On two-dimensional Bessel functions

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 39 (6) doi:1.188/35-447/39/48/8 On two-dimensional Bessel fnctions H J Korsch, A Klmpp 1 and D Witthat FB Physik, Technische Uniersität Kaiserslatern, D Kaiserslatern, Germany korsch@physik.ni-kl.de Receied 8 Agst 6, in final form October 6 Pblished 15 Noember 6 Online at stacks.iop.org/jphysa/39/14947 Abstract The general properties of two-dimensional generalized Bessel fnctions are discssed. Varios asymptotic approximations are deried and applied to analyse the basic strctre of the two-dimensional Bessel fnctions as well as their nodal lines. PACS nmber:.3.gp (Some figres in this article are in color only in the electronic ersion) 1. Introdction Generalized Bessel fnctions depending on seeral ariables were introdced in 1915 for a finite [1] and also an infinite nmber of ariables []. They hae ery similar properties as the ordinary Bessel fnctions bt are mch less familiar. More recently, howeer, two-ariable Bessel fnctions hae fond an increasing nmber of applications in arios areas of physics (see, e.g. [3 13]). The basic theory of generalized Bessel fnctions is described in a monograph by Dattoli and Torre [14]. Or own interest into the properties of these fnctions is cased by or recent stdies of qantm dynamics in periodic strctres [11, 1], in particlar in stdies of transport and dynamic localization [15]. In most cases these applications were restricted to the case of two ariables, and. Then the generalized Bessel fnctions Jn p,q (, ) are labelled by three integer indices n, p, q. The theory of these fnctions has been discssed, within the framework of a grop theoretic treatment, in the book by Dattoli and Torre [14] and in [16]. The special case (p, q) = (1, ) has been considered p to now almost exclsiely [5, 8, 1, 14, 17 19]. Here we will analyse the two-dimensional Bessel fnctions Jn p,q (, ) for general indices p and q (see [] for a well-written introdction to the case of infinite ariables; it shold also be mentioned that one-ariable fnctions Jn p,q (x), with p, q relatiely prime integers, hae been discssed already in 1964 [1]). 1 Present address: Institt für Physik, Uniersität Kassel, D-3419 Kassel, Germany /6/ $3. 6 IOP Pblishing Ltd Printed in the UK 14947

2 14948 HJKorschet al We will derie the fndamental properties of the two-dimensional Bessel fnctions and analyse their basic strctre for small and large argments in the following sections. It will be seen that the two-dimensional Bessel fnctions show a rich oscillatory strctre with regions of ery different behaior. We will analyse these strctral featres with special attention to the nodal lines which are of considerable importance for recent applications to localization phenomena in qantm dynamics [15].. Basic properties In this section we will collect the basic properties of the generalized Bessel fnctions Jn p,q (, ) with integer indices n, p, q and two real argments,. Most of the reslts in the literatre (see, in particlar, appendix B of [5] and chapter of [14]) hae been deried for the special case of Jn 1, (, )..1. Definition The two-dimensional Bessel fnctions can be defined by the generating fnction e i( sin pt+ sin qt) = Jn p,q (, ) e int, (1) n= also known as a Jacobi Anger expansion, or, somewhat more general, as ( exp (zp z p ) + ) (zq z q ) = Jn p,q (, )z n. () n= Integration of (1) oer t sing +π dt eint representation = πδ(n) immediately leads to the integral Jn p,q (, ) = 1 +π dt e i( sin pt+ sin qt nt), π (3) a generalization of the integral representation of the well-known ordinary Bessel fnction J n (x) = 1 +π dt e i(x sin t nt). π (4) From the properties of Forier series we find the bonds p,q J (, ) 1 and n (, ) 1/ for n. (5) As an immediate conseqence of (3) the integers p and q can be assmed to be coprime becase Jn p,q anishes otherwise or it can be redced to sch a coprime case. This is seen as follows (we assme that µ is an integer, µ 1): πj µp,µq n (, ) = = π πµ i( sin µpt + sin µqt nt) dt e ds µ ei( sin ps+ sin qs ns/µ) = [ µ ] π = e iπ(m 1)n/µ m=1 µ m=1 πm π(m 1) ds sin ps+ sin qs ns/µ) ei( µ ds µ ei( sin ps+ sin qs ns/µ). (6)

