Resonances of a Potential Well with a Thick Barrier

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. APPL. MATH. c© 2013 Society for Industrial and Applied MathematicsVol. 73, No. 4, pp. 1489–1512

RESONANCES OF A POTENTIAL WELL WITH A THICK BARRIER∗

D. C. DOBSON† , F. SANTOSA‡ , S. P. SHIPMAN§ , AND M. I. WEINSTEIN¶

Abstract. This work is motivated by the desire to develop a method that allows for easy andaccurate calculation of complex resonances of a one-dimensional Schrodinger’s equation whose poten-tial is a low-energy well surrounded by a thick barrier. The resonance is calculated as a perturbationof the bound state associated with a barrier of infinite thickness. We show that the corrector to thebound-state energy is exponentially small in the barrier thickness. A simple computational strategythat exploits this smallness is devised and numerically verified to be very accurate. We also providea study of high-frequency resonances and show how they can be approximated. Numerical examplesare given to illustrate the main ideas in this work.

Key words. resonances for Schrodinger’s equation, asymptotic analysis of resonances, approx-imate methods for calculating resonances

AMS subject classifications. 35Pxx, 24L16, 65L15

DOI. 10.1137/120883694

1. Introduction. Resonances are important in the study of transient phenom-ena associated with the wave equation. In particular, long-time behavior of the solu-tion of the wave equation is well described using resonances when radiation losses aresmall, as in the case of a potential well surrounded by a thick barrier (Figure 1.1).

Consider the wave equation

∂2u

∂t2=∂2u

∂x2− V (x)u,(1.1)

u(x, 0) = u0(x),∂u

∂t(x, 0) = u1(x).(1.2)

Here V (x) is a low-energy potential surrounded by a barrier and supported in [−L,L].We assume that u0 ∈ H1([−R,R]) and u1 ∈ L2([−R,R]), where R > 0. To solve(1.1)–(1.2), we first determine resonances and quasi modes which are solutions to thenonlinear eigenvalue problem

− ψ′′ + V (x)ψ = k2ψ,(1.3)

ψ′ + ikψ = 0 for x = L,(1.4)

ψ′ − ikψ = 0 for x = −L.(1.5)

There will be an infinity of resonances kn and their associated quasi modes ψn(x).We note that Im kn < 0 and assuming that the resonances are simple. Furthermore,for any A > 0, we can write [13]

∗Received by the editors July 6, 2012; accepted for publication (in revised form) April 29, 2013;published electronically July 16, 2013.

http://www.siam.org/journals/siap/73-4/88369.html†Department of Mathematics, University of Utah, Salt Lake City, UT 84112 (dobson@math.

utah.edu). This author’s work was partly supported by NSF grant DMS-0537015.‡School of Mathematics, University of Minnesota, Minneapolis, MN 55455 ([email protected]).

This author’s work was partly supported by NSF grant DMS-0807856.§Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803 (shipman@

math.lsu.edu). This author’s work was partly supported by NSF grant DMS-0807325.¶Appied Physics and Applied Mathematics Department, Columbia University, New York,

NY 10027 ([email protected]). This author’s work was partly supported by NSF grantDMS-1008855.

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1490 D. DOBSON, F. SANTOSA, S. SHIPMAN, AND M. WEINSTEIN

Fig. 1.1. A one-dimensional potential with a well of width a and height V0 surrounded by awall of thickness (L − a). The potential vanishes for |x| > L.

(1.6) u(x, t) =∑

Imkn>−Acje

−ikjtψj(x) + rA(x, t).

The remainder, rA, decays in a local energy sense, i.e., for any K > 0

‖rA(t, ·)‖H1([−K,K]) ≤ C(R,K)e−At(‖u0‖H1 + ‖u1‖L2).

The importance in the representation (1.6) is that for large t, the solution is welldescribed by

u(x, t) ∼ cj0e−ikj0 tψj0(x),

where j0 is the index for which kj has the smallest negative imaginary part, i.e., kj0is closest to the real axis. As we shall see below, the size of the negative imaginarypart is exponentially small in the barrier thickness L− a (Figure 1.1).

We refer the reader to Bindel and Zworski [1, 13] for a discussion of theoreti-cal, intuitive, and computational aspects of resonances with ample references to theliterature. For computational approaches, there are several in literature. Wei, Ma-jda, and Strauss [11] report an algorithm for computing acoustic resonances for abounded scatterer by relating time samples at a point in space to a quasi-normalmode expansion and using Prony’s method to extract the resonant exponents. Themethod of complex scaling of the spatial variable of Simon [8] can be used to convertresonances into eigenvalues of a modified non-self-adjoint operator—see also Datchev[3] and Zworski [13]. An introduction to this method and its relation to method ofperfectly matched layers (PML) can be found in [3]. The potential we consider in thispaper can be thought of as a well in an island potential similar to that of section 11of [5], except that we send the width of the island to infinity.

In this work, we look specifically at computing resonances in certain asymptoticregimes: (1) a potential wall surrounding a well becomes infinitely thick (L → ∞with a fixed), keeping k bounded, and (2) L fixed and �k → ∞, high frequencies.As a prototypical example, consider the one-dimensional Schrodinger equation whosepotential V L(x) is shown in Figure 1.1. We will assume that V L(x) ≥ 0 consists ofa low-energy well surrounded by a barrier of finite thickness, in this case a wall ofheight V0 and width (L− a). Denote by V∞(x), the limiting potential well (L = ∞),for which V∞(x) = V0 for |x| > a. Introduce

HL = −∂2x + V L(x) and H∞ = −∂2x + V∞(x).

For finite L, HL has no bound states. We give a short proof in Appendix A. Ifa < L < ∞, the spectrum of HL is continuous and occupies the nonnegative realline: σ(HL) = [0,∞). As explained above, a solution with spatially localized initialconditions will decay in the local energy sense as time advances. We are particularly

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RESONANCES OF SCHRODINGER’S EQUATION 1491

interested in the case where the wall thickness is large and the leakage of energy isslow. We shall quantify this leakage rate by relating it to the scattering resonanceeigenvalue problem for V L(x).

As L becomes large, two classes of resonances of HL emerge:(a) A finite number of low-frequency resonances, Eres,i(L), i = 1, . . . ,M , whose

real parts lie in the interval (0, V0) ⊂ σ(HL) and which converge, as L ↑,exponentially fast to the real eigenvalues, E∞,i, i = 1, . . . ,M , of H∞ locatedin the real interval (0, V0), below the continuous spectrum [V0,∞) of H∞.Within the wall in [−L,L], these resonant modes resemble the bound statesof H∞, being concentrated in the well in [−a, a]; outside of the wall, theygrow at a slow exponential rate.

(b) An infinite family of discrete scattering resonances, Ec-res,m(L), m ≥ 1,which for each fixed L < ∞ lies along an unbounded curve in the lowerhalf plane, emanating from the real point V0, with Re

√Ec-res,m ∼ πmL and

Im√Ec-res,m ∼ − 1

L log(mL

), m � 1. These discrete points become increas-

ingly clustered along this curve, which approaches the continuous spectrum[V0,∞), of the limit operator H∞. As L increases, these resonances becomeincreasingly insensitive to the well in [−a, a]. See Regge [7], Zworski [12], andSjostrand and Zworski [10].

