Asymptotics for 2D whispering gallery modes in optical micro-disks with radially varying index
AASYMPTOTICS FOR 2D WHISPERING GALLERY MODES IN OPTICALMICRO-DISKS WITH RADIALLY VARYING INDEX
ST ´EPHANE BALAC, MONIQUE DAUGE, AND ZO¨IS MOITIERA
BSTRACT . Whispering gallery modes [WGM] are resonant modes displaying special fea-tures: They concentrate along the boundary of the optical cavity at high angular frequenciesand they are associated with (complex) scattering resonances very close to the real axis. As aclassical simplification of the full Maxwell system, we consider two-dimensional Helmholtzequations governing transverse electric [TE] or magnetic [TM] modes. Even in this 2Dframework, very few results provide asymptotic expansion of WGM resonances at high an-gular frequency. In this work, using multiscale expansions, we design a unified procedure toconstruct asymptotic quasi-resonances and associate quasi-modes that have the WGM struc-ture in disk cavities with a radially varying optical index. We show using the black-boxscattering approach that quasi-resonances are asymptotically close to true resonances. Morespecifically, using a Schr¨odinger analogy we highlight three typical behaviors in such opti-cal micro-disks, leading to three distinct asymptotic expansions for the quasi-resonances andquasi-modes. C ONTENTS
1. Introduction 21.1. Helmholtz equations for optical micro-disks 21.2. Circular cavity with constant optical index 31.3. Radially varying index: Main results 61.4. Organization of the paper 82. Families of resonance quasi-pairs 93. Classification of the three typical behaviors by a Schr¨odinger analogy 104. Case ( A ) Half-triangular potential well 114.1. Statements 114.2. Proof: General concepts 134.3. Proof: Specifics in case ( A ) 165. Case ( B ) Half-quadratic potential well 235.1. Statements 245.2. Proof 256. Case ( C ) Quadratic potential well 286.1. Statements 296.2. Proof 307. Proximity between quasi-resonances and true resonances 317.1. Separation of quasi-resonances, quasi-orthogonality of quasi-modes 317.2. Spectral-like theorems for resonances 327.3. Application of the spectral-like theorems to disks with radially varying index 34Appendix A. Technical lemmas 34A.1. Explicit solutions to some differential equations 34 Mathematics Subject Classification.
Key words and phrases.
Whispering gallery modes, Optical micro-disks, Scattering resonances, Asymptoticexpansions, Axisymmetry, Schr¨odinger analogy, Quasi-modes.The authors acknowledge support of the Centre Henri Lebesgue ANR-11-LABX-0020-01. a r X i v : . [ m a t h . SP ] M a r ST ´EPHANE BALAC, MONIQUE DAUGE, AND ZO¨IS MOITIER
A.2. Half harmonic oscillator 35A.3. Borel’s Theorem 37A.4. Additional result 38References 38
1. I
NTRODUCTION
Helmholtz equations for optical micro-disks.
The motivation of our work is the studyof light-wave propagation in optical micro-resonators. These optical devices, with micro-metric size, came to be important components in the photonic toolbox. They are basicallycomposed of a dielectric cavity coupled to waveguides or fibers for light input and output[10]. When resonance conditions are met, it is possible to confine light-waves in the cavityand to access a wide range of optical phenomena. The study of such scattering resonancesis the subject of this paper. More specifically, we focus our study on “Whispering GalleryModes” that are modes essentially localized inside the cavity and concentrated in a boundarylayer, and that are associated with scattering resonances close to the real axis.If a complete, 3-dimensional, modeling would require to solve the full Maxwell sys-tem, in many situations, the solution of 2-dimensional scalar harmonic equations bringsinsight in resonance phenomena and constitutes a reference configuration in the literaturein optics. Moreover, the 2-dimensional model can be obtained as an approximation of the3-dimensional ones by using an approach referred in the optical literature as the effectiveindex method [26].In the 2-dimensional model, the optical cavity is represented by a bounded plane domain,that we denote by Ω . This cavity is associated with an optical index n > which is a regularfunction of the position in the closure Ω of Ω . Outside Ω , the index n is equal to . There aretwo relevant scalar Helmholtz equations associated with such a configuration, correspondingto Transverse Electric (TE) modes or
Transverse Magnetic (TM) modes: For such a bi-dimensional optical cavity Ω with index n , the resonance pairs ( k, u ) are obtained by solvingthe following problem where p = 1 for TM modes and p = − for TE modes: Find k ∈ C , u (cid:12)(cid:12) Ω ∈ H (Ω) , u (cid:12)(cid:12) R \ Ω ∈ H loc ( R \ Ω) s.t. − div ( n p − ∇ u ) − k n p +1 u = 0 in Ω and R \ Ω[ u ] = 0 across ∂ Ω[ n p − ∂ ν u ] = 0 across ∂ Ω (1.1a)with a radiation condition at infinity. This radiation condition imposes that the solution u toproblem (1.1a), outside any disk D (0 , R Ω ) which contains Ω , has an expansion in terms ofHankel functions of the first kind H (1) m in the following form in polar coordinates: u ( x, y ) = (cid:88) m ∈ Z C m e i mθ H (1) m ( kr ) ∀ θ ∈ [0 , π ] , r > R Ω . (1.1b)Owing to Rellich theorem (see [16] for instance) the radiation condition (1.1b) implies thatthe imaginary part of k is negative and the modes, exponentially increasing at infinity. Suchwave-numbers k are the poles of the extension of the resolvent of underlying Helmholtzoperators when coming from the upper half complex plane.We note that, in the case when n is constant inside Ω , problem (1.1a)–(1.1b) appearsalso as a modeling of scattering by a transparent obstacle (see M OIOLA and S
PENCE [17,Remark 2.1] for a discussion of the models in acoustics and electromagnetics). In contrastwith impenetrable obstacles (see S J ¨ OSTRAND and Z
WORSKI [25]), transparent obstaclesor dielectric cavities may have resonances super-algebraically close to the real axis due to
SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 3 almost total internal reflection on a convex boundary (see P
OPOV and V
ODEV [23], andalso G
ALKOWSKI [8]). These resonances correspond to the whispering gallery modes we areinterested in.This work is motivated by the following observation found in the literature in optics:Whispering Gallery Mode (WGM) resonators are in most cases formed of dielectric materi-als with constant optical index n but this constitutes a potential limit in their performance andin the range of their applications. Resonators with spatially varying optical index, that fallunder the category of “graded index” structures [9], offer new opportunities to improve andenlarge the field of applications of these devices and start to be investigated in optics. Amongthe graded index structures investigated in the literature in optics, we can quote a modifiedform of the “Maxwell’s fish eye”, that can be implemented using dielectric material, wherethe optical index varies with the radial position in the micro-disk resonator as [4, 19] n ( r ) = α (cid:18) r R (cid:19) − where α > and R is the disk radius. In [28] the authors consider a micro-cavity made of aquadratic-index glass doped with dye molecules and the refractive index is written as n ( r ) = α − βr , where α, β > . In [31], an analysis of hollow cylindrical whispering gallerymode resonator is carried out where the refractive index of the cladding varies according to n ( r ) = β/r , β > R .Motivated by the above mentioned examples, we found interesting to investigate the casewhen Ω is a disk (with radius denoted by R ), and the optical index is a radial functionof the position: In polar coordinates ( r, θ ) centered at the disk center, n = n ( r ) . Takingadvantage of the invariance by rotation of equations (1.1), it is easily proved that any solution u associated with a p ∈ {± } and a resonance k ∈ C can be expanded as a Fourier sum u ( x, y ) = (cid:88) m ∈ Z w m ( r ) e i mθ and that each term u m ( x, y ) := w m ( r ) e i mθ is a solution of problem (1.1) associated with thesame p and the same k . Hence it is sufficient to solve, for any m ∈ Z , problem (1.1) with u of the form w m ( r ) e i mθ . Here m ∈ Z is referred as the polar mode index . The radial problemsatisfied by w : r (cid:55)→ w ( r ) when u = w ( r ) e i mθ is plugged into problem (1.1) is the followingradial problem (1.2a)–(1.2b) ≡ (1.2) depending on m : Find k ∈ C , w (cid:12)(cid:12) (0 ,R ) ∈ H ((0 , R ) , r d r ) , w (cid:12)(cid:12) ( R, ∞ ) ∈ H loc ([ R, ∞ ) , r d r ) s.t. − r ∂ r ( n p − r∂ r w ) + n p − (cid:16) m r − k n (cid:17) w = 0 in (0 , R ) and ( R, + ∞ )[ w ] = 0 for r = R [ n p − w (cid:48) ] = 0 for r = Rw (0) = 0 if m (cid:54) = 0 (1.2a)with the following outgoing wave condition at infinity deduced from (1.1b): w ( r ) = C H (1) m ( kr ) when r > R, with a constant C (cid:54) = 0 . (1.2b)1.2. Circular cavity with constant optical index.
As a fundamental illustrative example,let us consider a circular cavity with constant optical index n ≡ n > in Ω . Thoughapparently simple, this case is indeed already very rich. The use of partly analytic formulasprovides a lot of information on the resonance set and the associated modes. Let us sketch ST ´EPHANE BALAC, MONIQUE DAUGE, AND ZO¨IS MOITIER this now. For both TM and TE modes, solutions w of (1.2) have the form w ( r ) = J m ( n kr ) if r ≤ R J m ( n kR ) H (1) m ( kR ) H (1) m ( kr ) if r > R. (1.3)Here J m refers to the order m Bessel function of the first kind and H (1) m refers to the order m Hankel function of the first kind. In both TE and TM cases, the resonance k is obtained as asolution to the following non-linear equation, termed modal equation (recall that p = 1 forTM modes and p = − for TE modes) n p J (cid:48) m ( n Rk ) H (1) m ( Rk ) − J m ( n Rk ) H (1) m (cid:48) ( Rk ) = 0 . (1.4)For each value of the polar mode index m , the modal equation (1.4) has infinitely many solu-tions k ∈ C . We denote by R p [ n , R ]( m ) this set. Because J − m ( ρ ) = ( − m J m ( ρ ) and thesame for H (1) m , the two integers ± m provide the same resonance values, which reflects a de-generacy of resonances for disk-cavities. Thus, we can restrict the discussion to nonnegativeinteger values of m .In Figure 1, we show the complex roots of equation (1.4) when n = 1 . , R = 1 , and ≤ m ≤ , the values of m being distinguished by a color scale. This figure displays clearlythe general features of the set of resonances. For each m the set of resonances R p [ n , R ]( m ) can be split into two parts, see [3], • an infinite part R p, inner [ n , R ]( m ) made of inner resonances for which the modes areessentially supported inside the disk Ω • a finite part R p, outer [ n , R ]( m ) made of outer resonances for which the modes areessentially supported outside the disk Ω .The sets of inner and outer resonances are given by R p, inner [ n , R ] = (cid:91) m ∈ N R p, inner [ n , R ]( m ) and R p, outer [ n , R ] = (cid:91) m ∈ N R p, outer [ n , R ]( m ) respectively. It appears that for TM modes ( p = 1 ) there exists a negative threshold τ suchthat the outer resonances satisfy Im k < τ , and the inner ones, Im k ≥ τ . We can clearlysee on Fig 1 some organization in sub-families, not indexed by m (i.e., m varies along thesefamilies). Observation of the associated modes shows that these families depend on anotherparameter, j , which can be called a radial mode index : This is the number of sign changes (ornodal points) of the real part of an associated mode. For inner resonances, the sign changesoccur inside the disk. For outer resonances, there is no such interpretation in term of signchanges but they can be linked to the resonances of the exterior Dirichlet problem: Onecan see in [2, Eq. (49)] that when m → + ∞ , outer resonances tend to zeros of the Hankelfunction k (cid:55)→ H (1) m ( Rk ) .Inner resonances will be denoted by k p ; j ( m ) , with p = 1 and p = − according to the TMand TE cases respectively, and with m and j the polar and radial mode indices, respectively.Then there exist two distinct asymptotics for k p ; j ( m ) according to j → ∞ or m → ∞ .On the one hand, direct calculations yield, [18, Section 3.3.1], when p = ± and j → + ∞ k p ; j ( m ) ∼ jπRn + (2 m + 2 − p ) π Rn + i2 n ln (cid:18) n − n + 1 (cid:19) . Thus, as j → ∞ , the imaginary part of k p ; j ( m ) tends to the negative value n ln( n − n +1 ) .For the example displayed in Figure 1, this value is − . . This same value can also befound in the physical literature, see [6, Eq. (13)] for example. SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 5 ( A ) p = +1 ( B ) p = − ( C ) p = +1 ( D ) p = − F IGURE
1. Roots of (1.4) when p ∈ {± } , n = 1 . , R = 1 , and ≤ m ≤ . The first row gives a global view, whereas the second row provides azoom on inner resonances.On the other hand, using asymptotic expansions of Bessel’s functions involved in themodal equation (1.4), one obtains that resonances k p ; j ( m ) for a given radial index j satisfythe following asymptotic expansion when m → + ∞ : k p ; j ( m ) = mRn (cid:34) a j (cid:18) m (cid:19) − n p n − (cid:18) m (cid:19) + 3 a j (cid:18) m (cid:19) − a j n p (3 n − n p )12( n − (cid:18) m (cid:19) + O (cid:0) m − (cid:1) (cid:35) . (1.5)For details, we refer to [15] where computations were carried out for a sphere and thereforespherical Bessel’s functions appears in this latter case in the modal equation instead of cylin-drical Bessel’s functions. In the asymptotic expansion (1.5), < a < a < a < · · · are thesuccessive roots of the flipped Airy function A : z ∈ C (cid:55)→ Ai ( − z ) where Ai denotes the Airyfunction. It is important to note that the terms in the asymptotic expansions are real : Hencethe imaginary part of k p ; j ( m ) is contained in the remainder. This part of the resonance setcorrespond to typical whispering gallery modes. ST ´EPHANE BALAC, MONIQUE DAUGE, AND ZO¨IS MOITIER m = 5 m = 10 m = 20 m = 40 . − i 10 − . . − i 10 − . . − i 10 − . . − i 10 − . . − i 10 − . . − i 10 − . . − i 10 − . . − i 10 − . . − i 10 − . . − i 10 − . . − i 10 − . . − i 10 − . F IGURE
2. Plots of real parts of TM modes u in a circular cavity of radius R = 1 and index n = 1 . (computed by solving the modal equation (1.4)using complex integration [22]). Below each plot, we give values of computedresonances k ∈ C . Each row corresponds to a distinct value of j , from to .We observe the following whispering gallery mode features when j is chosen and m getslarge, see Fig. 2:(1) The analytic resonances obtained by solving the modal equation (1.4) have a negativeimaginary part which tends to zero rapidly (exponentially) when m → ∞ .(2) The analytic modes u = w m ( r ) e i mθ with w m given by (1.3) and k solution of themodal equation (1.4) concentrate around the interface between the disk and the exte-rior medium.1.3. Radially varying index: Main results.
The proof of the formula (1.5) given in [15]relies on the modal equation (1.4) and makes use of asymptotic formulas for Bessel functions[20, 1]. Such an approach is specific to disks with constant optical index, and the number ofterms in the expansion is limited by the asymptotics as m → ∞ of Bessel functions availablein the literature.In this paper we develop a more versatile approach, based on multiscale expansions andsemiclassical analysis. The idea is to consider h = m as small parameter and to take advan-tage of the factor m in front of r in the first equation of (1.2a) to transform this equationinto a semiclassical 1-dimensional Schr¨odinger operator with a singular potential V . Gener-ically, V will have a potential well at r = R . We perform an asymptotic construction ofquasi-resonances and quasi-modes in the vicinity of r = R , in such a way that we can rely SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 7 on general arguments to deduce the existence of true resonances close to quasi-resonancesmodulo O ( m −∞ ) , i.e., more rapidly than any polynomial in m . Unless explicitly mentioned,we suppose the following. Assumption 1.1.