3 On two-dimensional Bessel fnctions Using µ { µ for n/µ Z e iπ(m 1)n/µ = (7) else m=1 and, for n/µ Z, π ds µ ei( sin ps+ sin qs ns/µ) = 1 µ n/µ (, ) (8) one obtains { p,q J J n µp,µq n/µ (, ) for n/µ Z (, ) = (9) else. In the following, we will therefore assme that the integers p and q hae no common diisor... Decomposition in terms of ordinary Bessel fnctions A representation in terms of ordinary Bessel fnctions can be deried from the integral representation (3). Inserting the generating fnction for the ordinary Bessel fnctions e ix sin s = J n (x) e ins (1) n= for both s = pt and s = qt into (3), we obtain Jn p,q (, ) = 1 +π i( sin pt+ sin qt nt) dt e π = J µ ()J ν () 1 +π dt e i[(µp+νq n)t]. (11) π µ,ν The integral is only different from zero if n = µp + νq is satisfied. If p and q hae no common diisor as assmed here, a soltion (µ, ν) = (M, N) of this Diophantine eqation always exists and can be fond systematically by, e.g., the Eclid algorithm []. Moreoer there is an infinite nmber of soltions µ = M qk,ν = N + pk, k =, ±1, ±,...becase of n = pm + qn = pm + qn + pqk pqk = p(m qk) + q(n + pk). We therefore hae n (, ) = J M qk ()J N+pk (), (1) where (M, N) is an arbitrary soltion of n = pm + qn. For the case p = 1 this reads (M = n and N = ) Jn 1,q (, ) = J n qk ()J k (). (13) These fnctions, denoted as Jn q (, ), appear as a by-prodct of the theory of Hermite Bessel fnctions. Jn p,q (, ) can be written in terms of Jn q (, ) according to the identity Jn p,q (, ) = J p n k (ν, )J q k ( ν, ), (14) where ν plays the role of a dmmy ariable. In most of the preios applications one enconters the case q = and in these cases one sally simplifies the notation by dropping the p, q indices, i.e. one defines J n (, ) = Jn 1, (, ). (15)

4 1495 HJKorschet al.3. Addition theorems The addition theorem n ( 1 +, 1 + ) = n k ( 1, 1 ) k (, ) (16) can be easily proed starting from (3)sing(1): Jn p,q ( 1 +, 1 + ) = 1 +π dt e i( 1 sin pt+ 1 sin qt nt) e i( sin pt+ sin qt) π = 1 +π dt e i( 1 sin pt+ 1 sin qt nt) k (, ) e ikt π l= = = k (, ) 1 π +π k (, ) n k ( 1, 1 ). dt e i( 1 sin pt+ 1 sin qt (n k)t) The Graf addition theorem for ordinary Bessel fnctions, + [ ] n τ l x x 1 /τ J l (x 1 )J n+l (x ) = Jn [g(x 1,x ; τ)] (17) x l= x 1 τ with g(x 1,x ; τ) = ( x1 + x x 1x (τ +1/τ) ) 1/, (18) can also be generalized to the two-dimensional case, at least for p = 1intheform + + [ ] n ql [ τ l J 1,q l ( 1, 1 )J 1,q n+l ( 1 /τ 1 /τ q ] l, ) = 1 τ 1 τ q l= J n ql [g( 1, ; τ)]j l [g( 1, ; τ q )] (19) (see Dattoli et al [17, 14] for the special case q = ). The generalized Graf addition theorem (19) can be deried in a straightforward calclation expressing first the two-dimensional Bessel fnctions as a sm oer ordinary ones (see eqation (13)) and sing the Graf addition theorem (17) for ordinary Bessel fnctions: + τ l J 1,q l ( 1, 1 )J 1,q n+l (, ) = τ l J l qk ( 1 )J k ( 1 )J l+n qj ( )J j ( ) l,j,k l= = j,k J k ( 1 )J j ( )τ qk l τ l J l ( 1 )J n+q(k j)+l ( ) = j,k J k ( 1 )J j ( )τ qk J n+q(k j) (g( 1, ; τ)) [ ] n+q(k j) 1 /τ 1 τ = l [ ] n ql 1 /τ J n ql (g( 1, ; τ)) 1 τ k τ qk J k ( 1 )J l+k ( ) = l [ ] n ql 1 /τ 1 τ with g(x 1,x ; τ) as defined in (18). [ 1 /τ q 1 τ q ] l Jn ql (g( 1, ; τ))j l (g( 1, ; τ q )) ()