Our main results are as follows:1. Theorem 2.2 states that for sufficiently large but finite wall thickness L, there

is in a neighborhood of every discrete eigenvalue of H∞ a nearby scatteringresonance in the lower half plane for HL. These resonances converge to theeigenvalues of H∞ exponentially fast as L → ∞. The scattering resonancesare fixed points of a complex-valued function whose iterates converge andprovide a numerical scheme for computing the resonances.

2. A second approach to the construction and computation of near-bound-statescattering resonances is discussed in section 3. In this approach, the reso-nances are obtained by viewing H∞ as the unperturbed operator and HL

as the perturbed operator. Such a perturbative approach is nontrivial, sinceV L(x)−V∞(x) = −V0 1[L,∞)(x) is a noncompact perturbation of V∞; whilethe eigenvalues of V∞(x) are real with L2 eigenfunctions, those for HL arecomplex non-L2 (only L2

loc) eigenstates. We numerically implement this per-turbative approach and demonstrate, for L large, that it gives excellent agree-ment with very accurate direct numerical calculations of the resonances ofHL.

3. For computing high-frequency resonances, we offer in section 4 an approxi-mate iterative numerical scheme based on leading terms of the high-E asymp-totics of solutions of the ODE. The scheme excludes terms that depend onthe particular form of the well. In two simple cases, V (x) = V0 (no well)and V (x) = 0 (square well) in |x| < a, we are able to compare the resultsof the scheme with the exact values of the resonances. In the latter caseand for small L, there is appreciable imprecision in the imaginary part of thecomputed resonances. This error vanishes as L increases, indicating that theshape of the well has a negligible effect as L, the width of the wall, increases.

2. Near bound-state resonances. Assuming E > 0, let us put E = k2, withRe k > 0. For simplicity, we consider a symmetric potential V (−x) = V (x). Inthis case all quasi-normal modes are even or odd. We restrict our attention to evenquasi-normal modes (scattering resonances) for which the pair (ψ(x), k) satisfies theeigenvalue problem

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HLψ := −ψ′′ + V L(x)ψ = k2ψ, 0 < x <∞,(2.1)

ψ(0) = 1, ψ′(0) = 0,(2.2)

ψ′(L)− ikψ(L) = 0,(2.3)

in which the prime indicates differentiation with respect to x. The potential is of thetype illustrated in Figure 1.1:

(2.4) V L(x) =

⎧⎨⎩v(x) for 0 ≤ x < a,V0 for a ≤ x < L,0 for x ≥ L,

where 0 ≤ v(x) ≤ V0. The initial conditions (2.2) imply that ψ(x) can be continued toa symmetric function on R satisfying outgoing radiation conditions at ±∞. Condition(2.3) is the outgoing radiation condition. Due to the boundary condition at x = L,the boundary-value problem (2.1)–(2.3) is non-self-adjoint. Thus, its eigenvalues canbe expected to be complex and in fact form a discrete subset of the open lower halfplane; see Appendix B.

2.1. The resonance condition. We next derive a characterization of scatteringresonance energies, k, as zeros of a function defined on a subset of C. Denote byψ(x, k) the unique solution of the initial-value problem (2.1)–(2.2). For V (x) ≡ V0,for a ≤ x ≤ L, we will also be able to make use of the explicit form of the solution inthis region,

(2.5) ψ(x) = Aei√k2−V0 (x−a) +Be−i

√k2−V0 (x−a),

where A, B are unknown constants. For x > L, V (x) = 0, and applying the radiationboundary condition (2.3), we obtain

(2.6) ψ(x) = Ceik(x−L), x > L,

where C is also unknown.We first briefly deal with the case L = ∞. The condition forH∞ that is consistent

with the outgoing condition in the case L <∞ is that ψ′(a)− i√k2 − V0 ψ(a) = 0, or

ψ(x) = Aei√k2−V0 (x−a), x > a.

For k2 < V0, this condition implies decay of the solution. Since ψ(a, k) �= 0 thecondition for a bound state can be written as

(2.7)√k2 − V0 − ψ′(a, k)

i ψ(a, k)= 0.

Equation (2.7) has a finite number of solutions (a proof is given in Appendix C) givingrise to eigenvalues of H∞:

(2.8) E∞,i = k2∞,i ∈ (0, V0), i = 1, . . . ,M.

These eigenvalues are simple, corresponding to the zeros, kB, of the expression in(2.7) being simple. We present a proof of this in Appendix D. For the remainder wewill denote by kB a generic bound-state frequency satisfying (2.7).

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For L finite, we now derive the extension of (2.7) to resonances. Continuity of ψand ψ′ at x = a and (2.5) imply

ψ(a) = A+B and ψ′(a) = i√k2 − V0 (A−B).

Continuity of ψ and ψ′ at x = L implies

Aei√k2−V0(L−a) +Be−i

√k2−V0(L−a) = C,

i√k2 − V0

(Aei

√k2−V0(L−a) −Be−i

√k2−V0(L−a)

)= ikC.

Eliminating A, B, and C, we arrive at an equation for the resonance energies E = k2:(ψ(a, k) +

ψ′(a, k)i√k2 − V0

)(1−

√k2 − V0k

)ei

√k2−V0(L−a)

+

(ψ(a, k)− ψ′(a, k)

i√k2 − V0

)(1 +

√k2 − V0k

)e−i

√k2−V0(L−a) = 0.(2.9)

Multiplication of (2.9) by

(k√k2 − V0)e

i√k2−V0(L−a)

ψ(a, k)(k +√k2 − V0)

yields the equivalent equation

(2.10) FB(k) + g(k)e2i√k2−V0(L−a) = 0 (resonance condition),

where

FB(k) ≡√k2 − V0 − ψ′(a, k)

iψ(a, k),(2.11)

g(k) ≡(√

k2 − V0 +ψ′(a, k)iψ(a, k)

)k −√

k2 − V0

k(k +√k2 − V0)

.(2.12)

By (2.7), the bound-state energies k2B of H∞ are characterized by FB(kB) = 0 and0 < k2B < V0 . The scattering resonance energies are solutions of the perturbedequation (2.10).

Remark 2.1. Equation (2.10) for the scattering resonance energies is equivalentto

W(k;L) ≡ φ1(x, k)∂xφ2(x; k) − φ2(x; k)∂xφ1(x; k) = 0,

where W(k;L) is the Wronskian of the two solutions φj(x, k), j = 1, 2, satisfyingHLφ = k2φ with φ1 satisfying the boundary condition (2.2) and φ2 satisfying theboundary condition (2.3) and φ2(L; k) = 1. The Green function for −∂2x + V − k2

explicitly given by variation of constants has poles (scattering resonances) in Im k < 0,at the zeros of W(k;L).

2.2. Resonances converging to eigenvalues as L → ∞. We now constructthe complex scattering resonances for HL and prove that they converge exponentiallyto the discrete eigenvalues {kB} of H∞ as L→ ∞ .