The radial function n : r (cid:55)→ n ( r ) satisfies the following properties(1) n ( r ) = 1 if r > R ;(2) The function r (cid:55)→ n ( r ) belongs to C ∞ ([0 , R ]) and n ( r ) > for all r ≤ R .This assumption motivates the following notations. Notation 1.2.
Let n ( R ) , n (cid:48) ( R ) , n (cid:48)(cid:48) ( R ) denote the limit values of n ( r ) and its derivatives as r (cid:37) R ; We set n = n ( R ) , n = n (cid:48) ( R ) , n = n (cid:48)(cid:48) ( R ) . (1.6)Let ˘ κ be the effective adimensional curvature defined by ˘ κ := R (cid:18) R + n n (cid:19) . (1.7)In this paper we prove expansions of resonances as m → ∞ in three distinct cases dis-criminated by the sign of ˘ κ . Such expansions are “modulo O ( m −∞ ) ” in a sense definedbelow. Notation 1.3.
Let ( a m ) m ∈ N be a sequence of numbers: a m = O ( m −∞ ) means that ∀ N ∈ N , ∃ C N such that | a m | ≤ C N m − N , ∀ m ∈ N . Theorem 1.A.
Assume that the radial function n satisfies Assumption and ˘ κ > . (1.8) Choose p ∈ {± } and denote by R p [ n, R ] the resonance set solution to problem (1.1) . Thenfor any j ∈ N , there exists a smooth real function K p ; j ∈ C ∞ ([0 , t (cid:55)→ K p ; j ( t ) definingdistinct sequences k p ; j ( m ) = m K p ; j (cid:16) m − (cid:17) , ∀ m ≥ (1.9) that are close modulo O ( m −∞ ) to the resonance set R p [ n, R ] , i.e. for each m , there exists k m ∈ R p [ n, R ] such that k p ; j ( m ) − k m = O ( m −∞ ) . Let K (cid:96)p ; j be the coefficients of the Taylorexpansion of K p ; j at t = 0 . We have, with numbers a j being the successive roots of theflipped Airy function, K p ; j = 1 Rn , K p ; j = 0 , K p ; j = 1 Rn a j κ ) , (1.10) All coefficients K (cid:96)p ; j are calculable, being the solution of an explicit algorithm involvingmatrix products and matrix inversions in finite dimensions. We refer to Section 4.1 for more details. As a consequence of Theorem 1.A, for eachchosen p and j , there exists a sequence of true resonances m (cid:55)→ k p ; j ( m ) ∈ R p [ n, R ] suchthat k p ; j ( m ) = m (cid:34) N − (cid:88) (cid:96) =0 K (cid:96)p ; j (cid:18) m (cid:19) (cid:96) + O (cid:18) m (cid:19) N (cid:35) ∀ N ≥ . (1.11)This clearly generalizes (1.5).When ˘ κ is zero, the powers of m − are replaced by powers of m − . ST ´EPHANE BALAC, MONIQUE DAUGE, AND ZO¨IS MOITIER
Theorem 1.B.
Assume that the radial function n satisfies Assumption and that ˘ κ = 0 , and ˘ µ := 2 − R n n > . (1.12) Then for any j ∈ N , there exists a smooth real function K p ; j ∈ C ∞ ([0 , t (cid:55)→ K p ; j ( t ) defining distinct sequences k p ; j ( m ) = m K p ; j (cid:16) m − (cid:17) , ∀ m ≥ (1.13) that are close modulo O ( m −∞ ) to the resonance set R p [ n, R ] . The first coefficients of theTaylor expansion of K p ; j at t = 0 are K p ; j = 1 Rn , K p ; j = 0 , K p ; j = 1 Rn (4 j + 3) √ ˘ µ . (1.14)We refer to Section 5.1 for more details. Note that, in contrast to the case ˘ κ > , thecoefficients K (cid:96)p ; j are not calculable (except the first four of them) in the sense that theirdetermination needs the inversion of infinite dimensional matrices.Unlike the two previous cases for which the quasi-modes are localized near the interface r = R , in the third case the quasi-modes are localized near an internal circle r = R withsome R < R . Theorem 1.C.
Assume that the radial function n satisfies Assumption and that ˘ κ < .Let R ∈ (0 , R ) such that R n (cid:48) ( R ) n ( R ) = 0 and assume further that ˘ µ := 2 − R n (cid:48)(cid:48) ( R ) n ( R ) > . (1.15) Then a similar statement as in Theorem holds. The first coefficients of the Taylor expan-sion of K p ; j at t = 0 are now, instead of (1.14) , K p ; j = 1 R n ( R ) , K p ; j = 0 , K p ; j = 1 R n ( R ) (2 j + 3) √ ˘ µ . (1.16)We refer to Section 6.1 for more details. Note that in this case the coefficients K (cid:96)p ; j are allcalculable in the sense introduced in Theorem 1.A. Remark . In all cases covered by Theorems 1.A–1.C, the quasi-resonances k p ; j ( m ) arereal. We will see in the proofs that the associated quasi-modes u p ; j ( m ) are localized in theradial variable (close to the interface r = R in the fisrt two cases and close to the internalcircle r = R in the third one). The couples ( k p ; j ( m ) , u p ; j ( m )) are in fact quasi-pairs fora transmission problem in a larger bounded domain D containing Ω . Such a problem canbe viewed as self-adjoint. This explains why that the k p ; j ( m ) are real. They can be bridgedwith the true complex resonances solution of problem (1.1) through general results by T ANG and Z
WORSKI [29], and S
TEFANOV [27].1.4.
Organization of the paper.
In section 2, we make precise the notion of quasi-pairs,quasi-resonances, and quasi-modes. In section 3, by means of the Schr¨odinger analogy, weclassify the three main types of localized resonance modes that can be observed in circulardielectric cavities. In sections 4–6, we construct quasi-pairs associated with localized res-onances in these three cases. Finally, in section 7, we show that the quasi-resonances justconstructed are asymptotically close to true resonances of the cavity, hence ending the proofof Theorems 1.A–1.C.The set of non-negative integers is denoted by N and the set of positive integer by N ∗ . Wedenote by L (Ω) the space of square-integrable functions on the open set Ω , and by H (cid:96) (Ω) SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 9 the Sobolev space of functions in L (Ω) such that their derivatives up to order (cid:96) belong to L (Ω) . Finally, S ( I ) denotes the space of Schwartz functions on the unbounded interval I .2. F AMILIES OF RESONANCE QUASI - PAIRS
Inspired by quasi-pair constructions used to investigate ground states in semiclassical anal-ysis of Schr¨odinger operators (see S
IMON [24] for instance), we are going to construct fami-lies of resonance quasi-pairs for problem (1.1). Here appears a fundamental difference: Thequasi-pair construction in semiclassical analysis consists in building approximate eigenpairs ( λ h , u h ) that solve A h u h = λ h u h with increasingly small error as h → where A h is forinstance the operator − h ∆ + V . In our case, we do not have any given semiclassical param-eter h . However, the term m r in the first equation of (1.2a) may play the role of a confiningpotential in a semiclassical framework if we set h = 1 | m | . This means that an internal frequency parameter m can be viewed as a driving parameter foran asymptotic study. This leads to the next definition for quasi-pairs, adapted to our problem. Definition 2.1.
Choose p ∈ {± } . A family of resonance quasi-pairs F p for problem (1.1)is formed by a sequence K p = ( k ( m )) m ≥ of real numbers called quasi-resonances and asequence U p = ( u ( m )) m ≥ of complex valued functions called quasi-modes, where for each m ≥ , the couple ( k ( m ) , u ( m )) is a quasi-pair for problem (1.1) with an error in O ( m −∞ ) when m → ∞ . More precisely, we mean that(1) For any m ≥ , the functions u ( m ) belong to the domain of the operator and arenormalized, u ( m ) ∈ H p ( R , Ω) and (cid:107) u ( m ) (cid:107) L ( R ) = 1 where H p ( R , Ω) = (cid:8) u ∈ L ( R ) (cid:12)(cid:12) u (cid:12)(cid:12) Ω ∈ H (Ω) , u (cid:12)(cid:12) R \ Ω ∈ H ( R \ Ω) , [ u ] ∂ Ω = 0 , and [ n p − ∂ ν u ] ∂ Ω = 0 (cid:9) . (2.1)(2) We have the following quasi-pair estimate as m → + ∞ , (cid:13)(cid:13) − div (cid:0) n p − ∇ u ( m ) (cid:1) − k ( m ) n p +1 u ( m ) (cid:13)(cid:13) L ( R ) = O (cid:0) m −∞ (cid:1) . (2.2)(3) Uniform localization: There exists a function X ∈ C ∞ ( R ) , ≤ X ≤ , such that (cid:107) X u ( m ) (cid:107) L ( R ) ≥ and (2.2) holds with X u ( m ) replacing u ( m ) . (4) Regularity with respect to m : There exist a positive real number β and a smoothfunction K ∈ C ∞ ([0 , t (cid:55)→ K( t ) such that k ( m ) m = K( m − β ) ∀ m ≥ . (2.3)If the cut-off function X in item (3) can be taken as any function that is ≡ in a neighborhoodof ∂ Ω for m large enough, we say that the family F p is a family of whispering gallery type . Remark . By Taylor expansion of the function K at t = 0 , we obtain that a consequenceof (2.3) is the existence of coefficients K (cid:96) , (cid:96) ∈ N , and constants C N such that ∀ N ≥ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ( m ) m − N − (cid:88) (cid:96) =0 K (cid:96) m − (cid:96)β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N m − Nβ . (2.4)Note that the asymptotics (1.5) satisfies such an estimate with β = and N = 6 . Remark . The estimate (2.2) implies a bound from below for the resolvent of the un-derlying operator, compare with [17, § blow up of theresolvent .3. C LASSIFICATION OF THE THREE TYPICAL BEHAVIORS BY A S CHR ¨ ODINGERANALOGY
In our way to prove Theorems 1.A–1.C, we transform the family of problems (1.2a) when m spans N ∗ into a family of 1-dimensional Schr¨odinger operators depending on the semi-classical parameter h = m .Namely, choosing a polar mode index m ∈ N ∗ and coming back to the ODE contained inproblem (1.2a) divided by n p +1 , we obtain the equation: − rn p +1 ∂ r ( n p − r∂ r w ) + m r n w − k w = 0 (3.1)As a start, we write a quasi-resonance as (compare with (1.5)) k ( m ) = m Λ where the number Λ depends on h and has to be found. Multiplying (3.1) by h = 1 /m , wefind that (3.1) takes the form of a one dimensional semiclassical Schr¨odinger modal equation − h H w + W w = Λ w, (3.2)where H is the second order differential operator H = 1 rn p +1 ∂ r ( n p − r∂ r ) (3.3)and W is the potential W ( r ) = (cid:18) r n ( r ) (cid:19) . (3.4)The operator − h H + W is self-adjoint on L ( R + , n p +1 r d r ) . We note that lim r (cid:37) R W ( r ) = (cid:18) R n (cid:19) and lim r (cid:38) R W ( r ) = (cid:18) R (cid:19) . (3.5)Since n > , we have a potential barrier at r = R . The first and second derivatives of W on (0 , R ] are given by W (cid:48) ( r ) = − (cid:18) r n ( r ) (cid:19) (cid:20) r + n (cid:48) ( r ) n ( r ) (cid:21) , (3.6a) W (cid:48)(cid:48) ( r ) = 2 (cid:18) r n ( r ) (cid:19) (cid:34) (cid:18) r + n (cid:48) ( r ) n ( r ) (cid:19) − n (cid:48) ( r ) rn ( r ) − n (cid:48)(cid:48) ( r ) n ( r ) (cid:35) . (3.6b)The local minima (potential wells) of W cause the existence of resonances near these energylevels and their asymptotic structure as h → is determined by the Taylor expansion of W at its local minima. Let us recall that ˘ κ = R (cid:0) R + n (cid:48) ( R ) n ( R ) (cid:1) . The sign of ˘ κ (if it is positive, zero,or negative) discriminates three typical behaviors in which case we will be able to constructfamilies of resonance quasi-pairs (see Theorems 1.A, 1.B, 1.C):( A ) ˘ κ > . Then W is decreasing on a left neighborhood of R and has a local minimumat R . In a two-sided neighborhood of R , W is tangent to a half-triangular potentialwell , see Fig. 3 ( A ). SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 11
R r ( A ) ˘ κ > R r ( B ) ˘ κ = 0 RR r ( C ) ˘ κ < F IGURE
3. The three typical local behaviors of the potential W : half-triangular potential well ( ˘ κ > ), half-quadratic potential well ( ˘ κ = 0 ) andquadratic potential well ( ˘ κ < ).( B ) ˘ κ = 0 . In this case, we assume that W (cid:48)(cid:48) ( R ) > , which is ensured by the condition R − n (cid:48)(cid:48) ( R ) n ( R ) > with R + n (cid:48) ( R ) n ( R ) = 0 . (3.7)Then W has a local minimum at R . In a two-sided neighborhood of R , W is tangentto a half-quadratic potential well , see Fig. 3 ( B ).( C ) ˘ κ < . Then W has no local minimum at R . But, since lim r → + W ( r ) = + ∞ ,it has at least one local interior minimum R over (0 , R ) . Now we assume that W (cid:48)(cid:48) ( R ) > , which is ensured by the condition R − n (cid:48)(cid:48) ( R ) n ( R ) > with R + n (cid:48) ( R ) n ( R ) = 0 . (3.8)Then W has a local non-degenerate minimum at R where it is tangent to a quadraticpotential well , see Fig. 3 ( C ).4. C ASE ( A ) H ALF - TRIANGULAR POTENTIAL WELL
The case ˘ κ > is in a certain sense the most canonical one, since it includes constantoptical indices n ≡ n inside Ω . In this section, after stating the result, we perform thedetails of construction of families of resonance quasi-pairs.4.1. Statements.
Recall that Assumption 1.1 is supposed to hold and Notation 1.2 is in use.We give now, in the case when ˘ κ is positive, the complete description of the quasi-pairs thatwe construct in the rest of this section. This statement has to be combined with Theorem 7.Dto imply Theorem 1.A. Theorem 4.A.
Choose p ∈ {± } . If ˘ κ > , there exists for each natural integer j , a familyof resonance quasi-pairs F p ; j = ( K p ; j , U p ; j ) of whispering gallery type (cf Definition )for which the sequence of numbers K p ; j = ( k p ; j ( m )) m ≥ and the sequence of functions U p ; j = ( u p ; j ( m )) m ≥ have the following properties:(i) The regularity property (2.3) – (2.4) with respect to m holds with β = : There existcoefficients K (cid:96)p ; j for any (cid:96) ∈ N , and constants C N such that ∀ N ≥ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k p ; j ( m ) m − N − (cid:88) (cid:96) =0 K (cid:96)p ; j m − (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N m − N/ . (4.1) The coefficients K p ; j (degree ) are all equal to Rn , the coefficients of degree are zero,and the coefficients of degree are all distinct with j , see (4.5) . (ii) The functions u p ; j ( m ) forming the sequence U p ; j have the form u p ; j ( m ; x, y ) = w p ; j ( m ; r ) e i mθ (4.2) where the radial functions w p ; j ( m ) have a boundary layer structure around r = R withdifferent scaled variables σ as r < R and ρ as r > R : σ = m (cid:16) rR − (cid:17) if r < R and ρ = m (cid:16) rR − (cid:17) if r > R. (4.3) This means that there exist smooth functions Φ p ; j ∈ C ∞ ([0 , , S ( R − )) : ( t, σ ) (cid:55)→ Φ p ; j ( t, σ ) and Ψ p ; j ∈ C ∞ ([0 , , S ( R + )) : ( t, ρ ) (cid:55)→ Ψ p ; j ( t, ρ ) such that w p ; j ( m ; r ) = X ( r ) (cid:16) r
The asymptotics of k p ; j ( m ) starts as k p ; j ( m ) = mRn (cid:34) a j (cid:18) κm (cid:19) − n p (cid:112) n − (cid:18) κm (cid:19) + k p ; j (cid:18) κm (cid:19) + k p ; j (cid:18) κm (cid:19) + O (cid:0) m − (cid:1)(cid:35) (4.5)where, as before, the a j are the successive roots of the flipped Airy function and the coeffi-cients k p ; j and k p ; j are given by k p ; j = a j (cid:18) − κ + 1˘ κ (cid:18) − R n n (cid:19)(cid:19) , k p ; j = − a j n p (cid:112) n − (cid:18) n − n p n − − κ + 2˘ κ (cid:18) − R n n (cid:19)(cid:19) . Remark . Note that the second term of (4.5) separates the families F p ; j , while the thirdterm distinguishes the TM ( p = 1 ) and TE ( p = − ) modes.4.1.2. Modes.