5 On two-dimensional Bessel fnctions Symmetries, special cases and nmerical examples From the definition (1) one erifies (by taking the complex conjgate and changing ariables t t) that the Jn p,q (, ) are real aled and satisfy Jn p,q (, ) = δ n. (1) The symmetry relations Jn p,q (, ) = Jn q,p (, ) () n (, ) = n (, ) = Jn q,p (, ) = Jn p, q (, ) (3) follow directly from the definition. For n = these eqations imply the symmetries (, ) = (, ) = J q,p (, ). (4) A frther direct reslt is a symmetry relation for een ales of one of the p, q-indices, say q. UsingJ n ( z) = ( 1) n J n (z) we get Jn p,q (, ) = J M qk ( )J N+pk () = ( 1) M J M qk ()J N+pk () = ( 1) n n (, ). (5) The last eqality holds becase of ( 1) n = ( 1) pm+qn = ( 1) M for q een and p odd. This symmetry implies Jn p,q (,)= ( 1) n Jn p,q (,) Jn p,q (,)= forn odd and q een. (6) If both pper indices are odd, their difference mst be een. This leads to another symmetry relation n (, ) = = J M qk ( )J N+pk ( ) ( 1) M+N+(p q)k J M qk ()J N+pk () = ( 1) M+N Jn p,q (, ) = ( 1) n Jn p,q (, ) for p, q odd. (7) Here the last eqality is based on the fact that for odd p, q-indices, p = j + 1 and q = k +1, we hae n = pm + qn = M + N +(j + k). In the case p = q the two-dimensional Bessel fnctions simplify and redce to ordinary Bessel fnctions if n is an integer mltiple of p: +π Jn p,p (, ) = 1 dt e i((+) sin pt nt) = 1 π π { Jn/p ( + ) for n/p N = else. +pπ pπ ds p ei((+) sin s ns/p) Another relation between the generalized and ordinary Bessel fnctions can be obsered if the index n is a mltiple of one of the pper indices, e.g. n = mq, m integer. Then we get mq (,)= 1 π = 1 π +π +π dt e i( sin qt mqt) = 1 πq qπ qπ i( sin s ms) ds e (8) ds e i( sin s ms) = J m () (9)

6 1495 HJKorschet al Figre 1. Color map of the two-dimensional Bessel fnction Jn 1, (, ) for n =,n= 1,n= (from left to right) Figre. Color map of the two-dimensional Bessel fnction Jn 1,3 (, ) for n =,n= 1,n= (from left to right). and, as a special case, Jn p,1 (,)= J n (). (3) According to (9) the two-dimensional Bessel fnction Jn p,q (,)is redced to an ordinary Bessel fnction J m () for n = mq. Otherwise the fnction anishes on the -axis as can easily be seen from (1): Jn p,q (,)= J M qk ()J N+pk () = δ M,qk J N+pk () =, (31) if M is not a mltiple of q or, eqialently, n = pm + qn is not an integer mltiple of q. We therefore hae Jn p,q (,)= if n mq, m Z (3) as a generalization of (6). Let s look at a few examples of two-dimensional Bessel fnctions calclated nmerically sing the representation (1) in terms of ordinary Bessel fnctions (similar graphs can be fond, e.g., in [14, 18]). Figres 1 and show color maps of Jn 1, 1,3 (, ) and Jn (, ) for n =,n = 1 and n = sing a re-normalization to nit maximm in each case (the regions of positie ales are colored red, of negatie ales ble). For Jn 1, in figre 1 we see that the symmetry relations J 1, (, ) = J 1, (, ), Jn 1, (, ) = ( 1)n Jn 1, (, ) (33) are satisfied (compare eqations (4) and (5)). The last relation implies a nodal line for n = 1 along the -axis, J 1, 1 (,)=. For the case Jn 1,3 (, ) in figre we hae the symmetry Jn 1,3(, ) = ( 1)n Jn 1,3 (, ) (compare eqation (8)).

7 On two-dimensional Bessel fnctions Sm rles and Kapteyn series The simple sm rle n= n (, ) = 1 (34) is a direct conseqence of the generating fnction (1) fort =. A ariety of sm rles for special cases can be obtained by choosing t in (1) appropriately. For example, for the important special case (p, q) = (1, ) another sm rle is fond by setting t = π/: i n Jn 1, (, ) = ei. (35) n= Similar sm rles can be obtained for other special cases. Another sm rle, ( p,q J k (, ) ) = 1 (36) follows from the addition theorem (16) for the special case n =, 1 = =, 1 = = sing (1). We frthermore note withot proof the Kapteyn type series [3, 4] n=.6. Frther generalizations n (n, n) = 1, 1 p q p + q < 1. (37) As already stated in the introdction, the nmber of ariables in the Bessel fnction can be extended. Different types of generalizations are, howeer, also possible. Modified higher dimensional Bessel fnctions can be constrcted, e.g. by replacing one of the ordinary Bessel fnctions in (1) by a modified one [9]. In addition, two-ariable, one-parameter Bessel fnctions [9, 5] can be defined as a generalization of (1): n (, ; τ) = J M qk ()J N+pk ()τ k. (38) (Let s recall that (M, N) are arbitrary soltions of n = pm + qn.) Here again we find Jn p,q (, ; τ) = δ n. In particlar the case τ = e iδ is of interest [9, 15] for applications in physics. We confine orseles to the most important case p = 1, i.e. Jn 1,q (, ; e iδ ) = J n qk ()J k () e ikδ. (39) Following the lines in the deriations aboe, one can easily show that these fnctions are generated by e i( sin t+ sin(qt+δ)) = Jn 1,q (, ; e iδ ) e int, (4) n= which leads to the integral representation Jn 1,q (, ; e iδ ) = 1 +π dt e i( sin t+ sin(qt+δ) nt). (41) π