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Write k = kB + κ, recalling that kB is real, and anticipate that κ is complex.From (2.10) we have

(2.13) FB(kB + κ) + g(kB + κ) exp{−2√V0 − (kB + κ)2(L − a)} = 0.

We shall seek solutions to (2.13) in the complex κ plane. These turn out to lie in thelower half plane and, for L large, exponentially close to the real zeros of FB(k).

Since FB(kB) = 0, and kB is a simple zero of FB , we have

(2.14) FB(kB + κ) = κ

∫ 1

0

F ′B(kB + θκ) dθ ≡ κ f1(kB + κ).

F ′B(kB) �= 0 and f1(kB + κ) is nonvanishing for |κ| sufficiently small. Thus (2.13),

may be rewritten as

(2.15) κ = G(κ) := − g(kB + κ)

f1(kB + κ)exp{−2

√V0 − (kB + κ)2(L− a)}.

Theorem 2.2. Let kB be such that FB(kB) = 0, and thus 0 < k2B < V0 is abound-state energy of H∞.

1. There are positive constants C, α, and L′ such that if L > L′, then the scat-tering resonance eigenvalue problem (2.1)–(2.3) has a solution with scatteringresonance eigenvalue k = kB + κ and

|κ| ≤ Ce−2α(L−a).

2. Im κ < 0 and

(2.16) κ ∼ Ke−2√V0−k2B L (L→ ∞)

with K = −g(kB)f1(kB)−1 exp(2a√V0 − k2B ), in the sense that the ratio of

the two sides of the expression tends to 1.Proof. To prove the theorem we use the following claim, proved below.Claim. f1(kB + κ) and g(kB + κ) are analytic in a neighborhood of κ = 0 and

there exist positive constants C, α, and r such that

(2.17)

∣∣∣∣ g(kB + κ)

f1(kB + κ)

∣∣∣∣ ≤ C and Re√V0 − (kB + κ)2 ≥ α for |κ| < r.

The theorem then is proved as follows. Let L′ be such that for L ≥ L′,

Ce−2α(L−a) < r and

∣∣∣∣dG(κ)dκ

∣∣∣∣ ≤ 0.99 for |κ| < r.

The bound on ∂G/∂κ comes from the analyticity of g/f1 and the exponential factorin (2.15). It follows that for |κ| < r, we have

(2.18) |G(κ)| ≤ Ce−2α(L−a) < r.

Thus G maps the disk {|κ| < r} into itself. Furthermore, G is a contraction on thisdisk because |G′(κ)| < 1 for |κ| < r. It follows that (2.15) has a unique solution in{|κ| < r}, which satisfies the bound (2.18). Therefore, to complete the proof, we needonly verify the above claim.

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Fig. 2.1. The potential used in the example in section 3.

To verify the claim it suffices to show that f1(kB + κ) does not vanish in someneighborhood kB . The expression for f1 in (2.14) suggests calculation of F ′

B(kB),using that F (kB) = 0. We obtain

d

dkFB(k)

∣∣∣∣k=kB

=d

dk(k +

√k2 − V0) ·

(√k2 − V0 − ψ′(a, k)

iψ(a, k)

) ∣∣∣∣k=kB

+

(kB +

√k2B − V0

)· d

dk

(√k2 − V0 − ψ′(a, k)

iψ(a, k)

)∣∣∣∣k=kB

=

(kB +

√k2B − V0

)· d

dk

(√k2 − V0 − ψ′(a, k)

iψ(a, k)

)∣∣∣∣k=kB

.(2.19)

We now observe that F ′B(kB) �= 0. Indeed, the second factor in (2.19) is the

derivative at k = kB of the function detecting bound states of H∞, which is non-zero (Appendix D). Thus, F ′

B(kB) �= 0 and by analyticity of FB(k) near kB and theexpression for f1 in (2.14), there exists r > 0 such that

|f1(kB + κ)| > 0 for |κ| < r.

Finally, let κ = κ1 + iκ2 and consider

V0 − (kB + κ)2 = (V0 − k2B − 2kBκ2 − κ21 + κ22)− 2i(kBκ2 + κ1κ2).

Since V0 > k2B , thereby perhaps taking r smaller but still positive, we have that for|κ| < r, the real part of (V0 − (kB + κ)2) > 0, and hence

Re√V0 − (kB + κ)2 ≥ α > 0,

for some α. This completes the proof of the claim and thus part 1 of Theorem 2.2.Part 2 comes from (2.15), the fact that κ→ 0 exponentially as L→ ∞, and the factthat g(kB) and f1(kB) are nonzero. That Im κ < 0 is standard; see Appendix B.

Remark 2.3. This result shows that for a resonance k near a bound state kB,k− kB is exponentially small in (L− a), or the thickness of the wall. Equation (2.15)suggests that for L− a� 1

(2.20) κ ≈ κ1 = − g(kB)

F ′B(kB)

exp{−2√V0 − (kB)2 (L− a)}.

2.3. A numerical illustration. To illustrate the conclusions of Theorem 2.2we consider the simple example of a piecewise constant potential (Figure 2.1):

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Table 2.1

The dependence of a resonance on the barrier thickness L. The barrier height is V0 = 4 andand the defect thickness is a = 1.

L k

3 1.02959049042796 − 0.00051708051625i4 1.02985760250922 − 0.00001677632024i5 1.02986623992970 − 0.00000054394364i6 1.02986651993964 − 0.00000001763573i

Table 2.2

The real and imaginary parts of κ for different L. Note the negative exponential dependenceon (L− a). Here, V0 = 4 and a = 1.

L− a Reκ Im κ

2 −2.76038894295727 × 10−4 −5.17080516251752 × 10−4

3 −8.92681303588105 × 10−6 −1.67763202402731 × 10−5

4 −2.89392557917267 × 10−7 −5.43943636731303 × 10−7

5 −9.38261446314925 × 10−9 −1.76357251517556 × 10−9

VL(x) =

⎧⎨⎩0 for 0 ≤ x < a,V0 for a ≤ x < L,0 for x ≥ L.

For this case,

ψ(x, k) = cos kx for 0 ≤ x < a.

The equation for the bound states (2.7) simplifies to

−k tan ka = i√k2 − V0,

which has solution(s) for k <√V0. We denote the bound state k by kB . The equation

for resonances (2.9) becomes

(cos ka− k sin kx

i√k2 − V0

)(1−

√k2 − V0k

)ei

√k2−V0(L−a)

+

(cos ka+

k sin ka

i√k2 − V0

)(1 +

√k2 − V0k

)e−i

√k2−V0(L−a) = 0.

Let us denote the resonances by k = kB+κ, which are complex numbers with negativeimaginary parts.

As an example, we set V0 = 4 and a = 1. We then use Newton’s method to solvefor kB and k for different barrier widths L. There are a finite number of solutions kBof FB(k) = 0. The lowest energy bound state is kB = 1.02986652932226. Table 2.1shows the dependence of the resonance k, which is a perturbation of kB, approachingkB as L↑∞ by Theorem 2.2. Note the exponential decay of κ in L.