The asymptotic expansions of the radial part of the quasi-modes w p ; j ( m ) in(4.2) starts as w p ; j ( m ; r ) = X ( r ) (cid:32) W p ; j ( m ; r ) + (cid:18) m (cid:19) W p ; j ( m ; r ) (cid:33) + O (cid:16) m − (cid:17) , (4.6)where, using the scaled variables σ = m ( rR − and ρ = m ( rR − W p ; j ( m ; r ) = (cid:40) A (cid:16) a j + (2˘ κ ) σ (cid:17) if r < R, if r ≥ R, (4.7)and W p ; j ( m ; r ) = − n p (2˘ κ ) (cid:112) n − (cid:40) A (cid:48) (cid:0) a j + (2˘ κ ) σ (cid:1) if r < R, A (cid:48) ( a j ) exp (cid:0) − √ n − n ρ (cid:1) if r ≥ R . (4.8)
SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 13
Special case of a constant optical index.
In the constant index case n ≡ n > , wehave ˘ κ = 1 and we find the following -term expansion of the resonances for the circularcavity. For an improved readability, we distinguish the TM and TE case and we denote k p ; j by k TM j if p = 1 and by k TE j if p = − . We have k TM j ( m ) = mRn (cid:34) a j (cid:18) m (cid:19) − n n − (cid:18) m (cid:19) + 3 a j (cid:18) m (cid:19) − a j n n − (cid:18) m (cid:19) + 10 − a j (cid:18) m (cid:19) + a j n ( n − n − (cid:18) m (cid:19) − a j (cid:18) a j n ( n − (cid:19) (cid:18) m (cid:19) + O (cid:0) m − (cid:1) (cid:35) (4.9)and k TE j ( m ) = mRn (cid:34) a j (cid:18) m (cid:19) − n ( n − (cid:18) m (cid:19) + 3 a j (cid:18) m (cid:19) − a j (3 n − n ( n − (cid:18) m (cid:19) + 18 (cid:18) − a j
350 + 1 n ( n − (cid:19) (cid:18) m (cid:19) + a j (3 n + 12 n − n − n + 8)80 n ( n − (cid:18) m (cid:19) − a j (cid:18) a j n − n + 12 n + 45 n − n ( n − (cid:19) (cid:18) m (cid:19) + O (cid:0) m − (cid:1) (cid:35) (4.10)4.2. Proof: General concepts.
As explained in Sect. 3, the problem under considerationhas the form (3.2) of the semi-classical Schr ¨odinger equation − h H w + W w = Λ w , where H is a modified Laplacian, W is a potential, discontinuous at the interface r = R , and h = m is the semiclassical parameter. Recall that in both cases ( A ) and ( B ), the potential W has alocal minimum at R , with the distinctive feature that for r < R , the shape of W is triangularin case ( A ), and quadratic in case ( B ). The rationale of the quasi-resonance construction is tolocalize equation around the well bottom r = R and to scale variables appropriately so thatequation (3.2) can be solved by a multiscale power expansion. In this section, we describethe general concepts of the proof, common to the two cases ( A ) and ( B ).4.2.1. Localization around the interface.
The localization starts with the introduction of thedimensionless variable ξ = rR − ∈ ( − , + ∞ ) for which the disk boundary is translated tothe origin. Accordingly, we denote by ˜ n the optical index function in this new variable, viz ˜ n : ξ (cid:55)→ n (cid:0) R (1 + ξ ) (cid:1) and, for all q ∈ N , we set ˜ n q = ˜ n ( q ) (0) , the q -th derivative of ˜ n at . Referring to Notation1.2, we have ˜ n q = R q n q and (cf (1.7) and (1.12)) ˘ κ = 1 + ˜ n ˜ n and ˘ µ = 2 − ˜ n ˜ n . (4.11)Since ˜ n = n , we will most often use the notation n . The minimum of W at r = R is its left limit W := lim r (cid:37) R W ( r ) = ( Rn ) − . Using thechange of variables r (cid:55)→ ξ , we set L ( ξ, ∂ ξ ) = R n H ( r, ∂ r ) , V ( ξ ) = R n ( W ( r ) − W ) , and (cid:101) Λ = R n (Λ − W ) , so that equation (3.2) is transformed into − h L v + V v = (cid:101) Λ v, (4.12a)with the new unknown function v ( ξ ) = w ( R (1 + ξ )) . We have L ( ξ, ∂ ξ ) = n ˜ n ( ξ ) ∂ ξ + n (cid:18) ξ ) ˜ n ( ξ ) + ( p −
1) ˜ n (cid:48) ( ξ )˜ n ( ξ ) (cid:19) ∂ ξ V ( ξ ) = (cid:18) n (1 + ξ ) ˜ n ( ξ ) (cid:19) − . Note that the potential V has as local minimum at ξ = 0 (well bottom).The unknown function v satisfies furthermore the following jump condition at ξ = 0 deduced from (1.2a) [ v ] { } = 0 and (cid:2) ˜ n p − ∂ ξ v (cid:3) { } = 0 . (4.12b)and the decay conditions in Schwarz spaces when ξ → ±∞ : v − := v (cid:12)(cid:12) R − ∈ S ( R − ) and v + := v (cid:12)(cid:12) R + ∈ S ( R + ) . (4.12c)Our concern now is to construct quasi-resonances and quasi-modes localized around the wellbottom ξ = 0 , solutions to (4.12a)–(4.12c) in an asymptotic sense.4.2.2. Principal part of the Schr¨odinger modal equation.
The structure of quasi-pairs isdetermined by the principal part of problem (4.12a)–(4.12c) defined as: − h L v + V v = Λ v (4.13a)where(1) The operator L = ( L − , L +0 ) = ( ∂ ξ , n ∂ ξ ) is the principal part of L frozen at ξ = 0 on the left and on the right,(2) The associated jump conditions are v − (0) = v + (0) and n p − ∂ ξ v − (0) = ∂ ξ v + (0) . (4.13b)and the decay condition is the same as above v − ∈ S ( R − ) and v + ∈ S ( R + ) . (4.13c)(3) The potential V = ( V − , V +0 ) is the first nonzero term in the left and right Taylorexpansions of V at ξ = 0 . In any of the cases ( A ) and ( B ), V +0 = n − > , whereas V − ( ξ ) = (cid:40) − κ ξ in case ( A ) ˘ µ ξ in case ( B ) . In order to cover both cases ( A ) and ( B ) in a unified way, we will assume moregenerally that V − ( ξ ) = γ | ξ | κ , ξ < , with γ > , κ > (4.13d)The system (4.13a)–(4.13c) can be solved by a formal series expansion according to thefollowing procedure: SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 15 (i) Scale the variable ξ differently on the left and on the right of the origin, introducing σ = ξ/h α for ξ < and ρ = ξ/h α (cid:48) for ξ > (4.14)with α and α (cid:48) > chosen in order to homogenize the operators − h L − + V − and − h L +0 + V +0 . We find that − h L − + V − becomes − h − α ∂ σ + γh κ α | σ | κ , and − h L +0 + V +0 becomes − h − α (cid:48) ∂ ρ + n − , implying to choose α = 22 + κ and α (cid:48) = 1 . (4.15)(ii) Expand the new functions ϕ ( σ ) := v ( ξ ) for ξ < and ψ ( ρ ) := v ( ξ ) for ξ > , (4.16)and Λ in series of type (cid:80) q ∈ N a q h qβ for some suitable β > . The jump condition(4.12b) being transformed into the following matching condition at σ = ρ = 0 ϕ (0) = ψ (0) and n p − h − α ∂ σ ϕ (0) = h − α (cid:48) ∂ ρ ψ (0) , (4.17)we find that α , α (cid:48) , and α (cid:48) − α should be integer multiples of β .4.2.3. Back to the full Schr¨odinger modal equation.
Now, we take advantage of the choicesmade in (i)–(ii) to treat the system (4.12a)–(4.12c) in its general form. Hence we knowthat α (cid:48) = 1 and leave α in equations for further determination. By the change of variables(4.14)–(4.15) and the change of functions (4.16), the equation (4.12a) is transformed into thefollowing two equations set on each side of the interface σ = ρ = 0 (cid:40) h − α ( − L − h ϕ + V − h ϕ ) = (cid:101) Λ ϕ, σ ∈ ( −∞ , − L + h ψ + V + h ψ = (cid:101) Λ ψ, ρ ∈ (0 , + ∞ ) (4.18a)with the matching condition ϕ (0) = ψ (0) and n p − h − α ∂ σ ϕ (0) = ∂ ρ ψ (0) , (4.18b)the decay condition ϕ ∈ S ( R − ) and ψ ∈ S ( R + ) , (4.18c)and where the operators L − h and L + h are defined by L − h = n ˜ n ( h α σ ) ∂ σ + h α n (cid:18) h α σ ) ˜ n ( h α σ ) + ( p −
1) ˜ n (cid:48) ( h α σ )˜ n ( h α σ ) (cid:19) ∂ σ L + h = n (cid:18) ∂ ρ + h hρ ∂ ρ (cid:19) (4.19)and the potentials V − h and V + h are given by V − h ( σ ) = h α − (cid:32)(cid:18) n (1 + h α σ ) ˜ n ( h α σ ) (cid:19) − (cid:33) ,V + h ( ρ ) = n (1 + hρ ) − . (4.20) Formal series of operators.
The next step is to associate a formal series of operatorsto the system (4.18a)–(4.18c), using a Taylor expansion at σ = ρ = 0 of their coefficients:For any smooth coefficient f , this association reads f ( h α σ ) ∼ (cid:88) (cid:96) ∈ N h α(cid:96) f ( (cid:96) ) (0) (cid:96) ! σ (cid:96) and f ( hρ ) ∼ (cid:88) (cid:96) ∈ N h (cid:96) f ( (cid:96) ) (0) (cid:96) ! ρ (cid:96) . This defines a formal series of operators in terms of powers of h β − L ± h + V ± h ∼ (cid:88) q ∈ N h qβ A ± q (4.21)inviting to look for ϕ , ψ , and (cid:101) Λ in the form of the formal series ϕ ( σ ) = (cid:88) q ∈ N h qβ ϕ q ( σ ) , ψ ( ρ ) = (cid:88) q ∈ N h qβ ψ q ( ρ ) , and (cid:101) Λ = (cid:88) q ∈ N h qβ (cid:101) Λ q . (4.22)Note that in the general framework (4.13d) we have A − = − ∂ σ + γ | σ | κ and A +0 = − n ∂ ρ + n − . (4.23)Finally, since we want to construct quasi-modes with a whispering gallery structure, we givepriority in (4.18a) to the equation in R − , which means that we look for an expansion of (cid:101) Λ starting as h − α . This motivates the introduction of λ = h α − (cid:101) Λ ∼ (cid:88) q ∈ N h qβ λ q , (4.24)so that Equations (4.18a) read (cid:40) − L − h ϕ + V − h ϕ = λϕ, σ ∈ ( −∞ , − L + h ψ + V + h ψ = h − α λψ, ρ ∈ (0 , + ∞ ) (4.25)still coupled with the matching condition (4.18b) and the decay condition (4.18c).4.3. Proof: Specifics in case ( A ). In case ( A ), ˘ κ is positive and the above general frame-work applies with the quantities κ = 1 , γ = 2˘ κ, α = , α (cid:48) = 1 , β = . This case is very close to the “toy model” considered in [5, Sec. III]. From expressions(4.19)–(4.21), we find that the first terms of the operator series A ± q are as follows A − = − ∂ σ + 2˘ κ | σ | , A − = 0 , A − = 2 ˜ n n σ ∂ σ − (cid:16) p − ˜ n n (cid:17) ∂ σ + c − σ , A +0 = − n ∂ ρ + n − , A +1 = 0 , A +2 = 0 , A +3 = − n ( ∂ ρ + 2 ρ ) , (4.26)(4.27) where c − = 3 + 4 ˜ n n + 3 ˜ n n − ˜ n n . For a comprehensive description of the general terms A ± q we need the introduction of polynomial spaces. Notation 4.2.
For q ∈ N , let P q denote the space of polynomials in one variable with degree ≤ q and P q ∗ the subspace of P q formed by polynomials P such that P (0) = 0 .A Taylor expansion at σ = ρ = 0 of the coefficients of L ± h and of V ± h allows to prove that Lemma 4.B.
For any integer q ≥ , there holds A − q = A − q ( σ ) ∂ σ + B − q ( σ ) ∂ σ + C − q ( σ ) with A − q ∈ P [ q ] , B − q ∈ P [ q ] − , C − q ∈ P [ q ]+1 A + q = B + q ( ρ ) ∂ ρ + C + q ( ρ ) with B + q ∈ P [ q ] − , C + q ∈ P [ q ] . SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 17
By the identifications (4.21)–(4.22), the system (4.25) with jump conditions (4.18b) isassociated with the formal series system of equations (cid:16)(cid:80) (cid:96) ∈ N h (cid:96) A − (cid:96) (cid:17) (cid:16)(cid:80) (cid:96) ∈ N h (cid:96) ϕ (cid:96) (cid:17) = (cid:16)(cid:80) (cid:96) ∈ N h (cid:96) λ (cid:96) (cid:17) (cid:16)(cid:80) (cid:96) ∈ N h (cid:96) ϕ (cid:96) (cid:17) in R − (cid:16)(cid:80) (cid:96) ∈ N h (cid:96) A + (cid:96) (cid:17) (cid:16)(cid:80) (cid:96) ∈ N h (cid:96) ψ (cid:96) (cid:17) = h (cid:16)(cid:80) (cid:96) ∈ N h (cid:96) λ (cid:96) (cid:17) (cid:16)(cid:80) (cid:96) ∈ N h (cid:96) ψ (cid:96) (cid:17) in R + (cid:16)(cid:80) (cid:96) ∈ N h (cid:96) ϕ (cid:96) (cid:17) = (cid:16)(cid:80) (cid:96) ∈ N h (cid:96) ψ (cid:96) (cid:17) at { } n p − h (cid:16)(cid:80) (cid:96) ∈ N h (cid:96) ϕ (cid:48) (cid:96) (cid:17) = (cid:16)(cid:80) (cid:96) ∈ N h (cid:96) ψ (cid:48) (cid:96) (cid:17) at { } (4.28)This system is equivalent to an infinite collection of systems obtained by equating the seriescoefficients: Namely, for q spanning N , (cid:80) q(cid:96) =0 A − (cid:96) ϕ q − (cid:96) = (cid:80) q(cid:96) =0 λ (cid:96) ϕ q − (cid:96) in R − (cid:80) q(cid:96) =0 A + (cid:96) ψ q − (cid:96) = (cid:80) q(cid:96) =2 λ (cid:96) − ψ q − (cid:96) in R + ϕ q (0) = ψ q (0) ψ (cid:48) q (0) = n p − ϕ (cid:48) q − (0) (4.29)where we agree that the right hand side of the second line is when q = 0 or q = 1 , and,likewise, the right hand side of the fourth line is when q = 0 .4.3.1. Initialization stage.