8 14954 HJKorschet al Figre 3. Color map of the two-dimensional, one-parameter Bessel fnction Jn 1, (, ; i) for n =,n= 1,n= (from left to right). The figres show the real part. The generalized Bessel fnctions Jn 1,q (, ; e iδ ) satisfy most of the properties of Jn 1,q (, ),as for example the bonds (5), the addition theorem (16) and the sm rles (34) and (36). These fnction are, howeer, complex aled. Figre 3 shows the real part of the Bessel fnctions Jn 1, (, ; i) for n =, 1,. A comparison with figre 1 shows that the strctre of the fnctions is strongly altered by the angle parameter δ..7. Differential eqations and recrrence relations Finally, we will derie recrrence relations for Jn p,q (, ) and their deriaties. Differentiating the generating fnction () with respect to leads to 1 (zp z p ) e (zp z p )+ (zq z q) = 1 Jn p,q (, )(z n+p z n p ) = Jn p,q (, )z n. n= n= (4) Eqating the coefficients of z n, we find Jn p,q (, ) = Jn p(, p,q ) Jn+p(, p,q ) (43) and similarly Jn p,q (, ) = Jn q(, p,q ) Jn+q p,q (, ). (44) If we differentiate () with respect to z and compare the coefficients, we find the recrrence eqation p ( Jn p(, p,q ) + Jn+p(, p,q ) ) + q ( Jn q(, p,q ) + Jn+q p,q (, ) ) = njn p,q (, ). (45) These are generalizations of the relations deried by Reiss [5] for the case Jn 1, (, ). Similarly one can show that the deriaties of the generalized Bessel fnctions (39) are gien by Jn 1,q (, ; e iδ ) = J 1,q n 1 (, ; eiδ ) J 1,q n+1 (, ; eiδ ) (46) Jn 1,q (, ; e iδ ) = e iδ Jn q(, 1,q ; e iδ ) e iδ Jn+q(, 1,q ; e iδ ). Using these relations one can show that the two-dimensional Bessel fnctions sole a ariety of linear partial differential eqations, depending on their indices (p, q). These differential eqations can be constrcted systematically by adding p deriaties of different order sch that the different terms Jn p,q cancel each other for eery ale of n. For example the ordinary two-dimensional Bessel fnctions with q = p sole the wae eqation ( ) J 1,1 n (, ) =, (47)

9 On two-dimensional Bessel fnctions while the generalized Bessel fnctions with (p, q) = (1, ) and δ = π/ sole the timedependent Schrödinger eqation [18, 6] i Jn 1, (, ; i) = ( 1) Jn 1, (, ; i). (48) Frthermore, a repeated application of the differentiation rles (43), (44) and the recrrence relations (45) leads to the copled differential eqations [(p + q ) + (p + q ) + p + q n ]Jn p,q (, ) = pq ( Jn p+q(, p,q ) + Jn+p q(, p,q ) Jn p,q (, ) ) (49) for arbitrary indices (p, q). Except from the right-hand side this eqation is strctrally similar to the defining differential eqation of the ordinary one-dimensional Bessel fnctions. Eqations (49) can be decopled for q = νp, ν Z by applying again (45). For (p, q) = (1, ±1) this yields [( + ) + ( + ) + ( ± ) n ]Jn 1,±1 (, ) =, (5) while the respectie calclations for other ales of q lead to more complicated reslts. 3. Polynomial expansion for small argments We will first analyse the regime of small argments and and derie a leading order polynomial expansion. Following Wasiljeff [4], we expand the, -dependent part of the exponential fnction in (3) in a Taylor series: Jn p,q (, ) = 1 +π e i( sin pt+ sin qt nt) dt (51) π π = 1 π k= 1 k k! ( e ipt e ipt + e iqt e iqt ) k e int dt. (5) Using the polynomial formla (a + b + c + d) j = j! a α b β c σ d ζ α! β! σ! ζ!, (53) α,β,σ,ζ where the primed sm rns oer all indices with j = α+β +σ +ζ, one obtains after rearranging terms and carrying ot the integration the series expansion Jn p,q 1 α+β σ +ζ (, ) = j α!β!σ!ζ!. (54) j= α,β,σ,ζ Here the doble-primed sm incldes all nonnegatie integers with j = α + β + σ + ζ and n = (α β)p + (σ ζ)q. (55) The sm can be transformed into a more conenient form by introdcing l = α + β,f = l + α β,m = σ + ζ and g = m + σ ζ. After some elementary algebra, this yields Jn p,q (, ) = a (n,p,q) m l l,m (56) l+m l,m with a (n,p,q) l,m = f =,...,+l g=,...,+m n=p(f l)+q(g m) 1 f!(l f)!g!(m g)!. (57)