Table 2.2 shows the real and imaginary parts of κ = k − kB . It is clear from thetable that κ decays exponentially as a function of (L−a). These results illustrate theestimate in Theorem 2.2. They also illustrate that even for barriers that are not sothick (L = 3), the estimate of kB is very good—the exponential decay of κ providesvery quick convergence.

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3. Direct perturbation for near-bound-state resonances. We now takea direct approach to the construction of near-bound states as perturbations of truebound states. This leads to a computational method, explored in sections 3.2 and3.3. We remark that parts of this approach may be applicable to problems in higherdimensions.

3.1. Direct perturbation of bound states. For a compactly supported po-tential V L, it is possible to reduce the scattering resonance problem (2.1)–(2.3) to anonlocal (boundary integral) equation at x = L. We seek solutions of (2.1)–(2.3) inthe form

ψ(x; k) = ψB(x) + ξ(x), k = kB + κ,

where

(3.1) H∞ψB = k2BψB, ψB ∈ L2(R), 0 < k2B < V0 .

Substitution of (3.1) into (2.1)–(2.3) yields the following system for ξ(x) and κ:

−ξ′′ + V∞(x)ξ − (kB + κ)2ξ = (2kBκ+ κ2)ψB , 0 < x < L,(3.2)

ξ(0) = 0, ξ′(0) = 0,(3.3)

ξ′ − i(kB + κ)ξ = −ψ′B + i(kB + κ)ψB, x = L.(3.4)

In obtaining (3.2) we used that V L(x) = V∞(x) for 0 ≤ x ≤ L. One must determineκ such that (3.4) is satisfied.

We introduce f �→ Skf the solution operator of the inhomogeneous initial-valueproblem

−u′′ + V∞(x)u − k2u = f, 0 < x < L,

u(0) = 0, u′(0) = 0 .

Note that for each fixed f ∈ C0([0, L]), the mapping k �→ S[k]f varies smoothly withk. Equation (3.2)–(3.3) can now be solved in terms of the operator SkB+κ:

(3.5) ξ =(2kBκ+ κ2

)SkB+κψB.

Substitution of (3.5) into the outgoing radiation condition (3.4) yields a nonlocalequation for κ = κ(L), the correction to kB , due to finite barrier width. Anticipatingthat κ will be small, we organize the terms of this equation in ascending order in κ :

ψ′B − ikBψB + κ qB(L)

+ κ [2kB(∂ − ikB) {SkB+κ − SkB}ψB]+ κ2 [∂SkB+κψB − i(3kB + κ)SkB+κψB] = 0,(3.6)

in which

(3.7) qB(L) ≡ [2kB (∂ − ikB)SkBψB − iψB] .

In (3.6), we have used the following conventions: ∂Skf = ∂x (Skf)|x=L and allevaluations of ψB, SkBψB, and ∂SkBψB are at x = L and k = kB . By smoothness ofk �→ Skf , the terms on the second and third lines of (3.6) can be expressed, for κ ina neighborhood of zero, in the form −κΛ(κ, L), where Λ is smooth in κ.

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Thus we can reexpress (3.6) as

qB(L)κ = ikBψB(L)− ψ′B(L)− κΛ(κ, L)

or

κ =ikBψB(L)− ψ′

B(L)

qB(L)− κ

Λ(κ, L)

qB(L),(3.8)

where

Λ(κ, L) =[2kB(∂ − ikB)(SkB+κψB − SkBψB)]

+ κ[∂SkB+κψB − i(3kB − κ)SkB+κψB].(3.9)

We will show that (3.8) admits a solution and that it is exponentially small in L.Notice that V L(x) = V0 for x > a. Therefore, from (3.1) we have

ψB(x) = ψB(a)e−σB(x−a), x > a,

where σB =√V0 − k2B > 0. To study the properties of qB(L), consider uB(x) satis-

fying

−u′′B + σ2BuB = ψB(a)e

−σB(x−a), x > a.

This can be solved explicitly to give

uB(x) = C1eσB(x−a) + C2e

−σB(x−a) +ψB(a)

2σBxe−σB(x−a), x > a.

Observe that for x > a, uB(x) = SkBψB for some choice of C1 and C2. From (3.7), itis clear that qB(L) is dominated by the positive exponential terms in uB(x) as longas C1 �= 0. Therefore, we argue that for some L0 > 0 and C > 0,

(3.10) |qB(L)| ≥ CeσBL

for all L > L0. We show that C1 �= 0 in Appendix D.Theorem 3.1. For some r > 0 (3.8) admits a solution κ with |κ| < r. Moreover,

κ is exponentially small in L for sufficiently large L.Proof. We examine the iterative approach to solve (3.8). We write

κ0 =ikBψB(L)− ψ′

B(L)

qB(L),(3.11)

κn+1 = κ0 − κnΛ(κn, L)

qB(L).(3.12)

Our aim is to show that

|κn+1 − κn| ≤ C0|κn − κn−1|for some C0 < 1 and that the iterates remain within some small ball centered at theorigin.

Since L > a, we can use (3.1) to obtain

κ0 =(ikB − σB)ψB(a)e

−σB(L−a)

qB(L).

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This, together with (3.10), leads to

|κ0| ≤ C0e−2σBL

for some C0 independent of L. From (3.12), we have

|κn+1| ≤ |κ0|+ |κn| |Λ(κn, L)||qB(L)| .

Suppose

(3.13) |κn| ≤ 2C0e−2σBL;

then

(3.14) |κn+1| ≤ C0e−2σBL

(1 + 2

|Λ(κn, L)||qB(L)|

),

and we must show that |Λ(κn, L)|/|qB(L)| < 1/2 for sufficiently large L.In the definition of Λ(κ, L), (3.9), there are two terms. To obtain a bound for the

second term, we need to consider SkB+κψB. Let u(x) satisfy

(3.15) −u′′ + σ2u = ψB(a)e−σB(x−a), x > a,

where σ =√V0 − k2 and k = kB + κn. Then u(x) = SkψB and is given by

u(x) = C1eσ(x−a) + C2e

−σ(x−a) +ψB(a)e

−σB(x−a)

σ2 − σ2B

, x > a.

One can evaluate C1 and C2 using boundary conditions u(a) and u′(a). In doing so,one can see that the expression approaches [SkBψB](x) as σ → σB .

For k = kB + κ, we have

σ =√V0 − (kB + κ)2 = σB

√1− 2kBκ− κ2

σ2B

.

Suppose |κ| ≤ 2C0e−2σL as in (3.13); then we can conclude that

(3.16) Reσ > 0 and |σ| ≤ σB(1 + s|κ|) for |κ| < r,

for some s > 0. It is clear then that u(x) is dominated by the positive exponentialterm for large x and therefore

|[SkB+κψB](L)| = |u(L)| ≤ C3eσBLes|κ|L.

Similarly, we argue that

|[SkB+κψB](L)| = |u′(L)| ≤ C4eσBLes|κ|L.

To study the first term of Λ(κ, L), we need to consider SkB+κψB −SkBψB . SinceSkψB is a smooth function of k, we can use the fundamental theorem of calculus andwrite

SkB+κψB − SkBψB =

∫ 1

0

κ∂kSkB+τκψBdτ.

