Transmission Eigenvalues in One Dimension

Transcription

1 Transission Eigenvalues in One Diension John Sylvester Abstract We show how to locate all the transission eigenvalues for a one diensional constant index of refraction on an interval. 1 Introduction The scattering operator for the tie haronic Helholtz equation with copactly supported index of refraction n(x) aps the asyptotics of incident waves (Herglotzwave functions) v 0 totheasyptotics of(outgoing)scattered waves u +. Both u + and v 0 are defined in R n and satisfy ( +k 2 n 2 )u + = (1 n 2 )k 2 v 0 ( +k 2 )v 0 = 0 in all of R n. If the scattering operator has a zero eigenvalue, we say that k 2 is a transission eigenvalue [6], [4]. It follows fro Rellich s theore and unique continuation that the scattered wave u + is identically zero outside any doain D that contains the support of n, and therefore that u = u + and v = v 0 are a pair of nontrivial functions satisfying ( +k 2 n 2 )u = (1 n 2 )k 2 v in D (1.1) ( +k 2 )v = 0 in D (1.2) u u = 0 = 0 (1.3) D D ν Departent of Matheatics,University of Washington, Seattle, Washington 98195,USA. (sylvest@uw.edu). Research was supported in part by NSF grant DMS

2 Whenever such a nontrivial pair exists, we say that k 2 is an interior transission eigenvalue for the pair (D,n). Reark 1. If we scale v by introducing w = k 2 v, then the k 2 on the right hand side of equation (1.1) disappears, and we see that the interior transission eigenvalue proble is a true generalized eigenvalue proble which can be written as u (1 n 2 )w = k 2 n 2 u w = k 2 w with the sae boundary conditions as in (1.3) [9]. The analogous interior proble for obstacle scattering is the Dirichlet proble, ( +k 2 )v = 0 v = 0 D in D and the Dirichlet eigenvalues are fairly well understood. In particular, the Dirichlet eigenvalues of a one diensional interval of length L are exactly n 2 π 2 L 2. The one diensional interior transission eigenvalue proble, even for a constant index of refraction, is not as siple. In one diension, with the index of refraction n(x) equal to the positive constant σ, equations (1.1) (1.2) (1.3) becoe u +k 2 σ 2 u = k 2 (1 σ 2 )v (1.4) v +k 2 v = 0 (1.5) u(± L 2 ) = 0 u (± L 2 ) = 0 (1.6) so that k 2 is an interior transission eigenvalue if and only there exist a nontrivial pair of eigenfunctions (u, v) satisfying (1.4)(1.5)(1.6). Wewillshowinproposition4thatk 2 isaninteriortransissioneigenvalue if and only if k is a root of the equation (σ 1) 2 sin 2 ((σ +1) kl 2 ) (σ +1)2 sin 2 ((σ 1) kl 2 ) = 0 (1.7) See section 8 of [3], as well as [8], for ore details about the connection between transission eigenvalues and interior transission eigenvalues 2

3 and the two theores below describe the roots of (1.7), which we will call the interior transission wavenubers (positive square roots of interior transission eigenvalues). Theore 2. The interior transission wavenubers satisfy I(k) 2 3σ +1 log L σ 1 In theore 3 below, the notation j denotes the floor, the greatest integer less than or equal to j. The condition j = j+1 eans that there is no integer between j j+1 and, and Z denotes the integers. Theore 3. Let = σ+1, and let j Z j σ 1 j+1 1. If Z and Z, there are exactly two siple interior transission wavenubers in the strip 2jπ 2(j +1)π < Re(k) < (σ +1)L (σ +1)L (a) If j j+1, both are real (b) If j = j+1 they are coplex conjugates with nonzero iaginary parts j 2. If Z, there is a quadruple interior transission wavenuber at k = j and no others in the strip 2(j 1)π 2(j +1)π < Re(k) < (σ +1)L (σ +1)L Thetransissionwavenubers fortwoσ s, withl = 2, areplottedbelow. The vertical blue lines indicate the sets Re(k) = 2jπ. The roots that fall (σ+1)l on these lines are quadruple. 2 TEs for σ = TEs for σ = As these σ s are rational, the transission eigenvalues are periodic. Reark 5 below explains how we did the coputation. 3