10 14956 HJKorschet al The lowest order approximation in this expansion can be fond in explicit form for the case p = 1. Then the lowest order term in (54)isgienby ( ) n n n n q,,, if / N and q mod(n, q) q +1 q q q ( ) n n n (α,β,σ,ζ)=, n + q,, if / N and q mod(n, q) q +1 q q q (,, nq ), n if N. (58) q In most cases it is gien by a single term, as for example in J 1, 3 (, ) 1 1,, J 4 (, ) 1. (59) Note that, becase n = 3 is not a mltiple of q =, the Bessel fnction J 1, 3 is identically eqal to zero on the -axis (see eqation (3)); howeer, J 1, 3 and J 1, 4 do not anish on the -axis, where we hae Jn 1,(, ) = J n() (/) n /n! (see eqation (3)). In certain cases more terms of the same minimm order j appear. This happens if n q / N and q mod(n, q) = q + 1. One can easily check that this reqires that q is odd, q = ν +1 and n = µq + ν + 1 with ν, µ N. In this case, the lowest order approximation reads Jn 1,q ν µ ( ) (, ) ν+µ+1 ν!µ! ν (6) µ +1 This yields the (approximate) nodal line = µ +1 ν +1 (61) for small and. As an example, we note q = 5 and n = 3, i.e. ν = and µ = 4 and therefore J 1,5 3 (, ) 4 ( 7!4! 3 + ). (6) 5 4. Asymptotic approximations In the examples of two-dimensional Bessel fnctions Jn p,q (, ) shown in figres 1 and for (p, q) = (1, ) and (p, q) = (1, 3), respectiely, one obseres a rich oscillatory strctre which will be analysed in the following. The skeleton of this strctre and alable approximations can be obtained asymptotically by means of the stationary phase approximation π dt h(t)e ig(t) π ±g (t t s ) h(t s) e ig(ts)±iπ/4. (63) s The sm extends oer all contribting real stationary points and the ± sign is chosen so that ±g (t s ) is positie (see, e.g., [7] for more details). Preios stdies of asymptotic approximations for two-dimensional Bessel fnctions [3, 1, 8] hae been restricted to the case p = 1,q = and special regions of the index n and argments,. Information abot the oscillatory strctre of the mltiariable Bessel fnctions Jn p,q (, ) can be obtained from asymptotic approximations for large argments and/or large indices. Notation: x is the largest integer x, x is the smallest integer >xand mod(n, q) = n q n q.

11 On two-dimensional Bessel fnctions We will consider three of the nmber of possible limits: the case when both argments and are large, whereas n remains fixed, and the case where one argment,, and the index n are large for fixed ale of the argment. Finally we will consider the limit where both ariables as well as the index n are large. In all cases we assme small fixed ales of the indices p and q Basic strctre for large argments and We base or analysis on the integral representation (3) Jn p,q (, ) = 1 +π dt e i(φ(t) nt), φ(t) = sin pt + sin qt. (64) π Here we will consider the asymptotic limit of large argments and assming that n is fixed, i.e. we identify g(t) = φ(t) in an application of (63). The condition φ (t) = p cos pt + q cos qt = (65) determines the stationary points t s (note that there are always pairs of sch stationary points with different sign de to the symmetry of the cosine fnction). The integral (64) isthen approximately gien by Jn p,q (, ) = 1 t s π φ (t s ) ei(φ(t j ) nt s ±π/4), (66) where the ± sign is gien by the sign of φ (t j ). The main contribtion to the sm is proided by real-aled stationary points, complex ones lead to exponentially decaying terms. At the points where two stationary points coalesce when the argments and are aried, the second deriatie φ (t) = p sin pt q sin qt (67) anishes and the approximation dierges. Crossing these bifrcation points, the fnction changes its character. In the present case, the bifrcations are determined by the simltaneos soltion of p cos pt s = q cos qt s and p sin pt s = q sin qt s. (68) This can be most easily satisfied if both sides of one of the two eqations are eqal to zero. We distingish two cases. Case (i) sin pt s = sin qt s = : for coprime p and q, this implies t s = ort s = π and therefore (from p cos pt s = q cos qt s )wehaep = q or p = ( 1) p+q q, respectiely. We therefore obtain the bifrcation lines =±p/q if one of the p, q is een (69) = p/q else. (7) Case (ii) cos pt s = cos qt s = : this implies pt s = π/+jπ and qt s = π/+kπ with integer j and k or (k +1)p = (j +1)q and (for coprime p and q) p = j + 1 and q = k + 1. With sin pt s = sin(π/+jπ) = ( 1) j and sin qt s = sin(π/+kπ) = ( 1) k the second condition in (68) leads to = ( 1) j+k p /q p = j +1, q = k +1. (71) The examples in figre 1 show Bessel fnctions Jn 1, (, ) for arios ales of n. Here q is een and from eqation (7) we find the bifrcation lines =±/. (7)