For q = 0 , the system (4.29) reads A − ϕ = λ ϕ in R − A +0 ψ = 0 in R + ϕ (0) = ψ (0) ψ (cid:48) (0) = 0 (4.30)for which we look for solutions ϕ ∈ S ( R − ) and ψ ∈ S ( R + ) .Since the equation A +0 ψ = 0 with the Neumann condition at has no non-zero solutionin S ( R + ) , it is natural to take ψ = 0 in (4.30). Then we are left with the following Airyeigen-problem on R − for ϕ − ϕ (cid:48)(cid:48) ( σ ) − κ σϕ ( σ ) = λ ϕ ( σ ) for σ ∈ ( −∞ , , and ϕ (0) = 0 whose decaying solutions can be expressed in terms of the mirror Airy function A . Recallthat a j for j ∈ N , denote the successive roots of A . We obtain immediately: Lemma 4.C.
Let j ∈ N . The couple of functions ( ϕ , ψ ) and the number λ defined by ϕ ( σ ) = A (cid:0) a j + (2˘ κ ) σ (cid:1) , ψ ( ρ ) = 0 , and λ = a j (2˘ κ ) solve (4.30) in S ( R − ) × S ( R + ) .Remark . The quasi-mode construction requires a cut-off at infinity at some stage. Sucha cut-off will be harmless to the satisfied equations if the functions ( ϕ , ψ ) , and more gen-erally ( ϕ q , ψ q ) , are exponentially decreasing when σ → −∞ and ρ → + ∞ . It is easy to seethat any such solution of (4.30) is proportional to one of the solutions given in Lemma 4.C. Sequence of nested problems and recurrence.
Reordering the terms in the system(4.29) of rank q , we can write it in the following form ( R ( A ) q ) − ϕ (cid:48)(cid:48) q ( σ ) − (2˘ κσ + λ ) ϕ q ( σ ) = λ q ϕ ( σ ) + S ϕq ( σ ) σ ∈ R − − n ψ (cid:48)(cid:48) q ( ρ ) + (cid:0) n − (cid:1) ψ q ( ρ ) = S ψq ( ρ ) ρ ∈ R + ϕ q (0) = ψ q (0) ψ (cid:48) q (0) = n p − ϕ (cid:48) q − (0) (4.31a)(4.31b)(4.31c)(4.31d) with right hand terms S ϕq and S ψq defined as (recall that A +1 = 0 ) S ϕq = − A − q ϕ + q − (cid:88) (cid:96) =1 ( λ (cid:96) − A − (cid:96) ) ϕ q − (cid:96) and S ψq = q (cid:88) (cid:96) =2 ( λ (cid:96) − − A + (cid:96) ) ψ q − (cid:96) . (4.32) Proposition 4.D.
Choose j ∈ N and define ( ϕ , ψ , λ ) according to Lemma 4.C. Then thereexist, for any q ≥ , • a unique λ q ∈ R • unique polynomials P ϕq ∈ P q ∗ , Q ϕq ∈ P q − , and P ψq ∈ P q − such that setting ϕ q ( σ ) = P ϕq ( σ ) A (cid:0) a j + (2˘ κ ) σ (cid:1) + Q ϕq ( σ ) A (cid:48) (cid:0) a j + (2˘ κ ) σ (cid:1) ∀ σ ∈ R − ψ q ( ρ ) = P ψq ( ρ ) exp (cid:0) − ρ (cid:112) − n − (cid:1) ∀ ρ ∈ R + (4.33) the collection ( ϕ , . . . , ϕ q , ψ , . . . , ψ q , λ , . . . , λ q ) solves the sequence of problems ( R ( A ) (cid:96) ) introduced in (4.31) for (cid:96) = 0 , . . . , q .Proof. We proceed by induction on q . For q = 0 , lemma 4.C provides λ , ϕ , and ψ solutions to ( R ( A ) ) and we readily obtain the polynomials P ϕ = 1 , Q ϕ = 0 , and P ψ = 0 . Let q ≥ and suppose that ( λ (cid:96) ) ≤ (cid:96) ≤ q − , ( ϕ (cid:96) ) ≤ (cid:96) ≤ q − , and ( ψ (cid:96) ) ≤ (cid:96) ≤ q − are solutions to problems ( R ( A ) (cid:96) ) for (cid:96) = 0 , . . . , q − , and satisfy (4.33).Using the expression (4.32) of S ψq combined with Lemma 4.B, we deduce from the induc-tion assumption that there exists a polynomial E ψq ∈ P q − such that S ψq ( ρ ) = E ψq ( ρ ) exp (cid:0) − ρ (cid:112) − n − (cid:1) . From Lemma A.1 in Appendix, there exists a unique polynomial (cid:101) P ψq ∈ P q − ∗ such that thefunction (cid:101) ψ defined by (cid:101) ψ q ( ρ ) = (cid:101) P ψq ( ρ ) exp( − ρ (cid:112) − n − ) is solution to (4.31b). It followsthat the sought function ψ q is given by ψ q ( ρ ) = (cid:0) a + (cid:101) P ψq ( ρ ) (cid:1) exp (cid:0) − ρ (cid:112) − n − (cid:1) where the constant a is determined from Neumann condition (4.31d). This defines thepolynomial P ψq as a + (cid:101) P ψq and hence P ψq ∈ P q − as desired.Let us now consider equation (4.31a). Using Lemma 4.B combined with the relation A (cid:48)(cid:48) ( z ) = − z A ( z ) , we deduce from the expression (4.32) of S ϕq and the induction assumptionthat there exist polynomials R ψq ∈ P q and T ϕq ∈ P q − such that S ϕq ( σ ) = R ϕq ( σ ) A (cid:0) a j + (2˘ κ ) σ (cid:1) + T ϕq ( σ ) A (cid:48) (cid:0) a j + (2˘ κ ) σ (cid:1) . From Lemma A.2 there exist unique polynomials P ϕq ∈ P q ∗ and (cid:101) Q ϕq ∈ P q − such that thefunction given by (cid:101) ϕ q ( σ ) = P ϕq ( σ ) A (cid:0) a j + (2˘ κ ) σ (cid:1) + (cid:101) Q ϕq ( σ ) A (cid:48) (cid:0) a j + (2˘ κ ) σ (cid:1) SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 19 is a solution to the ODE (compare it with (4.31a)) − (cid:101) ϕ (cid:48)(cid:48) q ( σ ) − (2˘ κσ + λ ) (cid:101) ϕ q ( σ ) = S ϕq ( σ ) . If we define ϕ q as follows: ϕ q ( σ ) = λ q (2˘ κ ) A (cid:48) (cid:0) a j + (2˘ κ ) σ (cid:1) + (cid:101) ϕ q ( σ ) with λ q = (2˘ κ ) A (cid:48) ( a j ) ( ψ q (0) − (cid:101) ϕ q (0)) , then ϕ q solves (4.31a) and the continuity condition (4.31c). (Note that we have A (cid:48) ( a j ) (cid:54) = 0 for all j ∈ N , see [21, Sect. 9.9(ii)].) Finally, we set Q ϕq = λ q (2˘ κ ) − + (cid:101) Q ϕq ∈ P q − . (cid:3) Calculating for q = 1 in Proposition 4.D provides P ϕ = 0 and explicit values for Q ϕ , P ψ and λ , from which we deduce Lemma 4.E.
We have λ = − n p κ (cid:112) n − and, for ( σ, ρ ) ∈ R − × R + , ϕ ( σ ) = − n p (2˘ κ ) (cid:112) n − A (cid:48) ( a j + (2˘ κ ) σ ) , ψ ( ρ ) = − n p (2˘ κ ) A (cid:48) ( a j ) (cid:112) n − (cid:18) − ρ (cid:113) − n − (cid:19) . Remark . From the proof of Proposition 4.D, we see that, for each chosen j and q ≥ ,the four operators { λ , . . . , λ q − , P ψ , . . . , P ψq − } (cid:55)−→ E ψq { E ψq , P ϕq − , Q ϕq − } (cid:55)−→ P ψq { λ , . . . , λ q − , P ϕ , . . . , P ϕq − , Q ϕ , . . . , Q ϕq − } (cid:55)−→ { R ϕq , T ϕq }{ R ϕq , T ϕq , P ψq } (cid:55)−→ { P ϕq , Q ϕq , λ q } act between finite dimensional spaces and can be identified to matrices. They result into analgorithm that can be derived and implemented in a computer algebra system to obtain theexpression of λ q , ϕ q , ψ q for q ≥ , see [18, Annexe D]. The coefficients of the polynomials P ϕq , Q ϕq , P ϕq are rational functions of the quantities (2˘ κ ) , (cid:112) n − , a j , A (cid:48) ( a j ) , n p , and n (cid:96) for all (cid:96) ∈ { , . . . , q } .4.3.3. Convergence.
Choose p ∈ {± } and a natural integer j . In a last stage, we have toprove that the formal series (cid:88) q ∈ N λ q h q , (cid:88) q ∈ N ϕ q h q , and (cid:88) q ∈ N ψ q h q , (4.34)obtained from Proposition 4.D give rise to a family of resonance quasi-pairs in the senseof Definition 2.1. Note that, by construction, the functions ϕ q and ψ q are exponentiallydecreasing at infinity, thus belong to S ( R − ) and S ( R + ) , respectively. Relying on Borel’stheorem [14, Thm. 1.2.6] and its variant given in Lemma A.4 in Appendix, we obtain theexistence of smooth functions having ( λ q ) q , ( ϕ q ) q and ( ψ q ) q as Taylor terms at . Combinedwith Lemma A.5, this yields the following results for the remainders of truncated seriesexpansions of formal series (4.34). Lemma 4.F.
Let ( λ q ) q ∈ N , ( ϕ q ) q ∈ N and ( ψ q ) q ∈ N given by Proposition 4.D. There exist smoothfunctions λ ∈ C ∞ ([0 , , Φ ∈ C ∞ ([0 , , S ( R − )) and Ψ ∈ C ∞ ([0 , , S ( R + )) such thatfor all ( h, σ, ρ ) ∈ [0 , × R − × R + and for all integer N ≥ , we have the following finiteexpansions with remainders λ ( h ) = N − (cid:88) q =0 h q λ q + h N R λN ( h ) , with R λN ∈ C ∞ ([0 , (4.35a) Φ( h ; σ ) = N − (cid:88) q =0 h q ϕ q ( σ ) + h N R ϕN ( h ; σ ) , with R ϕN ∈ C ∞ ([0 , , S ( R − )) (4.35b) Ψ( h ; ρ ) = N − (cid:88) q =0 h q ψ q ( ρ ) + h N R ψN ( h ; ρ ) with R ψN ∈ C ∞ ([0 , , S ( R + )) (4.35c)Note that the remainders at rank N = 0 simply coincide with the original function. Definition 4.5.
Choose a real number δ ∈ (0 , ) and a smooth cut-off function χ , ≤ χ ≤ ,such that χ ( ξ ) = 1 for | ξ | ≤ δ and χ ( ξ ) = 0 for | ξ | ≥ δ . We define for any integer m ≥ with the notation h = m − , the quantities: k ( m ) = mRn (cid:113) h λ ( h ) ,v ( m ; ξ ) = χ ( ξ ) (cid:40) Φ( h ; h − ξ ) , ξ ≤ h ; h − ξ ) , ξ > ξ ∈ ( − , + ∞ ) u ( m ; r, θ ) = v (cid:0) m ; rR − (cid:1) e i mθ ( r, θ ) ∈ (0 , + ∞ ) × R / π Z . We now show that the sequence ( k ( m ) , u ( m )) m ≥ is a family of “almost” quasi-pairs inthe sense of the following lemma. A further correction will have to be made to transformthis family into a true family of resonance quasi-pairs in the sense of Definition 2.1. Lemma 4.G.