4 2 Leas and Proofs Because we are dealing with constant coefficients, the transission wavenubers k are the zeros of the analytic function on the left hand side of equation (1.7). This analytic function has two factors, and our proofs siplify if we treat each factor separately. Proposition 4. k is a transission wavenuber if, and only if, k satisfies or (σ 1)sin((σ +1) kl ) = (σ +1)sin((σ 1)kl 2 2 ) (2.1) (σ 1)sin((σ +1) kl ) = (σ +1)sin((σ 1)kl 2 2 ) (2.2) The wavenuber k satisfies (2.1) if there exists an eigenfunction pair (u, v) of odd functions of x, and k satisfies (2.2) if there exists an eigenfunction pair (u,v) of even functions. Proof. If a pair u,v satisfies ( ), then so do their odd and even parts, and the calculation is a little sipler for each of these parts. The odd parts satisfy ( ) with (1.6) replaced by u(0) = v(0) = 0 u( L 2 ) = u ( L 2 ) = 0 and the even parts satisfy ( ) with (1.6) replaced by u (0) = v (0) = 0 u( L 2 ) = u ( L 2 ) = 0 In the odd case, the boundary conditions at zero iply that and v = sin(kx) u = sin(kx)+asin(kσx) (2.3) for soe constant A. The boundary conditions at x = L 2 iply that sin(σk L 2 )cos(kl 2 ) = σsin(kl 2 )cos(σkl 2 ) (2.4) 4

5 Noting that the left and right hand sides are the beats representation of the cobinationofthefrequencies (σ+1) L 2 and(σ 1))L 2 suggeststhatwerewrite (2.4) as (σ 1)sin((σ +1)) L 2 k) = (σ +1)sin((σ 1)L 2 k) which is (2.1). A siilar calculation shows the existence of even transission eigenfunction pairs if and only if k satisfies (2.2). and If we introduce the new variables = σ +1 σ 1 (2.5) z = (σ 1)k L 2π then equations (2.1) and (2.2) becoe equations for z which depend on the single real paraeter. They becoe p o (z) := sin(πz) sin(πz) = 0 (2.6) p e (z) := sin(πz)+sin(πz) = 0 (2.7) where the superscripts refer to even and odd, respectively. Because σ > 0, > 1. Because equations (2.6) and (2.7) reain the sae if we change the sign of, we ay also assue that > 1, and will do so throughout. Reark 5. If = p is rational and we set w = eiπz q q, then (2.6) becoes a polynoial equation in w. w 2p 1 = p q ( w p+q w p q) (2.8) We can use a polynoial root finding algorith to find the solutions and then take logariths and rescale to find the interior transission wavenubers k. This is how we produced figure 1. With the notation introduced in (2.5), theore 2 becoes Alternatively, we could start with the assuption that σ > 1, and then use the relationship between the transission wavenubers of σ and those of 1 σ, k j( 1 σ ) = σk j(σ) to show that our forulas and estiates apply to σ < 1 as well. 5

6 Theore 6. If z satisfies (2.6) or (2.7), then I(z) log(2+1) π( 1) (2.9) Proof. If z = x+iy satisfies (2.6) so that e π y e π y 2 sinπz = 2 sinπz e π y +e π y e π y e π y +e π y +e π y Dividing both sides by e π y gives e ( 1)π y e (+1)π y ++e 2π y 2+1 Now taking the logarith of both sides yields (2.9) Locating the real parts of the roots of p (z) = p o (z)pe (z) will take ore work. We will show (roughly) that there are always two roots in certain strips of the coplex plane. We define I j S j = = { x { z j < x < j +1 } j j +1 < Re(z) < } A restateent of theore 3, using the notation of equation (2.5), is theore 7 below. Theore 7. Let > If neither j strip S j. nor j+1 are integers, then p has two siple roots in the (a) If the interval I j contains an integer k, both roots are real, and one is bigger and the other saller than k. (b) If the interval I j does not contain an integer, the roots are coplex conjugates (with nonzero iaginary parts). j 2. If is an integer, j is a quadruple root of p, and p has no other roots in S j or Sj 1. 6