12 14958 HJKorschet al We obsere that the strctre of the Bessel fnctions changes if one crosses these lines. In the left and right sectors, we hae only two stationary points, ±t 1, whereas in the pper and lower sectors we hae for stationary points, ±t 1 and ±t, and conseqently a richer interference pattern. This will be analysed in more detail below. For the two-dimensional Bessel fnction Jn 1,3 (, ) displayed in figre both pper indices are odd and the bifrcation lines are gien by eqations (7) and (71): = /3 and = /9. (73) The qalitatie difference to the behaior of J 1, n Let s now analyse the fnction J 1, n (, ) in figre 1 is obios. (, ) in more detail working ot explicitly the 1, (, ) = J n (, ) (3)we stationary phase approximation. In iew of the symmetry Jn 1, can assme >inthe following for simplicity. The stationary phase condition cos t = cos t = ( cos t 1) (74) can be soled for c = cos t with soltion c ± = 1 8 ( / ± (/) +3). (75) In the region </<+ the necessary condition c ± 1 is met by c + and ice ersa by c in the region </<+. Note that in the interal </<+ both soltions flfil c ± 1. With t ± = arccos c ± and sin t ± =± 1 c± we arrie at φ ± = sin t ± + sin t ± =±( +c ± ) 1 c± (76) φ ± = sin t ± 4 sin t ± = ( +8c ± ) 1 c± (77) and with the definitions ( F + (, ) = π φ + φ + n arccos c + π ) for 4 else < (φ F (, ) = π φ cos + n arccos c π ) for 4 else <+ (78) (79) the final reslt can be written as Jn 1, (, ) F +(, ) + F (, ). (8) Here one shold be aware of the fact that in the region both of the terms F ± (, ) proide a non-anishing contribtion. This so-called primitie stationary phase approximation dierges at the bifrcation lines =±. If desired, it can be improed by taking complex stationary points into accont and by taming the diergences by niformization methods. In the limit the asymptotic approximation (8) simplifies drastically. Only F + contribtes for >(F for <) and with t ± =±π/,φ ± = φ ± =±we find Jn 1, (, ) ( π cos n π π ), (81) 4

13 On two-dimensional Bessel fnctions J(,) J(,) Figre 4. Two-dimensional Bessel fnctions Jn 1, (, ) with = 1 for n = (left) and n = 1 (right) as a fnction of (fll cre) in comparison with the stationary phase approximation (8) (open circles). which agrees with the well-known asymptotic approximation of the ordinary Bessel fnction J n () for large argments [9]. In the alternatie limit both terms, F + and F contribte. With c ± = 8 ± 1,φ ± = ± and φ ± =±4 we obtain ( + cos (n +1) π ) cos n een Jn 1, (, ) 4 ( π sin (n +1) π ) sin (8) n odd, 4 as already deried in [14]. As an illstration of the asymptotic formla (8), figre 4 shows the two-dimensional Bessel fnction Jn 1, (, ) in comparison with the stationary phase approximation for = 1 and n =, 1. With the exception of the icinity of the diergences at =± =±, the agreement is excellent. This approximation can be sed in order to determine the nodal lines of the two-dimensional Bessel fnctions which is of interest for applications in physics [15]. 4.. Large argment and large index n In this section we will consider the regime where the index n and one of the argments, e.g., are large. In iew of (3), we can assme n. Following the analysis applied to the special case Jn 1, (, ) by Reiss and Kraino [1], we separate the integral representation (3) intoa fast, e ig(t), and a slowly oscillating part: Jn p,q (, ) = 1 π dt e i sin pt e ig(t), g(t) = sin qt nt, (83) π and ealate the integral approximately by the method of stationary phase or the saddle point method if the stationary points are complex aled (for details see, e.g., [7]). The stationary points t s of g(t) are obtained from g (t s ) = as cos qt s = n (84) q with real-aled soltions for n<q and complex soltions otherwise. We discss these two cases separately. (i) For n<q the stationary points are t s ± =±(t +πs/q), s =, 1,,..., t = 1 q arccos n q, (85) i.e. a finite nmber in the interal <t s ± π.

14 1496 HJKorschet al With sin qt ± s =±sin qt we hae g ( t ± s ) = sin qt ± s nt ± s =± sin t n(t +πs/q)] (86) g ( t ± s ) = q sin qt s = q sin qt (87) and, sing sin pt s ± =±sin p(t +πs/q), the final reslt is Jn p,q 1 (, ) e ±i[ sin p(t + πs πq q )+ sin qt n(t + πs q ) π 4 ] sin qt s,± [ ( = cos sin p t πq + πs ) ( + sin qt n t + πs ) π ]. sin qt q q 4 s (88) We will work ot the case p = 1 and q = in more detail. Here we find for stationary points ±(t + sπ) with s = and 1 and therefore { Jn 1, 1 [ (, ) cos sin t + sin t nt π ] πsin t 4 [ + ( 1) n cos sin t + sin t nt π ] } 4 ( + cos( sin t ) cos sin t nt π ) n een = ( 4 πsin t sin( sin t ) sin sin t nt π ) (89) n odd 4 with t = 1 arccos n, sin t = 1 n 1 4, sin t = n 4. (9) In comparison with the semiclassical approximation deried in section 4.1, the reslt (89) agrees approximately with (8) also for small ales of n, as for example J 1, (, ) shown in figre 4 for = 1. Eqation (89) misses howeer the strctral transition at = and cannot describe the region >. (ii) For n>q the stationary points (85) are complex, t s = x s +iy with real part { sπ/q, > x s =, s =, ±1, ±,... (91) (s +1)π/q, < with <x s +π. The imaginary part is the same for all s: y ± =± 1 ( ) n q arccosh. (9) q The integral is approximately carried ot by the saddle point integration, where the integration path is deformed to a steepest decent cre passing throgh the saddle points [7]: π dt h(t)e ig(t) π ig (t s s ) h(t s) e ig(ts). (93) The second deriatie is ig (t s ) = iq sin qt s = q sinh qy (94)