The sequence ( k ( m ) , u ( m )) m ≥ defined above has the following properties: (i) For all m , the function u ( m ) is supported in an annulus around the interface r = R supp( u ( m )) ⊂ B (0 , R (1 + 2 δ )) \ B (0 , R (1 − δ )) . (ii) For all m , the function u ( m ) is piece-wise smooth up to the interface r = R : u ( m ) (cid:12)(cid:12) Ω ∈ C ∞ (cid:0) Ω (cid:1) and u ( m ) (cid:12)(cid:12) R \ Ω ∈ C ∞ (cid:0) R \ Ω (cid:1) . (iii) We have the following estimates for the jumps across the interface when m → + ∞ [ u ( m )] ∂ Ω = O (cid:0) m −∞ (cid:1) and (cid:2) n p − ∂ ν u ( m ) (cid:3) ∂ Ω = O (cid:0) m −∞ (cid:1) . (iv) Defining the residuals ε ( m ) := div (cid:0) n p − ∇ u ( m ) (cid:1) + k ( m ) n p +1 u ( m ) (4.36) we have the following estimates in Ω and R \ Ω when m → + ∞(cid:107) ε ( m ) (cid:107) L (Ω) + (cid:107) ε ( m ) (cid:107) L ( R \ Ω) (cid:107) u ( m ) (cid:107) L ( R ) = O (cid:0) m −∞ (cid:1) . Proof. (i) and (ii) are obvious consequences of the definition of u ( m ) .(iii) From Definition 4.5, we have, for all θ ∈ R / π Z , and with h = m [ u ( m )] ∂ Ω ( θ ) = [ v ( m ; ξ )] { ξ =0 } e i mθ = (cid:16) Ψ( h ; 0) − Φ( h ; 0) (cid:17) e i mθ . Let N ≥ . From (4.35b)–(4.35c), we deduce that Ψ( h ; 0) − Φ( h ; 0) = N − (cid:88) q =0 (cid:0) ψ q (0) − ϕ q (0) (cid:1) h q + h N (cid:16) R ψN ( h ; 0) − R ϕN ( h ; 0) (cid:17) . (4.37) SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 21
Since by construction ψ q (0) − ϕ q (0) = 0 for any q ∈ N , cf (4.31), we deduce from (4.37)that Ψ( h ; 0) − Φ( h ; 0) = O ( h N ) . This statement is true for any N ≥ , hence [ u ( m )] ∂ Ω = O ( m −∞ ) .We proceed in a similar way for the second jump condition. We have (cid:2) n p − ∂ ν u ( m ) (cid:3) ∂ Ω ( θ ) = R − (cid:2) n p − ∂ ξ v ( m, ξ ) (cid:3) { ξ =0 } e i mθ and (cid:2) n p − ∂ ξ v ( m, ξ ) (cid:3) { ξ =0 } = h − (cid:16) ∂ ρ Ψ( h ; 0) − n p − h ∂ σ Φ( h ; 0) (cid:17) = N − (cid:88) q =1 (cid:0) ψ (cid:48) q (0) − n p − ϕ (cid:48) q − (0) (cid:1) h q − + h N − (cid:16) − n p − ϕ (cid:48) N − (0) + ∂ ρ R ψN ( h ; 0) − n p − h ∂ σ R ϕN ( h ; 0) (cid:17) . From the jump relation (4.31d) ψ (cid:48) q (0) − n p − ϕ (cid:48) q − (0) for any q ≥ , we deduce that the abovequantity is a O ( h N − ) for any N ≥ , hence a O ( m −∞ ) .(iv) In order to prove the estimates on the residuals, it is enough to prove that the L norm ofthe residual ε ( m ) on Ω and on R \ Ω is O ( m −∞ ) and that (cid:107) u ( m ) (cid:107) L ( R ) = γ m − + O ( m − ) (4.38)for some positive constant γ . Given a parameter t > , we introduce the following weighted L (semi) norm on any interval I ⊂ R : (cid:107) w (cid:107) [ t ]( I ) = (cid:90) I ∩ ( − δ/t, ∞ ) | w ( τ ) | t (1 + τ t ) d τ. Let us first prove (4.38). We readily obtain from Definition 4.5, having set L := 2 πR , (cid:107) u ( m ) (cid:107) ( R ) = L (cid:18)(cid:13)(cid:13)(cid:13) χ ( · h )Φ( h ; · ) (cid:13)(cid:13)(cid:13) [ h ]( R − ) + (cid:13)(cid:13)(cid:13) χ ( · h )Ψ( h ; · ) (cid:13)(cid:13)(cid:13) [ h ]( R + ) (cid:19) . (4.39)From (4.35b) and (4.35c) considered with N = 1 , we have Φ( h ; σ ) = A ( a j + (2˘ κ ) σ ) + h R ϕ ( h ; σ )Ψ( h ; ρ ) = h R ψ ( h ; ρ ) where R ϕ ∈ C ∞ ([0 , , S ( R − )) and R ψ ∈ C ∞ ([0 , , S ( R + )) . We deduce that (cid:12)(cid:12)(cid:12) (cid:107) u ( m ) (cid:107) L ( R ) − √ L (cid:13)(cid:13) χ ( · h ) A ( a j + (2˘ κ ) · ) (cid:13)(cid:13) L [ h ]( R − ) (cid:12)(cid:12)(cid:12) ≤ C h ( h + h ) ≤ C (cid:48) h for some constants C and C (cid:48) . We now have to estimate the quantity (cid:13)(cid:13) χ ( · h ) A ( a j + (2˘ κ ) · ) (cid:13)(cid:13) [ h ]( R − ) = (cid:90) R − (cid:12)(cid:12)(cid:12) χ ( σh ) A ( a j + (2˘ κ ) σ ) | (cid:12)(cid:12)(cid:12) h (1 + σh ) d σ . We split the integral according to I − I − I with the three positive integrals I = h (cid:90) R − (cid:12)(cid:12)(cid:12) A ( a j + (2˘ κ ) σ ) (cid:12)(cid:12)(cid:12) d σI = h (cid:90) R − (cid:16) − χ (cid:0) σh (cid:1) (cid:17) (cid:12)(cid:12)(cid:12) A ( a j + (2˘ κ ) σ ) (cid:12)(cid:12)(cid:12) d σI = h (cid:90) R − (cid:12)(cid:12)(cid:12) χ ( σh ) A ( a j + (2˘ κ ) σ ) (cid:12)(cid:12)(cid:12) | σ | d σ . Since a primitive function of A is x (cid:55)→ A (cid:48) ( x ) + x A ( x ) , we find that I = h (2˘ κ ) − A (cid:48) ( a j ) . Moreover, since A is exponentially decreasing over R − , we find that I ≤ C h . Finally,Lemma A.6 in Appendix shows that I = O ( h ∞ ) .Let us now show that the L norm on Ω and on R \ Ω of the residual ε ( m ) defined in (4.36)is O ( m −∞ ) . Revisiting all variable changes and problem reformulations we find (cid:13)(cid:13) ε ( m ) (cid:13)(cid:13) L (Ω) = √ L (cid:13)(cid:13)(cid:13) h (cid:16) − L − h + V − h − λ ( h ) (cid:17) (cid:16) χ ( · h )Φ( h ; · ) (cid:17)(cid:13)(cid:13)(cid:13) L [ h ]( R − ) (4.40a) (cid:13)(cid:13) ε ( m ) (cid:13)(cid:13) L ( R \ Ω) = √ L (cid:13)(cid:13)(cid:13)(cid:16) − L + h + V + h − h λ ( h ) (cid:17) (cid:16) χ ( · h )Ψ( h ; · ) (cid:17)(cid:13)(cid:13)(cid:13) L [ h ]( R + ) (4.40b)Introducing the commutators (cid:104) L − h , χ ( · h ) (cid:105) and (cid:2) L + h , χ ( · h ) (cid:3) of the differential operators L ± h with scaled cut-off functions, we deduce from (4.40a)–(4.40b) the inequalities (cid:13)(cid:13) ε ( m ) (cid:13)(cid:13) L (Ω) ≤ √ L h ( N ϕ + N (cid:48) ϕ ) (4.41a) (cid:13)(cid:13) ε ( m ) (cid:13)(cid:13) L ( R \ Ω) ≤ √ L ( N ψ + N (cid:48) ψ ) (4.41b)where N ϕ = (cid:13)(cid:13)(cid:13) χ ( h · ) (cid:0) − L − h + V − h − λ (cid:1) Φ( h ; · ) (cid:13)(cid:13)(cid:13) L [ h ]( R − ) (4.42a) N (cid:48) ϕ = (cid:13)(cid:13)(cid:13)(cid:104) L − h , χ ( h · ) (cid:105) Φ( h ; · ) (cid:13)(cid:13)(cid:13) L [ h ]( R − ) (4.42b) N ψ = (cid:13)(cid:13)(cid:13) χ ( h · ) (cid:16) − L + h + V + h − h λ (cid:17) Ψ( h ; · ) (cid:13)(cid:13)(cid:13) L [ h ]( R + ) (4.42c) N (cid:48) ψ = (cid:13)(cid:13)(cid:13)(cid:2) L + h , χ ( h · ) (cid:3) Ψ( h ; · ) (cid:13)(cid:13)(cid:13) L [ h ]( R + ) . (4.42d)Both operators χ ( h · ) (cid:0) − L − h + V − h (cid:1) and χ ( h · ) (cid:0) − L + h + V + h (cid:1) are differential operators inthe form a ± ∂ + a ± ∂ + a ± with coefficients a ± i ( h , · ) belonging to C ∞ bounded ([0 , × R ± ) ,see(4.19)–(4.20). Hence, the formal series (4.21) gives rise to the following sequences offinite expansions with remainders: For any N ≥ , − L ± h + V ± h = N − (cid:88) q =0 h q A ± q + h N R ± N ( h ; · ) (4.43)where the remainders R ± N are differential operators of order such that χ ( h · ) R − N ( h ; · ) and χ ( h · ) R + N ( h ; · ) have coefficients belonging to C ∞ bounded ([0 , × R ± ) . It follows that for anygiven N, N (cid:48) ∈ N (cid:0) − L − h + V − h − λ (cid:1) Φ( h ; · )= (cid:32) N − (cid:88) q =0 h q ( A − q − λ q ) + h N (cid:16) R − N ( h ; · ) − R λN ( h ) (cid:17)(cid:33) (cid:32) N (cid:48) − (cid:88) q =0 h q ϕ q + h N (cid:48) R ϕN (cid:48) ( h ; · ) (cid:33) = h N (cid:32) N − (cid:88) q =0 ( A − q − λ q ) R ϕN − q ( h ; · ) + (cid:16) R − N ( h ; · ) − R λN ( h ) (cid:17) R ϕ ( h ; · ) (cid:33) (4.44) SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 23 where the second equality is obtained from the relation (cid:80) q(cid:96) =0 ( A − (cid:96) − λ (cid:96) ) ϕ q − (cid:96) = 0 for all q ∈ N deduced from (4.29). In a similar way, we show that (cid:0) − L + h + V + h − h λ (cid:1) Ψ( h ; · ) = h N (cid:32) N − (cid:88) q =0 ( A + q − λ q − ) R ψN − q ( h ; · )+ (cid:16) R + N ( h ; · ) − R λN − ( h ) (cid:17) R ψ ( h ; · ) (cid:33) (4.45)where λ − = λ − = 0 . We deduce from from (4.44) and (4.45) that N ϕ + N ψ ≤ h N N (cid:88) q =0 (cid:18)(cid:13)(cid:13)(cid:13) F ϕq ( h ; · ) (cid:13)(cid:13)(cid:13) L [ h ]( R − ) + (cid:13)(cid:13)(cid:13) F ψq ( h ; · ) (cid:13)(cid:13)(cid:13) L [ h ]( R + ) (cid:19) with F ϕq ∈ C ∞ ([0 , , S ( R − )) and F ψq ∈ C ∞ ([0 , , S ( R + )) , and finally that N ϕ + N ψ ≤ C N h N (4.46)for some constant C N independent of h .Let us consider now the two commutators norms N (cid:48) ϕ and N (cid:48) ψ . We observe that the coef-ficients of the operators (cid:2) L − h , χ ( h · ) (cid:3) and (cid:2) L + h , χ ( h · ) (cid:3) are zero in the regions defined by − δh − ≤ σ ≤ and ≤ ρ ≤ δh − , respectively. This allows us to deduce that N (cid:48) ϕ + N (cid:48) ψ ≤ (cid:18)(cid:13)(cid:13)(cid:13) G ϕ ( h ; · ) (cid:13)(cid:13)(cid:13) L [ h ]( −∞ , − δh − ) + (cid:13)(cid:13)(cid:13) G ψ ( h ; · ) (cid:13)(cid:13)(cid:13) L [ h ]( δh − , + ∞ ) (cid:19) with functions G ϕ ∈ C ∞ ([0 , , S ( R − )) and G ψ ∈ C ∞ ([0 , , S ( R + )) . Lemma A.6 showsthat N (cid:48) ϕ + N (cid:48) ψ = O ( h ∞ ) . Combined with (4.46) true for all N ∈ N , this complete the proofof part (iv) of the Lemma. (cid:3) Proof of Theorem 4.A.
Choose p ∈ {± } and j ∈ N . In order to meet all the require-ments listed in Definition 2.1, we modify the sequence of functions ( u ( m )) m ≥ constructedin Definition 4.5, so that each such function satisfies the jump conditions in (2.1). To lift thejumps of u ( m ) , we define the “radial” function v ∗ ( m ; ξ ) = χ ( ξ ) − [ v ( m )] ξ =0 − n − p ξ [ n p − ∂ r v ( m )] ξ =0 ξ ≤ v ( m )] ξ =0 + ξ [ n p − ∂ ν v ( m )] ξ =0 ξ > (4.47)where χ can be taken as the same cut-off function used in Definition 4.5. We set u p ; j ( m ; r, θ ) := (cid:0) v (cid:0) m ; rR − (cid:1) − v ∗ (cid:0) m ; rR − (cid:1)(cid:1) e i mθ and k p ; j ( m ) := k ( m ) . Using (4.38), we can normalize the function u p ; j ( m ) in the L norm.Relying on Lemmas 4.F and 4.G it is easy to check that the family ( K p ; j , U p ; j ) where K p ; j =( k p ; j ( m )) m ≥ and U p ; j = ( u p ; j ( m )) m ≥ satisfies the four conditions of Definition 2.1.5. C ASE ( B ) H ALF - QUADRATIC POTENTIAL WELL
Our concern is now the case when ˘ κ = 0 and ˘ µ > . According to the same plan asbefore, we start with the complete description of the quasi-pairs that are constructed in therest of the section. The corresponding statement has to be combined with Theorem 7.D toimply Theorem 1.B. As mentioned earlier, Case ( A ) and Case ( B ) share general concepts inthe way the asymptotic expansion of quasi-pairs is obtained. Hence, we do not provide acomprehensive proof of Theorem 5.A but instead highlight the differences with the proof ofTheorem 4.A. Statements.Theorem 5.A.
Choose p ∈ {± } . Let Assumptions 1.1 be verified and according to (3.7) and notations (4.11) assume that ˘ κ = 0 and ˘ µ > . Then there exists for each j ∈ N , afamily of resonance quasi-pairs F p ; j = ( K p ; j , U p ; j ) of whispering gallery type (cf Definition ) with K p ; j = ( k p ; j ( m )) m ≥ and U p ; j = ( u p ; j ( m )) m ≥ .(i) The regularity properties (2.3) – (2.4) with respect to m holds with β = : There existcoefficients K (cid:96)p ; j for any (cid:96) ∈ N , and constants C N so that ∀ N ∈ N , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k p ; j ( m ) m − N − (cid:88) (cid:96) =0 K (cid:96)p ; j m − (cid:96)/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N m − N/ . (5.1) The coefficients K p ; j are all equal to ( R n ( R )) − , the coefficients of degree are zero, andthe coefficients of degree are all distinct with j , see (5.3) .(ii) The functions u p ; j ( m ) still have the form (4.2) with radial functions w p ; j ( m ) that havea boundary layer structure around r = R with the different scaled variables σ as r < R and ρ as r > R : σ = m (cid:0) rR − (cid:1) if r < R and ρ = m (cid:0) rR − (cid:1) if r > R. There exist smooth functions Φ p ; j ∈ C ∞ ([0 , , S ( R − )) : ( t, σ ) (cid:55)→ Φ p ; j ( t, σ ) and Ψ p ; j ∈ C ∞ ([0 , , S ( R + )) : ( t, ρ ) (cid:55)→ Ψ p ; j ( t, ρ ) such that w p ; j ( m ; r ) = X ( r ) (cid:16) r
Proof.
The general outline of the proof of Theorem 5.A is similar to the one of Theorem4.A in Case ( A ). In particular, the general framework of section 4.2 applies with κ = 2 , γ = ˘ µ, α = , α (cid:48) = 1 , β = . We provide now the part of the proof of Theorem 5.A specific to Case ( B ).From general expressions (4.19)–(4.21), we find that A − = − ∂ σ + ˘ µσ , A − = − ∂ σ + ( p − ∂ σ − σ (cid:18) ˜ n n + ˜ n n (cid:19) , A +0 = − n ∂ ρ + n − , A +1 = 0 , A +2 = − n ( ∂ ρ + 2 ρ ) . (5.8)(5.9) The analog of Lemma 4.B describing the coefficients of the formal series of operators(4.21) in terms of powers of h β = h reads as follows. Lemma 5.B.
For any integer q ≥ , A − q = A − q ( σ ) ∂ σ + B − q ( σ ) ∂ σ + C − q ( σ ) with A − q ∈ P q , B − q ∈ P q − , C − q ∈ P q +1 A + q = B + q ( ρ ) ∂ ρ + C + q ( ρ ) with B + q ∈ P [ q ] − , C + q ∈ P [ q ] . We proceed as in Section 4.3, associating to the system (4.25) a formal series system ofequations like (4.28), in which the powers of h are modified according to the values of α , α (cid:48) , β , and κ . As a matter of fact, equating the series coefficients, we obtain in Case ( B ) exactlythe same infinite collection of systems (4.29) as in Case ( A ), but with the new expressions ofoperators A ± q . The coefficients of the formal series expansions (4.22) are obtained by solving(4.29) for q spanning N .5.2.1. Initialization stage.
For q = 0 , the couple of functions ( ϕ , ψ ) and the number λ are obtained by solving (4.30) with A − and A +0 given in (5.8)–(5.9). Since the equation A +0 ψ = 0 with the Neumann condition at has no non-zero solution in S ( R + ) , it is naturalto take ψ = 0 . Then, we are left with the following harmonic oscillator problem on R − − ϕ (cid:48)(cid:48) ( σ ) − ˘ µ σ ϕ ( σ ) = λ ϕ ( σ ) for σ ∈ ( −∞ , , and ϕ (0) = 0 whose bounded solutions are generated by the odd Gauss-Hermite functions (cid:8) Ψ GH j +1 (cid:9) j ∈ N . Lemma 5.C.
Let j ∈ N . The couple of functions ( ϕ , ψ ) and the number λ defined by ϕ ( σ ) = Ψ GH j +1 (cid:16) ˘ µ σ (cid:17) , ψ ( ρ ) = 0 , and λ = (4 j + 3) (cid:112) ˘ µ solve (4.30) for A − and A +0 given in (5.8) – (5.9) . Sequence of nested problems and recurrence.
