, with the points eing numeril verges. Foromplex wves, the nol ensity to e lulte is ρ P ( ; ),efiney numer of nol points per unit re = k 4π ρ P( ; ) (7) inorportingthepreviously erive (Berry n Ronik 986)ulkensity k /(4π)for nol points of isotropi Gussin rnom funtions fr from ounries. Applition of ientil resoning to tht in I les to D ρ P ( ; ) = ( X BD K ). (8) B 3/ Figure 3 shows nol point ensities for severl vlues of, gin in goo greement with numerilly-ompute ensemle verges over smple wves of type (). The pek for = π/4will e isusse in etil in setion 3. The long-rnge ehviour of the nol ensities n e estlishe from the symptoti ehviour of orreltions (3)nexpressions (6)n(8). Strightforwr ut long lultions le to ( ; ) + os( 4 π) ( ) (9) π 3π n ρ P ( ; ) + os( 4 π) + ( ). () π 4π Figures 4 n 5 show how urtely these symptoti formule esrie the ensities, even for rthersmll.

5 Noes of rnom wves ner ounries 5965 ρ P ρ P ρ P ρ P Figure 3. As figure, for nol point ensities ρ P ( ; ) nthetheoretil formul (8) Figure 4. As figure, with the thin urves showing the symptoti formul (9). The most striking spet of the lrge- symptotis is the epenene of the leing nonosilltory orretion. This mens tht when the exess ρ isintegrtetogivethe men exess length of nol line or numer of nol points in strip of height,theresult iverges s.thus L ex ( ; ) η( (η; ) ) = sin ( 4 π) π log 3π + C L() ( ) ()

6 5966 M V Berry n H Ishio.5.5 ρ P ρ P Figure 5. As figure 3, with the thin urves showing the symptoti formul (). n N ex ( ; ) η(ρ P (η; ) ) = sin ( 4 π) π + log 4π + C P() ( ) () where the numerilly-etermine onstnts C L n C P re shown in figures n 3 n will e isusse lter. These ivergenes imply tht there is sense in whih the effet of the ounry onitions persists infinitely fr from the ounry, ft whose implitions for quntum illirs were isusse in I. Moreover, the oeffiients of the logrithmi terms re the sme for ll vlues of the ounry prmeter, implying tht, surprisingly, the nture of the ivergene resulting from the imposition of ounry onitions on Gussin rnom funtions is inepenent of the form of the ounry onition. Figures 6 n 7 show how well these logrithmi ivergenes esrie the ensemle verges over numerilly-ompute rnom wves. 3. Ghosts of the eprte Dirihlet nol line Now we onentrte on the smll- peks in figures ()n3(), orresponing to = π/4, n show tht these reflet migrte remnnts of the nol line (for oth rel n omplex wves) t = forthedirihlet se =. For smll n lose to, wve() n e expne in series whose leing terms re ψ(r; ) ( 3 3) (u (X) +iv (X)) 3 3 (u 3 (X) +iv 3 (X)) (3) where we employthenottion u (X) u (X, ; ) u 3 (X) u (X, ; ) (4) to enote o norml erivtives of the Dirihlet wve on the ounry, n similrly for v. Equting ψ to zero gives, for rel wves, the nol line whose eqution is = + ( u 3(X) u (X) ). (5)

7 Noes of rnom wves ner ounries 5967 L ex -.5 L ex L ex L ex Figure 6. As figure, for the exess line ensity L ex ( ; ), with the fine line showing the smoothe symptotis log/3π + C L (). N ex N ex N ex N ex Figure 7. As figure 3, for the exess point ensity N ex ( ; ), with the fine line showing the smoothe symptotis +log /4π + C P ().