15 On two-dimensional Bessel fnctions J(,) J(,) J(,) Figre 5. Comparison of the asymptotic approximation (97) (open circles) with the exact twodimensional Bessel fnction Jn 1, (, ) (fll line) for n = 3. Left: case (i) for = 64 sing eqation (89). Middle: case (ii) with = 1 sing eqation (97). Right: case (ii) with = +1 sing eqation (99). and the conditions for the integration path [7] can only be satisfied for the saddle points in the pper (lower) complex plane for > ( < ). With ig(t s ) = i( sin qt s nt s ) = sinh qy inx s ny (95) we obtain the reslt Jn p,q sinh qy n y e (, ) = e inx s e i sin p(xs+iy). πq sinh qy s (96) Let s again consider the special case p = 1,q = in more detail. Two saddle points contribte (x s = ±π/ with y for < orx s = and π with y + for > ) and eqation (96) simplifies. For <wefind Jn 1, sinh y ny ( e+ (, ) = cos cosh y n π ) sinh y (97) with y = 1 arccosh n, sinh y = n 1, cosh y = 4 in agreement with the reslt deried in [1]. For >the reslting approximation is non-oscillatory: Jn 1, sinh y ny { e cosh( sinh y) (, ) = 4π sinh y sinh( sinh y) with n n een n odd (98) (99) y = 1 arccosh n, sinh y = n 4 1. (1) Note that both asymptotic approximations (89) and (99) satisfy the symmetry relation J n (, ) = ( 1) n J n (, ) (cf eqation (5)). Figre 5 demonstrates the qality of the asymptotic approximation for n = 3 and = 64 (case (i)) or = 1 (case (ii)). Reasonable agreement is obsered for < 1. These simple approximations get worse in the icinity of n = q where they dierge. A finite reslt can be obtained sing an appropriate niformization techniqe, in the present case a Bessel niformization, e.g. a mapping onto an (ordinary) Bessel fnction [3] (see also[8] for an alternatie method).

16 1496 HJKorschet al Figre 6. Color map of the two-dimensional Bessel fnctions Jn 1, (, ) for n = 9 (left) and n = 3 (right). Note the different symmetries of these fnctions. Their oerall strctre can be explained by means of the bifrcation set shown in figre 7. The asymptotic approximations (89) and (97) proide explicit estimates for the nodal lines of Jn 1, (, ). Forn< we find { 1 n (j +1) π 4 = n een, j =, ±1, ±,... (11) jπ n odd and for n> we hae for <zeros at 1 n 4 = (j + n)π, j =, ±1, ±,... (1) These reslts are, of corse, in agreement with the zeros obsered in figre Large argments, and large index n As an example of the strctre of the two-dimensional Bessel fnctions for large indices, figre 6 shows Jn 1, (, ) for n = 9 and n = 3. These fnctions look qite similar, they are clearly distingished, howeer, by their symmetry property Jn 1,(, ) = ( 1)n Jn 1, (, ) (see eqation (5)), i.e. J 1, 3 is een and J 1, 9 is odd with respect to a reflection. Therefore J 1, 1, 1, 9 anishes on the -axis, J9 (,) =. The fnction J3 is symmetric on the -axis: Jn 1, 1, (, ) = Jn (,)(see eqation (3)), despite of the apparent asymmetry with respect to the reflection. In additions to the oscillatory pattern in the for sectors, we obsere a region close to the centre where the ales of the Bessel fnctions are small. This pattern can again be explained by a consideration of the asymptotic limit where both argments and the index n are large sing g(t) = sin pt + sin qt nt (13) in the stationary phase approximation (63). The stationary points t s are determined by g (t s ) = p cos pt s + q cos qt s n =. (14) The zeros of the second deriatie g (t) = p sin pt q sin qt (15) appearing in the denominator of (63) determine the bifrcation set of these soltions.