As in Case ( A ), reordering the termsin the system (4.29) taking into account Lemma 5.B, we obtain that the couple of functions ( ϕ q , ψ q ) and the number λ q for q ≥ are solutions to ( R ( B ) q ) − ϕ (cid:48)(cid:48) q ( σ ) + (˘ µσ − λ ) ϕ q ( σ ) = λ q ϕ ( σ ) + S ϕq ( σ ) σ ∈ R − − n ψ (cid:48)(cid:48) q ( ρ ) + (cid:0) n − (cid:1) ψ q ( ρ ) = S ψq ( ρ ) ρ ∈ R + ϕ q (0) = ψ q (0) ψ (cid:48) q (0) = n p − ϕ (cid:48) q − (0) (5.10a)(5.10b)(5.10c)(5.10d) with right hand side terms S ϕq and S ψq defined as (recall that A +1 = 0 ) S ϕq = − A − q ϕ + q − (cid:88) (cid:96) =1 ( λ (cid:96) − A − (cid:96) ) ϕ q − (cid:96) and S ψq = q (cid:88) (cid:96) =2 ( λ (cid:96) − − A + (cid:96) ) ψ q − (cid:96) . (5.11) Notation 5.2.
For a real number t let ω t : σ (cid:55)→ exp (2 t | σ | ) . We denote by L ( R − , ω t ) and H (cid:96) ( R − , ω t ) , the weighted Lebesgue and Sobolev spaces with measure ω t ( σ ) d σ . Proposition 5.D.
Choose j ∈ N and take ϕ , ψ , λ as given in Lemma 5.C. For any q ≥ ,there exist • a unique λ q ∈ R • a unique real number c q , a unique real sequence m ∈ N (cid:55)→ b iq such that b jq = 0 , anda unique polynomial P ψq ∈ P q − such that setting ϕ q ( σ ) = c q Ψ GH (cid:0) ˘ µ σ (cid:1) + (cid:101) ϕ q ( σ ) with (cid:101) ϕ q ( σ ) = (cid:88) i ∈ N b iq Ψ GH i +1 (cid:0) ˘ µ σ (cid:1) ∀ σ ∈ R − ψ q ( ρ ) = P ψq ( ρ ) exp (cid:0) − ρ (cid:112) − n − (cid:1) ∀ ρ ∈ R + (5.12) the collection ( ϕ , . . . , ϕ q , ψ , . . . , ψ q , λ , . . . , λ q ) solves the sequence of problems ( R ( B ) (cid:96) ) for (cid:96) = 0 , . . . , q . Moreover (cid:101) ϕ q ∈ H ( R − ) ∩ H ( R − , ω − q ) Proof.
The proof is quite similar to the one of Proposition 4.D and we will focus on the maindifferences. We proceed by induction on q . For q = 0 , Lemma 5.C provides λ , ϕ , and ψ solutions to ( R ( B ) ) and we readily obtain c = 0 , (cid:101) ϕ = Ψ GH j +1 (˘ µ · ) ∈ H ( R − ) , and P = 0 . Moreover, ϕ belongs to H ( R − , ω ) because Ψ GH j +1 is defined as the product of the (2 j + 1) -th order Hermite polynomial of degree j + 1 by x (cid:55)→ exp( − x ) .Let q ≥ and suppose that ( λ (cid:96) ) ≤ (cid:96) ≤ q − , ( ϕ (cid:96) ) ≤ (cid:96) ≤ q − , and ( ψ (cid:96) ) ≤ (cid:96) ≤ q − are solutions toproblems ( R ( B ) (cid:96) ) for (cid:96) = 0 , . . . , q − , and satisfy (5.12). Solving equation (5.10b) for ψ q proceed in a way very similar to (4.31b) in the proof of Proposition 4.D to show that thereexists P ψq ∈ P q − such that ψ q ( ρ ) = P ψq ( ρ ) exp (cid:0) − ρ (cid:112) − n − (cid:1) . Let us now consider equation (5.10b) for ϕ q . First of all, we obtain by induction that ϕ (cid:96) ∈ H ( R − , ω − q ) for all (cid:96) ∈ { , . . . , q − } . Then, using Lemma 5.B and (5.11), it followsthat S ϕq ∈ L ( R − , ω / − q ) . Note that the value of the constant q − in the exponentialweight is reduced by to − q in order to absorb the polynomials behavior. To solveequation (5.10a) with the non-homogeneous boundary condition (5.10c) we introduce asnew unknown (cid:101) ϕ q = ϕ q − c q Ψ GH (˘ µ · ) where c q = ψ q (0)Ψ GH (0) . It belongs to H ( R − , ω − q ) and the SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 27
Dirichlet problem (5.10a), (5.10c) becomes (cid:40) − (cid:101) ϕ (cid:48)(cid:48) q + (˘ µσ − λ ) (cid:101) ϕ q = λ q Ψ GH j +1 (˘ µ · ) + (cid:101) S ϕq ∀ σ ∈ ( −∞ , (cid:101) ϕ q (0) = 0 (5.13a)(5.13b) where (cid:101) S ϕq = S ϕq + 2(2 j + 1) √ ˘ µ c q Ψ GH (˘ µ · ) . Problem (5.13) has a solution only when theleft hand side function λ q Ψ GH j +1 (˘ µ · ) + (cid:101) S ϕq is orthogonal to Ψ GH j +1 (˘ µ · ) . It follows that wemust have λ q = − µ (cid:90) −∞ Ψ GH j +1 (˘ µ σ ) (cid:101) S ϕq ( σ ) d σ. (5.14)Finally, from Lemma A.3, there exists a unique (cid:101) ϕ q in H ( R − ) ∩ H ( R − , ω − q ) and orthogonalto Ψ GH j +1 solution to (5.13). The formula giving (cid:101) ϕ q is (cid:101) ϕ q = + ∞ (cid:88) i =0 , i (cid:54) = j i − j ) (cid:16) (cid:101) S ϕq , ς i Ψ GH i +1 (cid:17) L ( R − ) ς i Ψ GH i +1 (5.15)where ς i = (cid:107) Ψ GH i +1 (cid:107) − ( R − ) . (cid:3) Remark . In contrast to Case ( A ), we cannot deduce from Proposition 5.D a finite algo-rithm to compute the terms of the sequence ( ϕ q ) q ∈ N . The reason is that for q ≥ , the sumof the series (5.15) cannot be computed explicitly. However, a few terms are explicit: Weknow ( ψ , ϕ , λ ) so we can compute, first P ψ , then c , and, after this, S ϕ . With these latterquantities, we can deduce an explicit expression of (cid:101) S ϕ as a finite sum of polynomials timesGauss-Hermite functions. Now, from the definition of the Gauss-Hermite functions [11, Eq.1.3.8] and recurrence relations on Hermite polynomials [21, Sect. 18.9(i)], we deduce thefollowing recurrence relations for i ≥ and z ∈ R , ∂ z Ψ GH i ( z ) = (cid:0) i (cid:1) Ψ GH i − ( z ) − (cid:0) i +12 (cid:1) Ψ GH i +1 ( z ) , (5.16a) z Ψ GH i ( z ) = (cid:0) i (cid:1) Ψ GH i − ( z ) + (cid:0) i +12 (cid:1) Ψ GH i +1 ( z ) . (5.16b)Hence we can rewrite (cid:101) S ϕ as a finite sum of Gauss-Hermite functions and with this we cancompute explicitly λ given by (5.14). Nevertheless (cid:101) ϕ will be an infinite sum of Gauss-Hermite functions so, for q ≥ , λ q does not have a closed form. Lemma 5.E.
For all q ∈ N , we have ϕ q ∈ S ( R − ) and ψ q ∈ S ( R + ) .Proof. From the expression (5.12) of ψ q , it is obvious that it belongs to S ( R + ) .From Proposition 5.D we know that ϕ q and its derivatives of order ≤ are exponentiallydecaying as σ → ∞ . Concerning higher order derivatives ϕ ( i ) q , from the identity ϕ (cid:48)(cid:48) q =(˘ µσ − λ ) ϕ q − λ ϕ − S ϕq deduced from (5.10a), from (5.11) and Lemma 5.B, we find thatthere exists families of polynomials P (cid:96)q,i , Q (cid:96)q,i such that ϕ ( i ) q = q (cid:88) (cid:96) =0 (cid:0) P (cid:96)q,i ϕ (cid:96) + Q (cid:96)q,i ϕ (cid:48) (cid:96) (cid:1) . (5.17)Hence ϕ ( i ) q is exponentially decaying too, and we have proved that ϕ q belongs to S ( R − ) . (cid:3) Convergence.
The proof that the formal series (cid:88) q ∈ N λ q h q , (cid:88) q ∈ N ϕ q h q , and (cid:88) q ∈ N ψ q h q , (5.18)obtained from Proposition 5.D give rise to a family of resonance quasi-pairs in the sense ofDefinition 2.1 can be achieved exactly as in Section 4.3.3 for Case ( A ). Namely, Lemma 4.Fand Definition 4.5 are respectively replaced by the following Lemma 5.F and Definition 5.4. Lemma 5.F.
Let ( λ q ) q ∈ N , ( ϕ q ) q ∈ N and ( ψ q ) q ∈ N given by Proposition 5.D. There exist smoothfunctions λ ∈ C ∞ ([0 , , Φ ∈ C ∞ ([0 , , S ( R − )) and Ψ ∈ C ∞ ([0 , , S ( R + )) such thatfor all ( h, σ, ρ ) ∈ [0 , × R − × R + and for all integer N ≥ , we have the following finiteexpansions with remainders λ ( h ) = N − (cid:88) q =0 h q λ q + h N R λN ( h ) , with R λN ∈ C ∞ ([0 , (5.19a) Φ( h ; σ ) = N − (cid:88) q =0 h q ϕ q ( σ ) + h N R ϕN ( h ; σ ) , with R ϕN ∈ C ∞ ([0 , , S ( R − )) (5.19b) Ψ( h ; ρ ) = N − (cid:88) q =0 h q ψ q ( ρ ) + h N R ψN ( h ; ρ ) with R ψN ∈ C ∞ ([0 , , S ( R + )) (5.19c) Definition 5.4.
Choose a real number δ ∈ (0 , ) and a smooth cut-off function χ , ≤ χ ≤ ,such that χ ( ξ ) = 1 for | ξ | ≤ δ and χ ( ξ ) = 0 for | ξ | ≥ δ . We define for any integer m ≥ with the notation h = m − , the quantities: k ( m ) = mRn (cid:113) h λ ( h ) ,v ( m ; ξ ) = χ ( ξ ) (cid:40) Φ( h ; h − ξ ) , ξ ≤ h ; h − ξ ) , ξ > ξ ∈ ( − , + ∞ ) u ( m ; r, θ ) = v (cid:0) m ; rR − (cid:1) e i mθ ( r, θ ) ∈ (0 , + ∞ ) × R / π Z . One can show that the sequence ( k ( m ) , u ( m )) m ≥ is a family of “almost” quasi-pairs inthe sense of Lemma 4.G. The main difference with Case ( A ) in proving Lemma 4.G for thesequence ( k ( m ) , u ( m )) m ≥ introduced in Definition 5.4 is that we do not have anymore anexplicit expression for ϕ q but this does not prevent to obtain the same estimates as in Case( A ). We refer to [18] for details.5.2.4. Proof of Theorem 5.A.
A further correction will have to be made to transform the se-quence of functions ( u ( m )) m ≥ constructed in Definition 5.4 into a true family of resonancequasi-modes in the sense of Definition 2.1. We set u p ; j ( m ; r, θ ) := (cid:0) v (cid:0) m ; rR − (cid:1) − v ∗ (cid:0) m ; rR − (cid:1)(cid:1) e i mθ where v ∗ is defined as in (4.47) and k p ; j ( m ) := k ( m ) . Relying on Lemmas 5.F and theanalogous of 4.G for Case ( B ), one can check that the family ( K p ; j , U p ; j ) where K p ; j =( k p ; j ( m )) m ≥ and U p ; j = ( u p ; j ( m )) m ≥ satisfies the four conditions of Definition 2.1.6. C ASE ( C ) Q UADRATIC POTENTIAL WELL
We are now under Assumption (3.8). We recall that Case ( C ) corresponds to a situationwhere ˘ κ < and the potential W has no local minimum at R but has at least one local innerminimum R over (0 , R ) . This case falls into to the framework investigated by H ELFFER and
SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 29 S J ¨ OSTRAND [12]. Namely, the asymptotic expansions of quasi-resonances and quasi-modesare given in respectively Theorem 10.7 and Theorem 10.8 in [12]. Note however that theconstruction is not made explicit in [12]. Here, in contrast, we construct explicit families ofresonance quasi-pairs F p ; j localized around the circle r = R inside the cavity Ω . Note thatstrictly speaking, these families of resonance quasi-pairs are not of whispering gallery type.6.1. Statements.Theorem 6.A.
Choose p ∈ {± } . Let Assumptions 1.1 be verified and assume ˘ κ < . Let R ∈ (0 , R ) such that R n (cid:48) ( R ) n ( R ) = 0 and ˘ µ := 2 − R n (cid:48)(cid:48) ( R ) n ( R ) > , cf (1.15) . Then,for each j ∈ N , there exists a family of resonance quasi-pairs F p ; j = ( K p ; j , U p ; j ) with K p ; j = ( k p ; j ( m )) m ≥ and U p ; j = ( u p ; j ( m )) m ≥ .(i) The regularity property (2.3) – (2.4) with respect to m holds with β = , see (5.1) . Thecoefficients K p ; j are all equal to ( R n ( R )) − , the coefficients of degree are zero, and thecoefficients of degree are all distinct with j , see (6.2) .(ii) The functions u p ; j ( m ) still have the form (4.2) with radial functions w p ; j ( m ) thatare smooth in the scaled variables σ = m ( r/R − . There exists a smooth function Φ p ; j ∈ C ∞ ([0 , , S ( R )) : ( t, σ ) (cid:55)→ Φ p ; j ( t, σ ) such that w p ; j ( m ; r ) = X ( r ) Φ p ; j ( m − , σ ) (6.1) where X ∈ C ∞ ( R + ) , X ≡ in a neighborhood of R . These families of resonance quasi-pairs are not of whispering gallery type: The quasi-modes are strictly localized inside the cavity. The first terms of the asymptotic expansion of k p ; j are: k p ; j ( m ) = mR n ( R ) (cid:34) (cid:88) (cid:96) =1 k (cid:96)p ; j (cid:18) √ ˘ µm (cid:19) (cid:96) + O (cid:16) m − (cid:17)(cid:35) (6.2)with k p ; j = 0 , k p ; j = 2 j + 12 , k p ; j = 0 , and k p ; j = 164 (cid:20) − p + 8 p − p − µ − η − η µ − η µ + (2 j + 1) (cid:18) − µ + 10 η + η ˘ µ − η µ (cid:19)(cid:21) . where η = 6 + R n (3) ( R ) n ( R ) and η = 24 − R n (4) ( R ) n ( R ) . The asymptotic expansion of the quasi-modes starts with u p ; j ( ± m ; x, y ) = X ( r )Ψ GH j (cid:16) ˘ µ m (cid:16) rR − (cid:17)(cid:17) e ± i mθ + O (cid:16) m − (cid:17) . (6.3) Remark . As in Case ( B ), the quasi-resonances are organized in an asymptotic lattice withconstant step: The gap between two resonances with consecutive polar mode index m and m + 1 and the same radial mode index j is found to be k p ; j ( m + 1) − k p ; j ( m ) = 1 R n ( R ) + O (cid:0) m − (cid:1) , whereas when m is fixed and j is incremented by , the gap between two resonance is foundto be k p ; j +1 ( m ) − k p ; j ( m ) = √ ˘ µn ( R ) + O (cid:16) m − (cid:17) . Proof.