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Let u(x) satisfy the k-differentiated version of (3.15), i.e.,

−u′′ + σ2u = 2ku, x > a.

Then u(x) = ∂kSkψB and is given by

u(x) = D1eσ(x−a) +D2e

−σ(x−a)

− kx

σC1e

σ(x−a) +kx

σC2e

−σ(x−a) +2kψB(a)e

−σB(x−a)

σ2 − σ2B

.

One concludes that if C1 does not vanish,

|u(L)| ≤ C5eσBLes|κ|L.

The estimates for u(L), u′(L), and u(L) lead to the following estimate for Λ(κ, L):

|Λ(κ, L)| ≤ κC6eσLes|κ|L.

Considering the bound in (3.14) and using (3.10), we arrive at

(3.17)|Λ(κn, L)||qB(L)| ≤ C exp

[−L (2σB + 2sC0e−2σBL

)].

It is apparent that we can choose L sufficiently large so that the expression above isless than 1/2. Therefore, the iterates (3.12) start and stay in an exponentially small(in L) ball.

To show that the iterates converge, we rewrite

κn+1 − κn =− κnΛ(κn, L)

qB(L)+ κn−1

Λ(κn−1, L)

qB(L)

=− (κn − κn−1)Λ(κn, L)

qB(L)− κn−1

Λ(κn, L)− Λ(κn−1, L)

qB(L).

From the above bound on |Λ(κn, L)/|qB(L)| , we obtain

|κn+1 − κn| ≤ |κn − κn−1|(1

2+ 2Coe

−2σBL|Λ(κn, L)− Λ(κn−1, L)|qB(L)|κn − κn−1|

).

Convergence is guaranteed as long as |Λ(κn, L)−Λ(κn−1, L)|/(|qB(L)||κn − κn−1|) isbounded.

We omit the demonstration of such a bound since the method is very similarto that used to establish the (3.17), namely, the explicit use of the solution of theconstant coefficient differential equation for x > a. This completes the proof.

3.2. A computational framework for near-bound-state resonances. Theperturbative approach of section 3 suggests a computational approach to near-bound-state resonances, based on (3.8) and the method of successive approximations.

Step 1. Solve for a bound-state eigenpair (ψB, EB = k2B) of H∞ in (3.1), corre-

sponding to the limiting potential.Step 2. Solve the initial-value problem

−u′′B + V∞(x)uB − k2BuB = ψB(x), 0 < x < L,

uB(0) = 1, u′B(0) = 0,

for the unique solution, and denote it by SkBψB(x).

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Step 3. Define the correction κ0, based on the leading order term in (3.8), by

(3.18) κ0 ≡ (ikBψB(L)− ψ′B(L))

2kB (∂ − ikB)SkBψB|x=L − iψB(L).

Step 4. Approximate the resonance eigenpair with energy k near kB by

kapprox = kB + κ0,(3.19)

ψapprox(x) = ψB(x) + 2kBκ0SkBψB.(3.20)

In the next section, we implement this approximation scheme for a specific example.

3.3. An example perturbation calculation. We will again use the piecewiseconstant potential V (x) in Figure 2.1 as an example. Recall that the bound-stateeigenfunction is given by

ψB(x) =

{cos(kBx) for 0 ≤ x < a,

cos(kBa) e−√V0−k2B(x−a) for x > a.

To obtain κ0 in Step 3, we solve for φ := 2kBS[kB]ψB . There are two sections to thepotential V (x), and therefore

−u′′B − k2BuB = cos(kBx) for 0 < x < a ,

−u′′B + σ2BuB = cos(kBa)e

−√V0−k2B(x−a) for a < x < L .

Hence, we have

uB(x) = −x sin(kBx)2kB

for 0 ≤ x < a.

Letting A = −a sin(kBa)/(2kB), and B = −(sin kBa+ akB cos(kBa))/(2kB), we havefor a < x < L

uB(x) = uBh(x) + uBp(x),

where uBh(x) and uBp(x) are the homogeneous and particular solutions, given by

uBh(x) =1

2σBe−σB(x−a)(AσB −B) +

1

2σBeσB(x−a)(AσB +B)

and

uBp(x) = −[(

1

4σ2B

+(x− a)

2σB

)e−σB(x−a) − 1

4σ2B

eσB(x−a)]cos(kBa).

To obtain the approximation κ0, one puts uB = SkBψB into (3.18):

(3.21) κ0 =(ikB + σB) cos(kBa) e

−σB(L−a)

2kB(u′B(L)− ikBuB(L))− i cos(kBa) e−σB(L−a) .

We compare in Table 3.1 the true resonances k with the values kB + κ computedby the perturbative approach for different values of L. The agreement between the res-onance value and the approximated value via the perturbation approach is excellent;the approximation becomes more accurate as L increases.

We assess the accuracy of the approximation to the quasi-normal mode. Theagreement between the exact solution ψ(x, k) and ψapprox(x) for L = 4 is so goodthat we cannot visually discern between them. As a result, we plot the real andimaginary parts of ψ(x, κ) − ψapprox(x), which we show in Figure 3.1.

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Table 3.1

Comparison of the true resonance values k and the approximate value obtained obtained byperturbing the bound-state values kB . Note that accuracy increases with barrier thickness L.

L k kB + κ0

4 1.029857602509223 − 0.000016776320240i 1.029857602963758 − 0.000016776618066i5 1.029866239929701 − 0.000000543943637i 1.029866239930306 − 0.000000543944139i6 1.029866519939643 − 0.000000017635724i 1.029866519939644 − 0.000000017635725i

Fig. 3.1. The graph of the approximation error ψ(x, k) − ψapprox(x) when L = 4. The solidand dashed curves correspond to the real and imaginary parts of the expression.

4. High-frequency resonances. Having studied the properties of resonancesthat are near to bound states, we now turn our attention to the case where theresonance k has a large real part. The energies k2 are far away from the bound-stateenergies and lie in the lower half complex plane.

We will observe that for large k and L, the solution of equations (2.1)–(2.3)depends negligibly on the function v(x) describing the defect in the interval [0, a].First, let us consider the simplest case in which V (x) ≡ V0 in [0, L], a potential barrierof width L without the well. In this case, the solution is ψ(x) = cos

√k2 − V0 x, and

the radiation condition ψ′(L)− ikψ(L) = 0 yields

(4.1) e2iL√k2−V0 = −4k2

V0

(1− V0

2k(k +√k2 − V0)

)2

by direct computation or by putting ψ(x) = cos√k2 − V0x into (2.9). Thus

iL√k2 − V0 = πi(m+ 1/2) + log(k) + log(2

√V0) + log

(1− V0

2k(k +√k2 − V0)

),

in which m is an integer. Using the identity√k2 − V0 = k − V0/(k +

√k2 − V0) and

setting

c = − 1

πlog(2/

√V0),

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we obtain

k =π(m+ 1/2 + ic)

L− i

Llog(k) +

V0

k +√k2 − V0

− i

Llog

(1− V0

2k(k +√k2 − V0)

)(4.2)

=π(m+ 1/2 + ic)

L− i

Llog(k) +O(|k|−1) (m� 1).