7 Theore 7 is an iediate consequence of the analogous theores for p o and p e. These theores are a little ore coplicated to state, but easier to prove. Theore 8. Let > Suppose that neither j nor j+1 are integers. (a) If the interval I j contains an integer k and j + j is even, then p o has a real root in the interval ( j,k) and no other roots in S j and p e j+1 a root in (k, ) and no other roots in S j. If the interval I j contains an integer k and j + j is odd, then po root in the interval (k, j+1 ) and no other roots in S j root in ( j,k) and no other roots in S. j has a real and p e a (b) If the interval I j does not contain an integer and j + j is odd, p o has 2 siple coplex conjugate roots in S j and p e has none. If j + j is even, pe has 2 siple coplex conjugate roots in and p o has none. S j 2. If j is an integer, neither po nor pe have any roots in S or j Sj 1. If j + j is even, j is a triple root of po and a siple root of pe. If j + j is odd, j is a triple root of pe and a siple root of p o. Wewillprovetheore8(wewillonlyprovetheresultsforp o )bystudying a one paraeter faily of equations with fixed and paraeter β. p β (z) = sin(πz) βsin(πz) (2.10) We will use the fact that that the roots of the entire function p β (z) depend continuously on the paraeter β. We will also use the fact that, as β, the roots of p β ust approach those of sin(πz). This last fact follows fro the first by ultiplying (2.10) by α = 1, and the fact that the β roots of α sin(πz) sin(πz) depend continuously on α as α approaches zero. The continuity of the roots of analytic functions follows fro the fact that roots are isolated, together with the arguent principle. 7

8 Before giving the rigorous details, we give a rough outline of the proof of Theore 8. To siplify this outline we ignore the case where j is an integer. Theore 8 akes two assertions 1. It tells us the nuber of roots p o in each strip S j. This nuber is the sae for p β for all β > It tells us the nuber of real roots in each strip when β =. The nuber of real roots of p β changes as β increases fro zero to, but is the sae for all β. We establish the nuber of roots in each strip by observing that: 1. at β = 0 all roots are real and on the boundary of the strips S j, and the derivatives are nonzero and pointing in the direction of the correct strip (lea 9), so that, for sall positive β, the correct nuber of roots have entered each strip(corollary 10). 2. Lea 11 tells us that no roots can cross the boundary of S j, so the correct nuber of roots reain in each strip for all β (corollary 12). We calculate the nuber of roots which are real at β = by observing that: 1. As β approaches infinity, a single root ust approach each integer (the roots of sinπz), and the other roots ust leave every copact subset of C. 2. A calculation(lea 13) shows that for β, all roots are siple. 3. This eans that any real roots of p β in S j at β = ust reain real for β and reain in S j, so they reain in the interval I j. But, as β approaches infinity, the roots of p β ust either approach the roots of sinπz or leave every copact subset of C. The roots in the S j s that don t contain an integer ust leave every copact set, so they cannot reain in the interval I j, so they cannot have been real when β =. 4. The S j s that contain an integer contain only one root for all β, so the root reains siple and therefore real at β =. That root approaches the integer as β approaches infinity. 8