The proof of Theorem 6.A can be seen as a simpler version of the proof ofTheorem 5.A since the driving operator A − = − ∂ σ + ˘ µσ of the asymptotic expansionis the same quadratic oscillator on both side of the potential well location R , i.e. A +0 = A − . Therefore, we will not detail the entire proof of Theorem 6.A but we will focus on aninteresting byproduct of our approach compared to the results of [12], viz a finite algorithmfor computing the terms of the asymptotic expansion of the resonance quasi-pairs.In the framework of the Schr¨odinger analogy introduced in Section 3, we start this timeby introducing the dimensionless variable ξ = rR − (instead of ξ = rR − as in the twoprevious cases) and the unknown v such that v ( ξ ) = w ( R (1 + ξ )) . This leads to the sameequation (4.12a) where (cid:101) Λ = R ˜ n (0) (Λ − W ) with ˜ n ( ξ ) = n ( R (1 + ξ )) . Compared to thegeneral framework introduced in Section 4.2 for Cases ( A ) and ( B ), the potential V is smoothat its local minimum at ξ = 0 . As a consequence, it is not anymore necessary to introducea different scaling on both side of ξ = 0 . Moreover, it is still possible to take advantage ofthe framework of Section 4.2, but taking into account the fact the variable σ = h − α ξ mustbe considered over R and not only over R − . This framework applies with the same relevantquantities as in Case ( B ) (the ones affecting L on R − ). Denoting by ϕ the new unknownsuch that ϕ ( σ ) = v ( ξ ) , equation (4.12a) become − L h ϕ + V h ϕ = λϕ , σ ∈ R , where theoperator L h and the potentials V h have the same expressions than L − h in (4.19) and V − h of(4.20) with ˜ n ( ξ ) = n ( R (1 + ξ )) . The decay condition is ϕ ∈ S ( R ) .We define a formal series of operators in terms of powers of h , similarly to (4.21), as − L h + V h ∼ (cid:80) q ∈ N h q A q and we look for a function ϕ and a scalar λ in the form of theformal series ϕ = (cid:80) q ∈ N h q ϕ q and λ = (cid:80) q ∈ N h q λ q . One can show that the coefficients A q , q ∈ N , satisfy Lemma 5.B (the statement on A − q ). Then, by the same arguments as inCases ( A ) and ( B ) that can equally apply here, we obtain that ( ϕ , λ ) is solutions to the fullharmonic oscillator equation (in opposition to the half harmonic oscillator of Case ( B )) − ϕ (cid:48)(cid:48) ( σ ) + ˘ µ σ ϕ ( σ ) = λ ϕ , σ ∈ R , ϕ ∈ S ( R ) , (6.4)and that for q ≥ , ( ϕ q , λ q ) are solutions to the sequence of problems ( R ( C ) q ) (cid:40) − ϕ (cid:48)(cid:48) q ( σ ) + (˘ µσ − λ ) ϕ q ( σ ) = λ q ϕ ( σ ) + S ϕq ( σ ) σ ∈ R ϕ q ∈ S ( R ) (6.5a)(6.5b) with the right hand side term S ϕq defined as S ϕq = − A q ϕ + (cid:80) q − (cid:96) =1 ( λ (cid:96) − A (cid:96) ) ϕ q − (cid:96) .Solutions to the full harmonic oscillator equation (6.4) are ϕ ( σ ) = Ψ GH j (cid:16) ˘ µ σ (cid:17) and λ = (2 j + 1) (cid:112) ˘ µ ( j ∈ N ) . (6.6)For q ≥ , the features of the solution ( ϕ q , λ q ) to problem ( R ( C ) q ) are detailed in thefollowing proposition. Its proof below also provides an algorithm to compute ϕ q and λ q . Proposition 6.B.
Let j ∈ N and let ( ϕ , λ ) given by (6.6) . Then there exist, for any q ≥ ,a unique λ q ∈ R and a unique ( b iq ) i ∈{ ,...,j +3 q } ∈ R j +3 q +1 with b jq = 0 such that by setting ϕ q ( σ ) = j +3 q (cid:88) i =0 b iq Ψ GH i (cid:16) ˘ µ σ (cid:17) , ∀ σ ∈ R , (6.7) the collection ( ϕ , . . . , ϕ q , λ , . . . , λ q ) solves the sequence of problems ( R ( C ) (cid:96) ) (cid:96) =0 ,...,q . SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 31
Proof.
The relations (5.16) combined with Lemma 5.B show that there exists a ( j + 3 q + 1) -tuple ( d iq ) i ∈{ ,...,j +3 q } ∈ R j +3 q +1 such that S ϕq ( σ ) = j +3 q (cid:88) i =0 d iq Ψ GH i (cid:16) ˘ µ σ (cid:17) ∀ σ ∈ R . (6.8)Namely, on one hand, the relations (5.16) indicate that the (cid:96) -th derivative of Ψ GH i or thefunction obtained by multiplying Ψ GH i by z (cid:96) can be expressed as a linear combination ofGauss-Hermite function up to order i + (cid:96) . On the other hand, Lemma 5.B indicates that A − q ϕ and ( λ (cid:96) − A − (cid:96) ) ϕ q − (cid:96) , for ≤ (cid:96) ≤ q − can respectively be expressed as a linearcombination of Gauss-Hermite function up to order j + q + 2 and j + 3 q − (cid:96) + 2 , these twonumbers being bounded by j + 3 q .Equation (6.5a) has a solution in L ( R ) if, and only if, λ q ϕ + S ϕq is orthogonal to Ψ GH j (˘ µ · ) ; This implies that λ q = − d jq . Moreover since the operator − ∂ σ + ˘ µ σ − (2 j +1) √ ˘ µ is diagonalizable and inversible on span(Ψ GH i (˘ µ · ) | i ≥ , i (cid:54) = j ) , we get b iq = d iq i − j ) for i ∈ { , . . . , j + 3 q } \ { j } and b jq = 0 . (cid:3) Remark . From the proof of Proposition 6.B we can deduce a finite algorithm for the com-putation of the terms in the asymptotic expansion of the resonance quasi-pairs because theexpression of S ϕq in (6.8) involves a finite sum and because the computation of the solution ( λ q , ϕ q ) is explicit form the coefficients of S ϕq .The proof that the formal series (cid:80) q ∈ N λ q h q and (cid:80) q ∈ N ϕ q h q obtained from Proposition6.B give rise to a family of resonance quasi-pairs in the sense of Definition 2.1 can beachieved exactly as in Section 5.2.3 for Case ( B ). Note that in order to use Borel’s The-orem and to obtain the required estimates, we have to show that ϕ q ∈ S ( R ) ∩ H ( R , e | σ | d σ ) .This properties can be deduced directly from Equation (6.7).Finally, we can conclude with the proof of Theorem 6.A in a way very similar to the oneof Theorem 5.A as detailed in Section 5.2.4.7. P ROXIMITY BETWEEN QUASI - RESONANCES AND TRUE RESONANCES
Separation of quasi-resonances, quasi-orthogonality of quasi-modes.
For the threecases ( A ), ( B ), and ( C ), cf Theorems 4.A, 5.A, and 6.A, we have exhibited families of res-onance quasi-pairs in the sense of Definition 2.1. Namely, for each j ≥ and m ≥ , wehave constructed a quasi-pair ( k p ; j ( m ) , u p ; j ( m )) where k p ; j ( m ) ∈ R + is a quasi-resonanceand u p ; j ( m ) ∈ H p ( R , Ω) is a compactly supported quasi-mode. Actually, to each quasi-resonance k p ; j ( m ) , we can associate two quasi-modes: u p ; j ( m ) and its conjugate. Thesequasi-modes are quasi-orthogonal with respect to j and m , as stated in the next lemma.We consider the Hilbert space L ( R , n ( x ) p +1 d x ) and denote its scalar product by (cid:10) f, g (cid:11) = (cid:90) R f ( x ) g ( x ) n ( x ) p +1 d x for f, g ∈ L ( R , n ( x ) p +1 d x ) . Lemma 7.A.
For all the three cases ( A ) , ( B ) , and ( C ) , and for all i, j ≥ and m, m (cid:48) ≥ ,we have (cid:10) u p ; i ( m ) , u p ; j ( m (cid:48) ) (cid:11) = 0 and (cid:10) u p ; i ( m ) , u p ; j ( m (cid:48) ) (cid:11) = if m = m (cid:48) and i = j, if m (cid:54) = m (cid:48) , O (cid:0) m −∞ (cid:1) if m = m (cid:48) and i (cid:54) = j. For any m ≥ and i, j ≥ , we have the separation property k p ; i ( m ) − k p ; j ( m ) = (cid:40) C ( A ) ij m + O ( m ) in Case ( A ) ,C ( X ) ij m + O ( m ) in Cases ( B ) , ( C ) , (7.1) with C ( X ) ij (cid:54) = 0 if i (cid:54) = j .Proof. The relation (cid:104) u p ; i ( m ) , u p ; j ( m (cid:48) ) (cid:105) = 1 , for all j ≥ and m ≥ , comes from thenormalization of the quasi-mode in Definition 2.1. The relations (cid:104) u p ; i ( m ) , u p ; j ( m (cid:48) ) (cid:105) = 0 ,for all i, j ≥ and m, m (cid:48) ≥ , and (cid:104) u p ; i ( m ) , u p ; j ( m (cid:48) ) (cid:105) = 0 , for all i, j ≥ and m (cid:54) = m (cid:48) , m, m (cid:48) ≥ , are deduced from the identity (cid:82) π e i qθ d θ = 0 for all integer q (cid:54) = 0 .For the last estimate, we consider i (cid:54) = j , i, j ≥ , and m ≥ . By construction, there exists R q ∈ L ( R ) , for q ∈ { i, j } , such that (cid:107) R q (cid:107) L ( R ) = O ( m −∞ ) and k p ; q ( m ) n p +1 u p ; q ( m ) = − div (cid:0) n p − ∇ u p ; q ( m ) (cid:1) − R q . (7.2)Using this identity, conjugated, for q = i , we deduce: k p ; i ( m ) (cid:90) R u p ; i ( m ) u p ; j ( m ) n p +1 d x = − (cid:90) R div (cid:16) n p − ∇ u p ; i ( m ) (cid:17) u p ; j ( m ) d x − (cid:90) R R i u p ; j ( m ) d x. Integrating by parts and using again (7.2), we get (cid:0) k p ; i ( m ) − k p ; j ( m ) (cid:1) (cid:10) u p ; i ( m ) , u p ; j ( m ) (cid:11) L ( R ) = (cid:90) R (cid:16) u p ; i ( m ) R j − R i u p ; j ( m ) (cid:17) d x. Taking the modulus and using Cauchy-Schwarz inequality, we obtain (cid:12)(cid:12) k p ; i ( m ) − k p ; j ( m ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:10) u p ; i ( m ) , u p ; j ( m ) (cid:11) L ( R ) (cid:12)(cid:12)(cid:12) ≤ (cid:107) R i (cid:107) L ( R ) + (cid:107) R j (cid:107) L ( R ) = O (cid:0) m −∞ (cid:1) . Then we use the separation property (7.1) (which is an obvious consequence of asymp-totic formulas for k p ; j ( m ) in each case) and finally get the estimate (cid:104) u p ; i ( m ) , u p ; j ( m (cid:48) ) (cid:105) = O ( m −∞ ) . (cid:3) Spectral-like theorems for resonances.
We have constructed well separated quasi-pairs for the operator P := − n − p − div( n p − ∇· ) with domain H p ( R , Ω) on the Hilbert space L ( R , n ( x ) p +1 d x ) . The operator P is self-adjoint and its spectrum Σ( P ) reduces to its essential spectrum, equal to [0 , + ∞ ) . If weapply the spectral theorem [13, Theorem 5.9] to our quasi-resonances for the operator P , weget that for each quasi-resonance k p ; j ( m ) there exists an interval I of length O ( m −∞ ) suchthat the intersection Σ( P ) ∩ I is non empty, which is useless, since we know already that Σ( P ) = [0 , + ∞ ) .If the operator P has been defined as the Dirichlet realization of − n − p − div( n p − ∇· ) on a bounded open set containing Ω , then its spectrum would have been discrete. In suchcase, the application of the spectral theorem would be more significant. Nevertheless, thisprocedure of cut-off would not inform us about resonances.That is why we need to use a spectral-like theorem for resonances. Two statements areavailable in the literature: one from T ANG and Z
WORSKI [29], and another from S
TEFANOV [27]. Those theorems lie in the black box scattering framework. We are going to present mainassumptions and results of these papers in a simplified way, convenient for our application.
SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 33
In dimension , the main ingredients are • A complex Hilbert space H with orthogonal decomposition (with positive (cid:37) ) H = H (cid:37) ⊕ L ( R \ B (0 , (cid:37) )) • A family of unbounded selfadjoint operators h (cid:55)→ P ( h ) on H with domain indepen-dent of h , whose projection onto L ( R \ B (0 , (cid:37) )) coincides with H ( R \ B (0 , (cid:37) )) .We introduce the following assumptions B (0 ,(cid:37) ) ( P ( h ) − i) − compact H → H (H1)and R \ B (0 ,(cid:37) ) P ( h ) u = − h ∆ u (cid:12)(cid:12) R \ B (0 ,(cid:37) ) . (H2)Then we choose (cid:37) (cid:29) (cid:37) and periodize P ( h ) outside B (0 , (cid:37) ) , obtaining an operator P (cid:93) ( h ) on the Hilbert space H (cid:93) = H (cid:37) ⊕ L ( M \ B (0 , (cid:37) )) with M = ( R /(cid:37) Z ) Denoting by N ( P (cid:93) ( h ) , I ) the number of eigenvalues in I , we write the third assumption as N ( P (cid:93) ( h ) , [ − λ, λ ]) = O (cid:16) ( λ/h ) n (cid:93) / (cid:17) λ → ∞ , for some n (cid:93) ≥ . (H3)Let us denote by Z ( P ( h )) the set of poles of the resolvent z (cid:55)→ ( P ( h ) − z ) − . In dimension , this set is a subset of the Riemann logarithmic surface, its elements satisfy arg z < withour convention for the definition of resonances.Now we can state a simplified version of the main result of [29]: Theorem 7.B ([29]) . Let P ( h ) satisfy hypotheses (H1) , (H2) , and (H3) . Assume that thereexists for any h ∈ (0 , h ] a quasi-pair ( E ( h ) , u ( h )) with E ( h ) ⊂ [ E − h, E + h ] for somereal E , and with u ( h ) normalized in H and compactly supported independently of h . Thequasi-pairs are supposed to satisfy the residue estimate (cid:107) ( P ( h ) − E ( h )) u ( h ) (cid:107) H = O ( h ∞ ) . Then for any h ∈ (0 , h (cid:48) ] with a positive h (cid:48) small enough, there exists a resonance pole z ( h ) ∈ Z ( P ( h )) such that | E ( h ) − z ( h ) | = O ( h ∞ ) . The result in [27] is more precise but requires one more hypothesis, according to whichthe number of resonance poles is not too large: For some positive integers N and N (cid:48) Card (cid:8) z ∈ Z ( P ( h )) , a ≤ | z | ≤ b , − Im z < h N (cid:9) ≤ C a ,b h N (cid:48) . (H4)Our simplified version of the main result of [27] follows: Theorem 7.C ([27]) . Let H be a infinite subset of (0 , with accumulation point at . Let P ( h ) satisfy hypotheses (H1) , (H2) , (H3) , and (H4) . Assume that, for any h ∈ (0 , h ] ∩ H ,there exists d quasi-pair ( E (cid:96) ( h ) , u (cid:96) ( h )) with E ( h ) = . . . = E d ( h ) ∈ [ a , b ] , and with u (cid:96) ( h ) compactly supported independently of h , and almost orthonormal: |(cid:104) u i ( h ) , u (cid:96) ( h ) (cid:105) H − δ i(cid:96) | = O ( h ∞ ) . The quasi-pairs are supposed to satisfy the residue estimate (cid:107) ( P ( h ) − E (cid:96) ( h )) u (cid:96) ( h ) (cid:107) H = O ( h ∞ ) , (cid:96) = 1 , . . . , d. Then for any h ∈ (0 , h (cid:48) ] ∩ H with a positive h (cid:48) small enough, there exists d resonance poles z (cid:96) ( h ) ∈ Z ( P ( h )) with repetition according to multiplicity, such that | E (cid:96) ( h ) − z (cid:96) ( h ) | = O ( h ∞ ) , (cid:96) = 1 , . . . , d. The distinction between the two latter theorems is the consideration of multiplicity inTheorem 7.C. The multiplicity of a resonance pole z is understood as the rank of the operator π (cid:90) | z − z | = ε ( P ( h ) − z ) − d z for ε > small enough to isolate the pole z and ( P ( h ) − z ) − is the meromophic extensionof the resolvent [7, Definition 4.6].7.3. Application of the spectral-like theorems to disks with radially varying index.