17 On two-dimensional Bessel fnctions I III V IV 4 II 4 4 Figre 7. Bifrcation cres of the stationary points for the two-dimensional Bessel fnction J 1, 3 (, ). Restricting orseles again to the case (p, q) = (1, ) these eqations simplify and can be soled in closed form: g (t s ) = cos t s +cos t s n = (16) with soltions c ± = cos t s = ( ) 8 ± n (17) 4 (here again each soltion c ± implies two stationary points t s becase of the symmetry of the cosine fnction). The bifrcation set the skeleton of the Bessel fnction is fond when g (t s ) = sin t s 4 sin t s = (18) is satisfied in addition to (16). Eliminating t s we find the soltions = (n ± )/ (19) (note that for n these straight lines agree with those stated aboe in eqation (7)) and the ellipse 16( + n/4) + n n = 1 (11) centred at (, ) = (, n/4) with half axes n and n/4. Inside this ellipse the stationary points (17) are complex, otside they are real. A brief calclation frthermore shows that the straight lines (19) are tangential to the ellipse (11). This bifrcation set is shown in figre 7. In the pper sector (I) between the bifrcation lines we hae 1 <c ± < +1, as well as in the lower sector (II) otside the ellipse. Hence we hae for real soltions t s in these regions and a corresponding oscillatory pattern. In the right sector (IV) two of these soltions become complex becase of c + > 1 and similarly in the left-hand sector (III) with c > 1. In the trianglar segment (V) in the lower sector aboe the ellipse, we hae c + > 1 and c > 1 and therefore no real stationary points. In the elliptic region with complex aled stationary points, the Bessel fnction is damped bt still oscillatory. An example is shown in figre 5 (right-hand side) which shows a ct throgh the Bessel fnction J 1, 3 (, ) shown in figre 6 for = 1 close to the elliptic bifrcation cre. A ct at = 64 (left-hand side) shows the oscillations in region (I). Note that the semiclassical approximations shown in figre 5 are the simplified ersions deeloped in section 4.. A more refined semiclassical analysis along the lines discssed aboe will proide a mch better agreement for larger ales of (compare also the treatment in [5]).

18 14964 HJKorschet al Acknowledgments Spport from the Detsche Forschngsgemeinschaft ia the Gradiertenkolleg Nichtlineare Optik nd Ultrakrzzeitphysik as well as from the Stdienstiftng des detschen Volkes is grateflly acknowledged. References [1] Appell P 1915 C. R. Acad. Sci [] Pérès J 1915 C. R. Acad. Sci [3] Nikisho A I and Rits V I 1964 So. Phys. JETP 1 59 [4] Wasiljeff A 1969 Z. Angew. Math. Phys. 389 [5] Reiss H R 198 Phys. Re. A 1786 [6] Becker W, Schlicher R R and Sclly M O 1986 J. Phys. B: At. Mol. Phys. 19 L785 [7] Becker W, Schlicher R R, Sclly M O and Wódkiewicz K 1987 J. Opt. Soc. Am. B [8] Dattoli G, Chiccoli C, Lorenztta S, Maino G, Richetta M and Torre A 199 J. Math. Phys [9] Paciorek W A and Chapis G 1994 Acta Cryst. A [1] Reiss H R and Kraino V P 3 J. Phys. A: Math. Gen [11] Keck F and Korsch H J J. Phys. A: Math. Gen. 35 L15 [1] Korsch H J and Mossmann S 3 Phys. Lett. A [13] Baer J 5 J. Phys. A: Math. Gen [14] Dattoli GandTorre A1996Theory and Applications of Generalized Bessel Fnctions (Rome: Aracne Editrice) [15] Klmpp A, Witthat D and Korsch H J 6 in preparation Preprint qant-ph/6817 [16] Dattoli G, Torre A, Lorenztta S, Maino G and Chiccoli C 1996 Noo Cimento B [17] Dattoli G, Giannessi L, Mezi M L and Torre A 199 Noo Cimento B [18] Dattoli G, Torre A, Lorenztta S, Maino G and Chiccoli C 1991 Noo Cimento B 16 1 [19] Dattoli G, Chiccoli C, Lorenztta S, Maino G, Richetta M and Torre A 1991 Noo Cimento B [] Lorenztta S, Maino G, Dattoli G, Torre A and Chiccoli C 1995 Rend. di Mat. Serie VII [1] Miller W 1964 Comm. Pre Appl. Math [] Mordell L J 1969 Diophantine Eqations (New York: Academic) [3] Dattoli G, Torre A, Lorenztta S and Maino G 1998 Compt. Math. Appl [4] Dattoli G 4 Integral Transforms Spec. Fnct [5] Dattoli G, Cesarano C and Sacchetti S Georgian Math. J [6] Dattoli G, Mari C, Torre A, Chiccoli C, Lorenztta S and Maino G 199 J. Sci. Compt [7] Marsden J E 1987 Basic Complex Analysis (New York: Freeman) [8] Lebner C 1981 Phys. Re. A [9] Abramowitz M and Stegn I A 197 Handbook of Mathematical Fnctions (New York: Doer) [3] Stine J R and Marcs R A 1973 J. Chem. Phys