9 The detailed proof follows: Lea 9. The roots of p 0 = sin(πz) are j. Each of these roots is siple, and thus continues to a unique root z j (β) of p β for sall β. Moreover, dz j dβ = ( 1) j+ j sin( jπ ) (2.11) β=0 Proof. At β = 0, the roots of sinπz are located at j and are all siple. The iplicit function theore iplies that, for sall β, they reain siple and depend analytically on β, so we can define z j (β) to be the continuation of that root. Differentiating (2.10) yields so that dz j dβ dz j dβ (πcos(πz j)+βπcos(πz j )) = sin(πz j ) = β=0 sin πj cosπj = ( 1) j sin jπ = ( 1) j+ j sin( jπ ) Knowledge of the derivatives tells us that the roots are in the correct strips for sall β. Corollary 10. For β > 0 and sufficiently sall, all roots of p β are real and: 1. Suppose that neither j nor j+1 are integers. contains an integer k, then p β has a real root in (a) If the interval I j the interval ( j,k) and no other roots in S j has a root in (k, j+1 ) and no other roots in S j (b) If the interval I j p β has no roots in S j if j+ j is even. p β if j + j is odd. does not contain an integer and j + j is odd,. If j+ j is even, p β has two roots in S j. 2. If j is an integer, j is a siple root of p β. If j+ j is even, there is a siple root in each of I j and Ij 1. If j + j is odd, there are no roots in I j or Ij 1. 9

10 Proof. Because sinz = sinz, the roots of (2.6) ust occur in conjugate pairs, and siple roots ust reain real for sall β. Lea 9 tells us that, z j starts at j and oves to the right if j is not an integer and j + j is even, or to the left if j is not an integer and j + j is odd. If I j does not contain an integer, then j+1 + j+1 is odd whenever j + j is even, and even whenever j + j is odd, so that both z j and z j+1 ove into I j when j + j is even, and away fro I j when j + j is odd, which establishes ite 1b. If I j fro I j contains an integer k, then z j oves into I j when j + j is even, and z j + 1 oves into I j when j+ j is odd. The root which oves into I j or (k, j+1 ) for sall β, which establishes ite 1a. and z j+1 oves away and z j oves away reains in either ( j,k) Finally, if j is an integer, both ters in (2.10) vanish, so that j reains j a root for all β, and reains siple at least for β sall. If is an integer, j 1 + j 1 = j + j j+1 2 and + j+1 = j + j + 1, so z j 1 oves into Ij 1 and z j+1 oves into Ij+1 if j + j is even, and both ove away fro those intervals if j + j is odd. This establishes ite 2 and finishes the proof. At a root of p β (z), the real parts of the sinπz ust equal the real part of βsinπz. The lea below identifies the sets on which the real parts of these ters vanish. Lea 11. Re(sin(z)) = 0 iff sin(re(z)) = 0 Proof. and cosh y never vanishes. sin(x+iy) = sinxcoshy +icosxsinhy A consequence of lea 11 is that p β has no roots on Re(z) = j or on Re(z) = k, when j or k are integers, unless j equals an integer. This is enough to show that the roots which entered each strip for sall β, ust reain there. 10

11 Corollary 12. For all 0 < β < 1. The conclusions in ites 1a and 1b of corollary 10, reain true. 2. If j is an integer and j + j is odd, j is a siple root of p β and p β j has no other roots in S j or Sj+1. If is an integer and j + j is even, p β has three roots in Sj 1 Sj, one of which is j. Proof. The hypothesis of ite 1, that j are not integers, cobine with lea 11 to guarantee that p β can have no roots on the boundary of S j. As the roots of the entire analytic function p β depend continuously on β, this iplies that no root can enter or leave S j. Thus, if S j contained no roots for sall β, it contains no roots for all β. If S j contained two roots for soe β, it contains two roots for all β. This establishes ite 1b for all β. To establish ite 1a, note that lea 11 forces the siple root to reain in k < Re(z) < j+1 or j < Re(z) < k. As the siple real root reains siple, it ust also reain real. and j+1 The two stateents in ite 2 follow siilarly fro the fact that no root can cross j+1 j 1 or, as these cannot be integers if j is an integer because > 1. Lea 13. For β >, all roots of p β are siple. For β =, all roots are siple except z = j in the case that j+ j is an even integer. In this case, j is a triple root of p β. Proof. All roots of p β that aren t real are siple, because there are no ore than three in any strip, and they ust occur in conjugate pairs. Suppose we have a double real root, then and sinπz = βsinπz (2.12) cosπz = βcosπz (2.13) Squaring both (2.12) and (2.13), and adding the results gives or sin 2 πz +cos 2 πz +( 2 1)cosπz = β 2 cos 2 πz = β (2.14) 11