Weapply the above theorems to our situation. We set P ( h ) = h P with P = − n − p − div( n p − ∇· ) on H = L ( R , n ( x ) p +1 d x ) . The subset H is { h = m , m ∈ N ∗ } . Hypotheses (H1) and (H2) are easy to check. Concern-ing (H3), by using the max–min principle for eigenvalues [11, Theorem 11.12] and com-paring the eigenvalues of P (cid:93) ( h ) with ( − h ∆) (cid:93) on a large torus M = ( R /(cid:37) Z ) for (cid:37) (cid:29) R ,we get that the counting function N ( P (cid:93) ( h ) , [ − λ, λ ]) = O ( λ/h ) for λ → ∞ , which yields n (cid:93) = 2 .Concerning (H4), we simply have to use the main theorem in [30] (with φ ( t ) = t and a = 1 + ε ).Then to bridge our families of resonance quasi-pairs with the formalism of [27], we set,for any chosen p ∈ {± } and any chosen j ∈ N : E (cid:96) ( h ) = h k p ; j ( h ) , h ∈ H , (cid:96) = 1 , with u ( h ) = u p ; j ( h ) and u ( h ) = u p ; j ( h ) , h ∈ H . Then, as h tends to , the energy E (cid:96) ( h ) converges to / ( Rn ( R )) in Cases ( A ) and ( B ), andto / ( R n ( R )) in Case ( C ). Applying Theorem 7.C and coming back to resonances by theformula k m = m (cid:113) z ( m ) and k (cid:48) m = m (cid:113) z ( m ) , m ≥ we have proved: Theorem 7.D.
For p ∈ {± } , j ∈ N , and m large enough, there exist two resonances k m and k (cid:48) m (counted with multiplicity) such that, as m → + ∞ , we have max (cid:0)(cid:12)(cid:12) k p ; j ( m ) − k m (cid:12)(cid:12) , (cid:12)(cid:12) k p ; j ( m ) − k (cid:48) m (cid:12)(cid:12)(cid:1) = O (cid:0) m −∞ (cid:1) . Remark . (i)
It is plausible that modes associated with the true resonances k m and k (cid:48) m have m as polar mode index. The proof of this would require to apply a spectral theorem tothe family of one dimensional resonance problems (1.2a)-(1.2b), which seemingly does notenter the general framework of [29] or [27]. Nevertheless, finite computations performedwith perfectly matched layers displayed numerical modes complying with the structure ofquasi-modes (see [18, Chapter 7] and a forthcoming paper of the authors). (ii) Throughout the paper we have assumed that p ∈ {± } , because of the physical moti-vation, but without any change, everything is true for p ∈ R .A PPENDIX
A. T
ECHNICAL LEMMAS
A.1.
Explicit solutions to some differential equations.Lemma A.1.
For all (cid:96) ∈ N , let denote by γ (cid:96) the mapping z ∈ R (cid:55)→ z (cid:96) e − z and for d ≥ by E d the set { γ (cid:96) ; (cid:96) = 0 , . . . , d } . The operator − ∂ z + 1 is a bijection from the vector-space span( E d \ { γ } ) to the vector-space span( E d − ) . SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 35
Proof.
For all (cid:96) ≥ , we readily obtain ( − ∂ z +1) γ (cid:96) = (cid:96) ( (cid:96) − γ (cid:96) − − (cid:96) γ (cid:96) − . Considering thevector basis ( γ , . . . , γ d ) and ( γ , . . . , γ d − ) of span( E d \ { γ } ) and span( E d − ) respectively,the matrix of − ∂ z + 1 from span( E d \ { γ } ) to span( E d − ) is an upper triangular matrix witha determinant equal to ( − d d ! (cid:54) = 0 . (cid:3) Lemma A.2.
For all (cid:96) ∈ N , let denote by α (cid:96) the mapping z ∈ R (cid:55)→ z (cid:96) A ( z ) and by β (cid:96) themapping z ∈ R (cid:55)→ z (cid:96) A (cid:48) ( z ) where A is the mirror Airy’s function. For all d ∈ N , let A d be the set { α (cid:96) , β (cid:96) ; (cid:96) = 0 , . . . , d } . The operator − ∂ z − z is a bijection from the vector-space span( A d \ { α } ) to the vector-space span( A d \ { β d } ) .Proof. From the definition of Airy’s function, we have A (cid:48)(cid:48) = − z A and therefore, for (cid:96) ≥ , ( − ∂ z − z ) α (cid:96) = − (cid:96) ( (cid:96) − α (cid:96) − − (cid:96) β (cid:96) − , ( − ∂ z − z ) β (cid:96) = − (cid:96) ( (cid:96) − β (cid:96) − + (2 (cid:96) + 1) α (cid:96) . Considering the vector basis ( β , α , β , . . . , α d , β d ) of span( A d \ { α } ) and the vector ba-sis ( α , β , α , β , . . . , α d ) of span( A d \ { β d } ) , the matrix of − ∂ z − z considered from span ( A d \ { α } ) to span ( A d \ { β d } ) is an upper triangular matrix with a determinant equalto ( − d (2 d + 1)! (cid:54) = 0 . (cid:3) A.2.
Half harmonic oscillator.
We recall that we denote by L ( R − , ω x ) and H (cid:96) ( R − , ω x ) the weighted Sobolev spaces with measure ω x ( σ ) d σ where ω x : σ (cid:55)→ exp (2 x | σ | ) for x realand that Ψ GH j +1 refers to the Gauss-Hermite function of order j + 1 , see [1, 20]. Lemma A.3.
Let β ∈ R , θ > , and j ∈ N . For any S ∈ L ( R − , ω β ) ∩ span(Ψ GH j +1 ) ⊥ thereexists a unique solution to the problem: Find w ∈ H ( R − ) ∩ span(Ψ GH j +1 ) ⊥ such that (cid:40) − w (cid:48)(cid:48) ( x ) + ( x − j − w ( x ) = S ( x ) ∀ x ∈ ( −∞ , w (0) = 0 . (A.1) Moreover, this solution belongs to H ( R − , ω β ) ∩ H ( R − , ω β − θ ) .Proof. Existence and unicity rely on the fact that the family (Ψ GH (cid:96) +1 ) (cid:96) ∈ N is a Hilbert basis of L ( R − ) and that the half harmonic oscillator operator is diagonalizable on span(Ψ GH (cid:96) +1 | (cid:96) ∈ N ) . The solution to problem (A.1) can be written as w = + ∞ (cid:88) (cid:96) =0 , (cid:96) (cid:54) = j (cid:96) − j ) ( S, ς (cid:96) Ψ GH (cid:96) +1 ) L ( R − ) ς (cid:96) Ψ GH (cid:96) +1 (A.2)where ς (cid:96) = (cid:107) Ψ GH (cid:96) +1 (cid:107) − ( R − ) and we clearly have w ∈ H ( R − ) ∩ span(Ψ GH j +1 ) ⊥ .We set J := (cid:112) j + 1) + 2 β − so that, for all x ≤ − J , we have V ( x ) := x − j − − β − ≥ . Let φ ∈ C ∞ ( R ) such that ≤ φ ≤ , φ ( x ) = 0 for all x ≤ , and φ ( x ) = 1 for all x ≥ , let b = (1 + 2 β ) max R | φ (cid:48) | and let a > J + 2 b . We define a cut-off function χ a ∈ C ∞ comp ( R − ) by χ a ( x ) = φ ( b − ( x + a )) · φ ( − b − ( x + J )) , ∀ x ∈ R − . We also define χ ( x ) = φ ( − b − ( x + J )) for x ∈ R − . Note that, for all a > J + 2 b , we have | χ (cid:48) a | ≤ C where C = (1 + 2 β ) − . Let also (cid:98) w := w ω β − and (cid:98) S := S ω β − . Multiplying bothsides of equation (A.1) by χ a w ω β and integrating over R − , yields (cid:90) −∞ (cid:16) w (cid:48) ( χ a w ω β ) (cid:48) + χ a (cid:0) x − j − (cid:1) (cid:98) w (cid:17) d x = (cid:90) −∞ χ a (cid:98) S (cid:98) w d x. Since w (cid:48) ω β − = (cid:98) w (cid:48) + 2 β − (cid:98) w and w (cid:48) ( x )( w ( x ) ω β ) (cid:48) = (cid:98) w (cid:48) ( x ) − β − (cid:98) w ( x ) , we deduce that (cid:90) −∞ χ a (cid:98) w (cid:48) + χ a V (cid:98) w d x + (cid:90) −∞ χ (cid:48) a (cid:98) w (cid:0) (cid:98) w (cid:48) + 2 β − (cid:98) w (cid:1) d x = (cid:90) −∞ χ a (cid:98) S (cid:98) w d x. (A.3)For the first term on the left hand side of (A.3), since χ a V ≥ χ a , we have (cid:90) −∞ χ a (cid:98) w (cid:48) + χ a V (cid:98) w d x ≥ (cid:90) −∞ χ a (cid:16) (cid:98) w (cid:48) + (cid:98) w (cid:17) d x. (A.4)Then, for the second term on the left hand side of (A.3), since χ (cid:48) a ≥ − C , ≥ χ a , and b issuch that C (1 + 2 β ) = 1 , we have (cid:90) −∞ χ (cid:48) a (cid:98) w (cid:0) (cid:98) w (cid:48) + 2 β − (cid:98) w (cid:1) d x ≥ − C (cid:90) −∞ (cid:98) w (cid:98) w (cid:48) + 2 β − (cid:98) w d x ≥ − C (cid:90) −∞ (cid:98) w (cid:48) + (1 + 2 β ) (cid:98) w d x ≥ − (cid:90) −∞ χ a (cid:16) (cid:98) w (cid:48) + (cid:98) w (cid:17) d x. (A.5)For the last term on the right hand side of (A.3), since χ a ≤ χ a , we have (cid:90) −∞ χ a (cid:98) S (cid:98) w d x ≤ (cid:107) (cid:98) S (cid:107) L ( R − ) (cid:18)(cid:90) −∞ χ a (cid:98) w d x (cid:19) ≤ (cid:107) (cid:98) S (cid:107) L ( R − ) N w ( a ) (A.6)where N w ( a ) = (cid:113)(cid:82) −∞ χ a (cid:0) (cid:98) w (cid:48) + (cid:98) w (cid:1) d x . Combining the estimates (A.4), (A.5), and (A.6)yields N w ( a ) ≤ (cid:107) (cid:98) S (cid:107) L ( R − ) N w ( a ) . The function a ∈ ( J + 2 , + ∞ ) (cid:55)→ N w ( a ) is not negative and not decreasing, so the functionis either always zero or positive for a large enough but in any cases we have N w ( a ) ≤ (cid:107) (cid:98) S (cid:107) L ( R − ) . By letting a tends towards + ∞ , we obtain that (cid:90) −∞ χ (cid:16) (cid:98) w (cid:48) + (cid:98) w (cid:17) d x ≤ (cid:107) (cid:98) S (cid:107) ( R − ) which implies that (cid:98) w belongs to H ( R − ) . It follows that w belongs to L ( R − , ω β ) . From therelation w (cid:48) ω β − = − β − (cid:98) w − (cid:98) w (cid:48) , we deduce that w ∈ H ( R − , ω β ) .Finally, using the relation w (cid:48)(cid:48) = ( x − j − w − S , we get (cid:90) −∞ w (cid:48)(cid:48) ω β − θ d x ≤ C (cid:90) −∞ (cid:0) w + S (cid:1) ω β d x where C = max x ∈ R − ( x − j − ω − β + (cid:101) θ ( x ) < + ∞ with (cid:101) θ = ln(1 − − θ ) / ln(2) . Thisshows that w ∈ H ( R − , ω β − θ ) . Note that the constant β is replaced by β − θ by the need totake into account the coefficient x − j − . (cid:3) SYMPTOTICS FOR WHISPERING GALLERY MODES IN OPTICAL MICRO-DISKS 37
A.3.
Borel’s Theorem.
Our construction of quasi-modes requires to find a smooth functiongiven its Taylor expansion. This can be achieved using a Borel’s like theorem on the spacesof Schwartz functions S ( R ± ) . We denote by p α,β ( f ) = sup x ∈ R ± | x α ∂ βx f ( x ) | , α, β ∈ N , theusual family of semi-norms over S ( R ± ) . Lemma A.4.
Let ( f q ) q ∈ N be a sequence of functions where f q ∈ S ( R ± ) for all q ∈ N . Thereexists f ∈ C ∞ ([0 , , S ( R ± )) such that ∂ qt f (0 , x ) = f q ( x ) , ∀ x ∈ R ± . Proof.
This proof is inspired by the proof of [14, theorem 1.2.6] where smooth functionswith compact support are replaced by Schwartz functions.Let g ∈ C ∞ comp ( R ) be a smooth cut-off function such that g ( t ) = 1 for all t ∈ [ − , . Foreach q ∈ N we introduce the function g q : ( t, x ) ∈ R × R ± (cid:55)−→ g (cid:0) ε − q t (cid:1) t q q ! f q ( x ) for some positive number ε q that will be specified later on. For all d, α, β ∈ N , we have x α ∂ dt ∂ βx g q ( t, x ) = ε q − dq G ( d ) q ( ε − q t ) x α f ( β ) q ( x ) , ∀ ( t, x ) ∈ R × R ± , (A.7)where G q ( s ) = g ( s ) s q q ! . It follows that (cid:12)(cid:12) x α ∂ dt ∂ βx g q ( t, x ) (cid:12)(cid:12) ≤ C α,βq,d ε q − dq where C α,βq,d = sup s ∈ R (cid:12)(cid:12) G ( d ) q ( s ) (cid:12)(cid:12) p α,β ( f q ) < + ∞ . By choosing ε q = min d + α + β d + α + β . Therefore, the sum f = (cid:88) q ≥ g q is well defined because the series converge absolutely. Its successive derivatives are equal tothe sum of the derivatives of g g ; As a consequence, f ∈ C ∞ ([0 , × R ± ) . Moreover, fromthe estimate p α,β (cid:0) ∂ dt f ( t, · ) (cid:1) ≤ d + α + β (cid:88) q =0 C α,βq,d ε q − dq + 2 − d − α − β , ∀ t ∈ [0 , , ∀ d, α, β ∈ N we obtain f ∈ C ∞ ([0 , , S ( R ± )) . From (A.7), we deduce that for all d ∈ N ∂ dt f (0 , x ) = + ∞ (cid:88) q =0 ε q − dq G ( d ) q (0) f q ( x ) , ∀ x ∈ R ± . Since g is constant equal to around t = 0 , we have G ( d ) q (0) = δ q,d where δ q,d is theKronecker symbol. This implies that ∂ dt f (0 , x ) = f d ( x ) . (cid:3) By Taylor’s formula with integral remainder we deduce immediately the following result.
Lemma A.5.
Let f be a function belonging to C ∞ ([0 , , S ( R ± )) . For all integer N ≥ there exists R N ∈ C ∞ ([0 , , S ( R ± )) such that f ( t, x ) = N − (cid:88) q =0 ∂ qt f (0 , x ) q ! t q + t N R N ( t, x ) , ∀ ( t, x ) ∈ [0 , × R ± . A.4.
Additional result.Lemma A.6.
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