8 5968 M V Berry n H Ishio Figure 8. Nol lines of u n v for smple Gussin rnom funtion with = π/ = Smll ots: theoretil formul (5) for nol lines of u; lrge ots: theoretil formul (6) for nol points of ψ;sheline: = + 3 /. X Foromplex wves the nol points {X, } re the simultneous solutions of = + { }) ψ3 (X) (+ 3 3 Re ψ (X) = + ( u ) (X) u 3 (X) + v (X) v 3 (X) u (X) + v (X). (6) { } ψ3 (X) Im = i.e. u 3 (X)v (X) u (X) v 3 (X) = ψ (X) Twoimplitions of these equtions re: nol struture lose to =, ntherefore prt of the physil wve >when is positive; n onentrtion of this nol struture within strip whose -with is of orer 3 tht is, the peks get shrper s gets smller. Figure 8 illustrtes how urtely this smll- symptoti theory esries the nol lines n points for n iniviul funtion from the ensemle (). To etermine the shpe of the smll- peks, it is onvenient to mesure from, in units of 3,efining η y + 3 η (7) n expning verges (3)forsmll. Tothelowestrelevnt orer, this gives ( (η ) B 6 ) + 44 D X 4 6 ( (η 6 D K 3 ( η) ) + 44 ) ( ). (8) For the nol line ensity, sustitution into (6), n further expnsion, gives the Lorentzin pek shpe ( + 3 η; ) 4 π 3 (+(η ) ) ( ) (9)

9 Noes of rnom wves ner ounries Figure 9. Smll- peks in ( ; ), for () = π/3, () = π/4, () = π/6, () = π/. Thik urves: ext formul (6); thin urves: symptoti formul (9) Figure. As figure 9, for ρ P ( ; ), with the ext formul (8) n the symptoti formul (). with mximum t η = /, tht is = + 3 /. As figure 9 shows, this gives n inresingly urte esription of the peks s ereses. For the nol point ensity, sustitution into (8)givestheistorte Lorentzin pek shpe ρ P ( + 3 η; ) +(η ) 3 (+(η ) ) 3/ ( ) () with mximum lose to η = / (tully η = /.4 598). As figure shows, this is urte for smll too. The heights of the peks sle s 3,with oeffiients ( + 3 ; ) 4 π 3 ( ) ()

10 597 M V Berry n H Ishio Figure. Nol ensities t = + 3 /, lose to the mximum for smll. Thik urve, rtio /theoretil vlue (); thin urve, rtio ρ P /theoretil vlue (). C L Figure. Constnt C L () inthe exess nol line symptotis (). C P Figure 3. Constnt C P () inthe exess nol point symptotis (). n ( ρ P + ) 3 ; ( ). () 3 Figure showsthe ury of these formule. For negtive, these nol ensity peks lie on the negtive- (virtul) sie of the fiel. As inreses through zero, the peks pper s δ-funtion ontriutions t =. Therefore the integrls of the nol ensities the nol exesses () n() re isontinuous t =, so the symptoti onstnts C L ()nc P ()re isontinuous t =. The onstnts re shown in figures n 3.

11 Noes of rnom wves ner ounries 597 For nol lines, (9)givestheisontinuity s C L lim (C L () C L ( )) + = lim 3 η ( + 3 η; ) =. (3) + The orretness of this result is immeitely onfirme y the oservtion tht the line ppering from = issimply the Dirihlet nol line long the -xis, whose length per unit X-with is of ourse unity, whih fter the sling (5)is. For nol points, () givestheisontinuity s 3 + C P lim (C P () C P ( )) = lim ηρ P ( + 3 η; ) + = ( ( ) 5 5 )E 4 π, ( 5+3) + ( ( ) 5 5+)E 4 π, ( 5+3) = (4) Here E enotes the inomplete ellipti integrl (the efinition is tht of Mthemti (Wolfrm 996)). An lterntive wy to onfirm the orretness of this result is to lulte the men X-ensity of the nol points s given y the seon eqution in (6), fter the sling (7): C P = 4π δ(u 3 v u v 3 ) X (u 3 v u v 3 ). (5) Alengthy lultion reproues (4). 4. Disussion We hve explore severl wys in whih ounry onitions long line ffet the sttistil properties of Gussin rnom funtions. Two unexpete phenomen emerge from this stuy. First, the effets of the ounry persist infinitely fr from the ounry, n re inepenent of the form of the ounry onition: the long-rnge nol line epletion n nol point enhnement re inepenent of the prmeter. Seon, for smll positive, there re short-rnge enhnements of the nol ensity, s the nol line for the Dirihlet se = migrtes to smll positive n gets istorte n, in the omplex se, roken into lineofnol points. We envisge tht the min pplition of these results will e to quntum illirs, in the form of perimeter orretions to the semilssil symptotis of nol sttistis for eigenfuntions. It shoul e possile in numeril experiments to etet oth the long-rnge nol effets (epletions n enhnements) n the short-rnge enhnement for smll. We re investigting this now. The ies employe here n e pplie more wiely. For exmple, in reent stuy, inepenent of ours, y Bies n Heller (), nol lines were investigte for lssilly hoti wvefuntions ner lines in the plne where prtiles re reflete y smooth potentil. They employe the wve ψ(r) = u(r) +iv(r) = J J Ai ( + Q ) j exp{i(qj X + φ j )} (6) j=