12 which is ipossible for β > because the right hand side is bigger than one. If β =, then (2.14) and (2.13) are possible, but only if cosπz = cosπz = ±1 which iplies that z and z are either both even or both odd integers. This is equivalent to the stateent that z = j and j + j is an even integer. One can check directly that the root is indeed triple in this case. Proof of Theore 8. We only prove the theore for p o, as the proof for p e is analogous. Ite 1a follows iediately fro ite 1 of corollary 12. Ite 1 of corollary 12 also iplies all of ite 1b, except the fact that the two roots are not real. To see this, note that, as β, the roots of p β ust either approach the roots of sinπz, which are integers, or leave every copact subset of the coplex plane. Since I j contains no integers, the two roots in S j ust leave every copact subset of C. As Re(z) ust reain between j j+1 and, the iaginary parts of these two roots ust grow unbounded as β. If they were real at β =, lea 13 would iply that they were siple and reained siple and real for all β, contradicting the fact that their iaginary parts grow unbounded as β. Hence they ust not have been real at β =. Ite 2 follows for a siilar reason. We can check directly that j is either a siple or a triple root, depending on the parity of j + j. Ite 2 of corollary 12 tells us that these are all of the roots in the closure of Sj 1 S j, so there are no other roots of p o in these strips. 3 Soe Liiting Cases and Conclusions We have shown that there are two transission wavenubers in each closed 2jπ 2(j+1)π strip Re(k) of width 2π, and two real transission (σ+1)l (σ+1)l (σ+1)l wavenubers in each of those strips which contains an integer. We show below that, in the liiting cases of a weak scatterer (σ 1), there are no longer any real transission wavenubers, and in the liiting case of a strong scatterer (σ ), all the transission wavenubers becoe real. The Born, or linear approxiation, is the liit as σ 1. We return to equation (1.7), set πz = kl, and let σ 1 to obtain 12

13 πz = ±sinπz (3.1) Introducing the paraeter β in front of the πz on the right hand side, and repeating the steps we used to prove theore 7, gives: Theore 14. For every integer j except j = 1 and j = 0, equations (3.1) have exactly two non-real coplex conjugate roots in the strip j Re(j) j+1, and theylie in the interior of the strip. The onlyrootin 1 Re(z) 1 is the triple root at z = 0. The iaginary parts of these roots grow logarithically. The precise asyptotics, which can be verified by substituting (3.2) into (3.1), are ( ) z ± j = (j + 1)±ilog(2 j + 1 )+O log(2 j+ 1 2 ) (3.2) 2 2 (j+ 1 2 ) so that k j = πz j satisfy L ( k j ± π L (j + 1 )±ilog(2 j + 1 )) 2 2 where, for the odd integers j, z j solve equation (3.1) with the plus sign, and, for even integers j, the z j solve (3.1) with the inus sign. The liit as σ is a little ore work to calculate. This is the liit as 1, so that equation (2.7) becoes or sinπz = sinπz sinπz = 0 (3.3) with roots at the integers. Thus, as σ, the even transission wavenubers k j 2πj σl have the sae asyptotics as the square roots of the eigenvalues of the Dirichlet proble u +k 2 σ 2 u = 0 u(± L 2 ) = 0 (3.4) 13