It seems natural to drop the O(|k|−1) term and use the remaining terms as an ap-proximation for an iterative method for computing the resonances,

(4.3) k ≈ π(m+ 1/2 + ic)

L− i

Llog k.

Then setting

(4.4) k =π(m+ 1/2 + ic)

L(1 + κ)

leads to

(4.5) κ = fap(κ) :=−i

π(m+ 1/2 + ic)

(log

(π(m+ 1/2 + ic)

L

)+ log(1 + κ)

)

and hence to the iterative scheme with initial approximation κ0 = 0 and recursionrule

(4.6) κn+1 = fap(κn).

Observe that fap does not depend on the defect in the interval [0, a]. Numericalcalculations with v(x) = V0 show that fixed points of fap are altered only negligiblyif the full formula in (4.2) is used, as demonstrated in Table 4.1.

If we put v(x) = 0 in [0, a], a < L, corresponding to a square well, the resonanceequation becomes

e2i√k2−V0(L−a) = −1 +

√k2−V0

k

1−√k2−V0

k

cos ka− i k√k2−V0

sin ka

cos ka+ i k√k2−V0

sin ka

(4.7)

= −k2

V0

(1 +

√k2 − V0k

)2 cos ka− i k√k2−V0

sin ka

cos ka+ i k√k2−V0

sin ka

= −4k2

V0

(1− V0

2k(k +√k2 − V0)

)2 e−ika − iV0

k2−V0+k√k2−V0

sin ka

eika + iV0

k2−V0+k√k2−V0

sinka.

Comparing this equation to (4.1), we see that for Im k < 0, which is the case for theresonances we seek, there is an additional factor on the left decaying like e−2|Imk|a.This factor is not obviously compensated on the right because of the exponentially in-creasing behavior of sin ka. When L is not too big, numerical computations show thatthe approximation (4.3) produces k-values whose imaginary parts do in fact deviateappreciably from those of the true resonances. We calculate the exact resonances bya recursion rule κn+1 = fex(κn) obtained from (4.7) without making any approxima-tions. As L increases, the approximate results become very good, as demonstratedin Table 4.1. Thus we see that for small L, the imaginary parts of the resonancesare sensitive to the form of the well. The error in the imaginary part in that table

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Table 4.1

A comparison of computations of resonances km. The first two values are km = (π(m+ 1/2 +ic)/L)(1 + κ), with κ = 0 and κ = κ∗, where κ∗ = fap(κ∗). The exact resonances kexact arecalculated by a recursion κn+1 = fex(κn) obtained from the resonance equations (4.1) for v(x) = V0and (4.7) for v(x) = 0 without making any approximations. Observe the deviation between theimaginary parts of kapprox and kexact for v(x) = 0 for various values of L (barrier width) overcomparable ranges of Re k. This deviation is significant for small L but diminishes as L increases.The values a = 1 (well width) and V0 = 5 (barrier height) are used throughout.

m π(m+ 1/2 + ic)/L kapprox kexact, v(x) = V0 kexact, v(x) = 0

L = 455 43.590 + 0.0279 i 43.585− 0.8600 i 43.674 − 0.8585 i 43.594 − 0.7153 i120 94.641 + 0.0279 i 94.638− 1.1096 i 94.664 − 1.1094 i 94.581 − 0.8873 i185 145.70 + 0.0279 i 145.69− 1.2175 i 145.71 − 1.2174 i 145.47 − 0.9736 i250 196.74 + 0.0279 i 196.74− 1.2926 i 196.75 − 1.2925 i 196.99 − 1.0343 i315 247.79 + 0.0279 i 247.79− 1.3503 i 247.80 − 1.3502 i 247.88 − 1.0802 i380 298.84 + 0.0279 i 298.84− 1.3971 i 298.85 − 1.3971 i 298.77 − 1.1176 i

L = 50600 37.731 + 0.0022 i 37.731− 0.0704 i 37.797 − 0.0703 i 37.794 − 0.0689 i1500 94.279 + 0.0022 i 94.279− 0.0887 i 94.306 − 0.0887 i 94.304 − 0.0869 i2400 150.83 + 0.0022 i 150.83− 0.0981 i 150.84 − 0.0981 i 150.84 − 0.0962 i3300 207.38 + 0.0022 i 207.38− 0.1045 i 207.39 − 0.1045 i 207.39 − 0.1024 i4200 263.93 + 0.0022 i 263.93− 0.1093 i 263.94 − 0.1093 i 263.93 − 0.1071 i5100 320.47 + 0.0022 i 320.47− 0.1132 i 320.48 − 0.1132 i 320.48 − 0.1109 i

L = 2002700 42.419 + 0.0006 i 42.419− 0.0251 i 42.423 − 0.0251 i 42.423 − 0.0246 i6200 97.397 + 0.0006 i 97.397− 0.0223 i 97.423 − 0.0223 i 97.423 − 0.0222 i9700 152.38 + 0.0006 i 152.38− 0.0246 i 152.39 − 0.0246 i 152.40 − 0.0244 i13,200 207.35 + 0.0006 i 207.35− 0.0261 i 207.37 − 0.0261 i 207.37 − 0.0260 i16,700 262.33 + 0.0006 i 262.33− 0.0273 i 262.34 − 0.0273 i 262.35 − 0.0272 i20,200 317.31 + 0.0006 i 317.31− 0.0282 i 317.32 − 0.0282 i 317.32 − 0.0281 i

for L = 4 even for large k is likely due at least in part to the discontinuity of thesquare-well potential. A full analysis of the accuracy of the iteration (4.6) is evidentlysubtle and delicate, and it is not our intention to carry out this analysis here.

Remark 4.1. It is tempting to examine the WKB solution of (2.1)–(2.2) with ageneral potential v(x) for 0 < x < a. We found that such an approach needs to be

treated with caution because of the subtlety related to an exponential factor e|Imk|

appearing in the WKB expression. See Appendix F for more details.

Tables 4.2 and 4.3 compare the approximation of k given by (4.4) with κ = 0 tothe value obtained by setting κ equal to the limit of the sequence κn defined by theiteration (4.6). When V0 = 4, we have c = 0, and the initial approximation of k isreal; this is shown in Table 4.2. The term 1/2+ic in fap is evidently not unimportant;it turns out that it contributes to a shift in the real parts of the computed resonancesthat is on the order of the spacing between the resonances km.

Figure 4.1 shows the computed value of k for V0 = 4 and for various values of mand L. One observes that the density of resonances increases linearly with the widthL of the potential wall, in accordance with theoretical results [10, 4, 9], and that theresonances lie on a curve in the lower complex half-plane that tends to the real lineas L→ ∞, a phenomenon that is also known to occur within the spectral bands of aone-dimensional layered medium as the number of layers increases [6].