12 597 M V Berry n H Ishio where J, the Q j re trnsverse wvenumers uniformly istriute over [, ]nthe φ j re rnom phses. This Airyfie Gussin rnom funtion stisfies the time-inepenent Shröinger eqution for the potentil (ll onstnts, inluing energy, n e remove y sling). The preeing theory n e pplie iretly to etermine the nol sttistis of (6), provie the verges in (3) re reple (up to n irrelevnt overll onstnt) y B() = D X ( ) = D ( ) = K() = Q Ai ( + Q ) QQ Ai ( + Q ) Q Ai ( + Q ) Q Ai( + Q ) Ai ( + Q ). (The first of these equtions, giving the men proility ensity of rnom wve ner smooth ounry, hs een erive efore (Berry 989).) The -epenene of the ensity of nol lines is given y (6), n the ensity of nol points y (8). Areferee hs suggeste tht the long-rnge effets reporte here re not present in higher imensions. This is orret. Wehve lulte the ensity of nol surfes for rel Gussin rnom wves in three imensions, with Dirihlet ounry onitions on the XZ plne =. The leing nonosilltory terms re proportionl to /(8 )for, so tht the totl exess nol re, use y the ounry, is finite rther thn logrithmilly ivergent. Aknowlegments We thnk Willim Bies n Eri Heller for sening us their pper efore pulition. HI knowleges QinetiQ for finnil support. (7) Referenes Berry M V 977 Regulr n irregulr semilssil wve funtions J. Phys. A: Mth. Gen Berry M V 989 Fringes eorting ntiustis in ergoi wvefuntions Pro. R. So. A Berry M V Sttistis of nol lines n points in hoti quntum illirs: perimeter orretions, flututions, urvture J. Phys. A: Mth. Gen Berry M V n Dennis M R Phse singulrities in isotropi rnom wves Pro. R. So. A Berry M V n Ronik M 986 Quntum sttes without time-reversl symmetry: wvefront islotions in nonintegrle Ahronov Bohm illir J. Phys. A: Mth. Gen Bies W E n Heller E J Nol struture of hoti eigenfuntions J. Phys. A: Mth. Gen. t press Blum G, Gnutzmnn S n Smilnsky U Nol omins sttistis riterion for quntum hos Phys. Rev. Lett Gustvson K n Ae T 998 The thir ounry onition ws it Roin s? Mth. Intelligener 63 7 Heller E J 99 Chos n Quntum Physis (Les Houhes Leture Series vol 5) e M-J Ginnoni, A Voros n J Zinn-Justin (North-Holln: Amsterm) pp O Connor P, Gehlen J n Heller E J 987 Properties ofrnom superpositions of plne wves Phys. Rev. Lett Sihev A I, Ishio H, Sreev A F n Berggren K-F Sttistis of interior urrent istriutions in two-imensionl open hoti illirs J. Phys. A: Mth. Gen. 35 L87 L93 Wolfrm S 996 The Mthemti Book (Cmrige: Cmrige University Press)