14 with odd (u(x) = u( x)) eigenfunctions. The solutions of (2.6), which describe the transission wavenubers with odd eigenfunctions, have a different asyptotic behavior, that will atch the asyptotics of a different self-adjoint boundary value proble. To see the liiting equation, we subtract sin πz fro both sides of (2.6) and divide by 1 sin(πz) sinπz 1 which becoes, as 1 πzcosπz = sinπz or sin(πz) sinπz = sinπz sinπz πz = tanπz The transission wavenubers satisfy = sinπz k(σ 1) L 2 = tank(σ 1)L 2 (3.5) and thus have the sae asyptotics, as σ, as the square roots of the eigenvalues of the self-adjoint boundary value proble. with odd eigenfunctions. v +k 2 σ 2 v = 0 (3.6) ± L 2 v (± L 2 ) = v(±l 2 ) Because these liiting wavenubers are the square roots of the eigenvalues of a self-adjoint proble, we see that, as σ, all transission wavenubers becoe real. In our discussion of the one diensional transission eigenvalue proble, we have observed certain relationships between transission eigenvalues, The wavenubersof (3.6) with even eigenfunctions satisfy equation (3.5) with tangent replaced by cotangent 14

15 Dirichlet eigenvalues, and the eigenvalues of another self-adjoint boundary value proble. We expect (optiistically) that soe of these relationships will persist in higher diensions. We know that, in the Born approxiation, there are no real transission eigenvalues [8], that a generic radially syetric index of refraction has coplex transission eigenvalues [7], and that there are infinitely any transission eigenvalues [8] [1]. All of these results are consistent with our one diensional conclusions. A consequence of theore 3 is that the counting function for the nuber of interior transission wavenubers in a strip of width W is (σ +1)LW N C (W;σ,L) = ±2 π which has the sae asyptotics as the counting function for Dirichlet wavenubers for an interval of the sae length with index of refraction n = σ +1. The counting function for real interior transission wavenubers in an interval of width W is (σ 1)LW N R (W;σ,L) = ±2 π which atches the asyptotics of the counting function for Dirichlet wavenubers for an interval of the sae length with index of refraction n = σ 1. It sees reasonable to conjecture that the relationship between the counting functions for the Dirichlet wavenubers and the transission wavenubers reains true for non-constant σ, and in higher diensions as well. References [1] Fioralba Cakoni, Drossos Gintides, and Housse Haddar. The existence of an infinite discrete set of transission eigenvalues. SIAM J. Math. Anal., 42(1): , [2] David Colton, Andreas Kirsch, and Lassi Päivärinta. Far-field patterns for acoustic waves in an inhoogeneous ediu. SIAM J. Math. Anal., 20(6): , This is correct as is for irrational σ. For rational σ we need to adopt the convention that every quadruple root (there will always be soe) is counted as two real roots and two non-real roots. 15

16 [3] David Colton and Rainer Kress. Inverse acoustic and electroagnetic scattering theory, volue 93 of Applied Matheatical Sciences. Springer, New York, third edition, [4] David Colton and Peter Monk. The inverse scattering proble for tieharonic acoustic waves in an inhoogeneous ediu. Quart. J. Mech. Appl. Math., 41(1):97 125, [5] David Colton, Lassi Päivärinta, and John Sylvester. The interior transission proble. Inverse Probl. Iaging, 1(1):13 28, [6] Andreas Kirsch. The denseness of the far field patterns for the transission proble. IMA J. Appl. Math., 37(3): , [7] Yuk-J Leung and David Colton. Coplex transission eigenvalues for spherically stratified edia. Inverse Probles, 28(7):075005, 9, [8] Lassi Päivärinta and John Sylvester. Transission eigenvalues. SIAM J. Math. Anal., 40(2): , [9] John Sylvester. Discreteness of transission eigenvalues via upper triangular copact operators. SIAM J. Math. Anal., 44(1): ,