Proposition 4.2 establishes the convergence of the iteration (4.6) for large enoughintegers m. The following simple estimate is needed in the proof. Let z1 and z2 becomplex numbers. Then

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Table 4.2

The values of the quantities in (4.4) for barrier width L = 4 and height V0 = 4 for which c = 0.The last column is the radius of D in Proposition 4.2.

m k π(m+ 1/2 + ic)/L |κ| |κ| bound55 43.5842 − 0.943732 i 43.5896 0.0216508 0.043299556 44.3697 − 0.948196 i 44.3750 0.0213681 0.042734457 45.1551 − 0.952582 i 45.1604 0.0210936 0.042185458 45.9406 − 0.956891 i 45.9458 0.0208268 0.041651959 46.7260 − 0.961128 i 46.7312 0.0205675 0.041133360 47.5115 − 0.965295 i 47.5166 0.0203152 0.040628761 48.2970 − 0.969393 i 48.3020 0.0200697 0.040137862 49.0824 − 0.973424 i 49.0874 0.0198307 0.039659963 49.8679 − 0.977392 i 49.8728 0.0195980 0.039194564 50.6533 − 0.981298 i 50.6582 0.0193712 0.038741065 51.4388 − 0.985144 i 51.4436 0.0191502 0.0382991

Table 4.3

The values of the quantities in (4.4) for barrier width L = 4 and height V0 = 10 for whichc �= 0. The last column is the radius of D in Proposition 4.2.

m k π(m+ 1/2 + ic)/L |κ| |κ| bound55 29.0569 − 0.485209 i 29.0597 + 0.0763576 i 0.0193247 0.038648556 29.5806 − 0.488185 i 29.5833 + 0.0763576 i 0.0190833 0.038165757 30.1042 − 0.491109 i 30.1069 + 0.0763576 i 0.0188485 0.037696258 30.6278 − 0.493982 i 30.6305 + 0.0763576 i 0.0186201 0.037239459 31.1515 − 0.496807 i 31.1541 + 0.0763576 i 0.0183979 0.036794960 31.6751 − 0.499585 i 31.6777 + 0.0763576 i 0.0181814 0.036362161 32.1987 − 0.502317 i 32.2013 + 0.0763576 i 0.0179707 0.035940662 32.7224 − 0.505005 i 32.7249 + 0.0763576 i 0.0177653 0.035529863 33.2460 − 0.507651 i 33.2485 + 0.0763576 i 0.0175651 0.035129464 33.7696 − 0.510255 i 33.7721 + 0.0763576 i 0.0173698 0.034739065 34.2932 − 0.512819 i 34.2957 + 0.0763576 i 0.0171794 0.0343582

L 6

L 5L 4

L 8

L 10

L 50

25 30 35 40 45 50 55

1.0

0.8

0.6

0.4

0.2

0.0

Fig. 4.1. Resonances km (plotted in the complex plane) computed by the approximate iterativealgorithm (4.4)–(4.5) with barrier height V0 = 4 and for various values of barrier width L. Thesquares show km for m = 65 and the circles for m = 80.

(4.8) | log(1 + z1)− log(1 + z2)| ≤ 4|z1 − z2| ∀z1, z2 ∈{|z| ≤ 1

2

}.

Proposition 4.2. If

8

π|m+ 1/2 + ic| < 1 and4

π|m+ 1/2 + ic|∣∣∣∣log π(m+ 1/2 + ic)

L

∣∣∣∣ < 1,

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then fap is a contraction of the disk

D =

{κ : |κ| < 2

π|m+ 1/2 + ic|∣∣∣∣log π(m+ 1/2 + ic)

L

∣∣∣∣}

into itself and therefore D contains a unique solution κ to (4.5).Consequently, there is a unique value of k solving (4.3) exactly such that∣∣∣∣k − π(m+ 1/2 + ic)

L

∣∣∣∣ < 2

L

∣∣∣∣log π(m+ 1/2 + ic)

L

∣∣∣∣ .Proof. To see that fap maps D into itself, assume κ ∈ D and use (4.8) and the

hypotheses of the theorem to obtain

|fap(κ)| ≤ 1

π|m+ 1/2 + ic|(∣∣∣∣log π(m+ 1/2 + ic)

L

∣∣∣∣+ | log(1 + κ)|)

≤ 1

π|m+ 1/2 + ic|(∣∣∣∣log π(m+ 1/2 + ic)

L

∣∣∣∣+ 4|κ|)

≤ 2

π|m+ 1/2 + ic|∣∣∣∣log π(m+ 1/2 + ic)

L

∣∣∣∣so that fap(κ) ∈ D.

To see that fap is a contraction in D, let κ1 and κ2 be in D and use (4.8) toobtain

|fap(κ1)− fap(κ2)| = 1

π|m+ 1/2 + ic| |log(1 + κ1)− log(1 + κ2)|

≤ 4

π|m+ 1/2 + ic| |κ1 − κ2| ≤ 1

2|κ1 − κ2|.

Since the solutions to (4.5), or f(κ) = κ, correspond to the solutions k of theresonance equation (4.3) through the relation (4.4), the unique solution of f(κ) = κin D yields a unique resonant value k satisfying the inequality claimed in the laststatement of the theorem.

5. Discussions. Motivated by the desire to devise a computational method forfast and accurate calculation of resonances, we studied the properties of the reso-nances of a Schrodinger’s equation with a potential well that has thick barriers. Weshowed that the resonances associated with such a potential are close to the defectmode eigenfrequencies of the limiting potential in which the barrier thickness goesto infinity. We established that the difference between the eigenfrequencies and thenearby resonances is exponentially small in the barrier thickness. Inspired by thisfinding, we developed a perturbational method for approximating resonances that aresimple to implement and accurate. To complete our study, we explored the high-frequency resonances of potentials with thick barriers and obtained a picture of theirbehavior as the barrier thickness becomes large.

The method developed here takes advantage of the one-dimensional nature ofthe problem. It relies on the theory of ODEs—the structure of solutions, the role ofthe Wronskian, existence and uniqueness of the initial-value problem, exact solutionsin the barrier, and an exact description of the outgoing condition. We expect thetechniques to be applicable to higher-dimensional problems with radial symmetry.

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It is not clear how this work can be extended to treat general higher-dimensionalproblems.

A natural extension of the present work is to the study of resonances of a one-dimensional finite (truncated) periodic structure with a defect and to compare themto localized defect modes of its infinite periodic counterpart. One simply replacesthe uniform barrier with a periodic one. By the Floquet theory, the solutions ofthe time-independent Schrodinger equation in the barrier are periodically distortedexponential functions. Thus, if one takes the end L of the barrier to tend to infinitythrough a sequence L = nd, where d is the period and n is an integer, the calculationsin this paper remain essentially unchanged. Now, instead of having bound states onlybelow the bottom of the continuous spectrum of the infinite-barrier problem, theremay be defect modes within the gaps of the spectrum created by the periodicity ofthe potential. Preliminary results for the case of a piecewise constant medium showthat the difference between the frequencies of the defect mode of the infinite periodicmedium and the resonances of its finite counterpart is exponentially small in thenumber of periodic layers.

Appendix A. HL has no bound states. If ψ satisfies −ψ′′+V (x)ψ = k2ψ andψ decays to 0 as x → ∞, then k2 = −η2 for some η > 0. Multiplying the differentialequation by ψ and integrating over the interval [a, b] yields

∫ b

a

(V (x) + η2)|ψ|2 =

∫ b

a

ψ′′ψ = −∫ b

a

|ψ′|2 + ψ′(b)ψ(b)− ψ′(a)ψ(a) .

Under the assumption that V (x) ≥ 0, taking a→ ∞ and b→ −∞ (full-line problem)or a → ∞ and b = 0 (half-line problem) causes the terms of evaluation at a and b tovanish, leaving ψ(x) ≡ 0.

Appendix B. Resonances are in the lower half plane. Let ψ satisfy equa-tions (2.1)–(2.3) with Re k > 0. Multiplying the ODE by the complex conjugate ψ(x)and integrating by parts from 0 to x yields∫ x

0

|ψ′(y)|2 dy − ψ′(x)ψ(x) +∫ x

0

V (y)|ψ(y)|2 dy − k2∫ x

0

|ψ(y)|2 dy = 0.

Using ψ(x) = Ceikx for x > L from the outgoing condition (2.3), the imaginary partof the later equation is

Re k |C|2 e−2 Im(k)x + Im(k2)

∫ x

0

|ψ|2 = 0.

Recalling that Re k > 0, we see that k = 0 =⇒ C = 0 =⇒ ψ = 0. If Im k > 0,then the left-hand term tends to zero as x → ∞, which implies that

∫∞0

|ψ|2 = 0, sothat ψ = 0. We conclude that Im k < 0.

Appendix C. Point spectrum is finite. We give a proof of the well-knownfact that the point spectrum for H∞(x) is finite. The proof is adapted from [2, Thm.3.3] and quantifies the number of bound states as the number of Neumann eigenvaluesof the well that are less than V0.

The self-adjoint operator −∂2x + V (x) on (0, a) with homogeneous boundary con-ditions ψ′(0) = 0 and ψ′(a) +

√V0 − k2 ψ(a) = 0 has form domain equal to H1(a, b)

on which its associated positive form is given by

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Ak(ψ, φ) :=

∫ a

0

(ψ′φ′ + V (x)ψφ

)dx +

√V0 − k2 ψ(a)φ(a).

A bound-state pair (kB , ψB) is characterized by the equation

(C.1) AkB (ψB , φ) = k2B

∫ a

0

ψBφ dx ∀φ ∈ H1(0, a).

The form Ak admits a sequence of eigenfunctions ψj(k;x) associated with eigenvalues{λj(k)}∞j=1 that tends to infinity, defined by

Ak(ψj(k), φ) = λj(k)

∫ a

0

ψφ dx ∀φ ∈ H1(0, a).

One shows that each function ψj(k;x) is a decreasing continuous function of k byexamining its characterization by means of the Rayleigh quotient,

λj(k) = minXj∈Sj(H1)

maxψ ∈ Xj

ψ �= 0

Ak(ψ, ψ)∫ a0 ψ

2 dx,

in which Sj(H1) denotes the set of j-dimensional subspaces of H1. Now, each bound-

state value kB satisfies, for some j,

k2B = λj(kB).

Since the λj(k) are decreasing and continuous, the number of such relations for k <√V0 is equal to the number of eigenvalues of A√

V0that are less than V0. Since the

boundary term in A√V0

vanishes, this form is the Neumann one. Thus the number of

bound states of V (x) is equal to the number of Neumann eigenvalues of −∂2x + V (x)on (0, a) that are less than V0, and this is finite.

Appendix D. Eigenvalues of H∞, zeros of FB(k), are simple. We differ-entiate the expression iFB(z), which is real for k2 < V0:

(D.1)d

dk(iFB(k)) = − d

dk

[ √V0 − k2 +

(ψ′(a, k)ψ(a, k)

) ].

Note that

∂

∂k

(ψ′(a, k)ψ(a, k)

)=ψ′(a, k)ψ(a, k)− ψ′(a, k)ψ(a, k)

ψ(a, k)2,

where ∂ψ∂k = ψ, and similarly for ∂ψ′

∂k = ψ′. Differentiating (2.1), we find that ψsatisfies

−ψ′′ + V (x)ψ = k2ψ + 2kψ in (0, a),(D.2)

ψ(0, k) = ψ′(0, k) = 0,

where ψ solves (2.1). Multiplying (D.2) by ψ and integrating by parts, we find

ψ′(a, k)ψ(a, k) =∫ a

0

ψ′ψ′ + (V (x) − k2)ψψ − 2kψ2 dx.

On the other hand, multiplying (2.1) by ψ and integrating by parts,

ψ′(a, k)ψ(a, k) =∫ a

0

ψ′ψ′ + (V (x)− k2)ψψ dx.

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Combining these expressions, we obtain

∂

∂k

(ψ′(a, k)ψ(a, k)

)=

−2k∫ a0ψ2 dx

ψ2(a, k).

Note that when k2 < V0, all parameters in (2.1) are real, and the solution ψ(x, k) ishence real. Further, because ψ(0, k) = 1 and ψ is continuous, we conclude

∫ a0ψ(x, k)2 dx >

0. Finally, we obtain

d

dk(iFB(k)) = k

(1√

V0 − k2+

2∫ a0 ψ

2(x; k) dx

ψ2(a, k)

)�= 0.

The factor of k is harmless because it disappears when taking the derivative withrespect to the eigenvalue E = k2 instead of with respect to k. (We have assumedthat v(x) > 0 so that kB > 0, although this is not necessary here.) Since ψ(x, kB) isexponentially decaying for x > a, ψ(a, kB) �= 0, and we conclude that

d

dk(iFB(kB)) > 0

at each eigenvalue kB.

Appendix E. Proof that C1 �= 0. Let ψ1 and ψ2 satisfy−ψ′′+V (x)ψ−k2ψ = 0with [

ψ1(0) ψ2(0)

ψ′1(0) ψ′

2(0)

]=

[1 0

0 1

],

so that W [ψ1, ψ2](x) = ψ1(x)ψ′2(x)− ψ2(x)ψ

′1(x) ≡ 1.

When the solution ψ1 satisfies the bound-state condition at x = a, it is equal tothe bound state ψB in [0, a]:

ψ′B(a) + σψB(a) = 0 (ψB = ψ1).

Consider the initial-value problem⎧⎨⎩−u′′ + V (x)u − k2u = ψB(x) ,

u(0) = u′(0) = 0 .

By the standard method of variation of parameters, the solution can be written as avariable-coefficient combination of ψB and ψ2 chosen such that

(E.1)

⎧⎨⎩u(x) = γ1(x)ψB(x) + γ2(x)ψ2(x) ,

u′(x) = γ1(x)ψ′B(x) + γ2(x)ψ

′2(x) ,

in which

(E.2) γ1(x) =

∫ x

0

ψB(y)ψ2(y) dy , γ2(x) = −∫ x

0

ψB(y)2 dy .

For x ≥ a, the bound state (for L = ∞) is ψB(x) = ψB(a)e−σ(x−a), and the

equation −u′′ + V (x)u − k2u = ψB(x) becomes

−u′′ + σ2u = ψB(a)e−σ(x−a) .

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The general solution of this equation is

u(x) = C1eσ(x−a) +

(C2 +

ψB(a)

2σx

)e−σ(x−a).

Let us obtain C1 in terms of u(a) and u′(a):

u(a) = C1 + C2 +ψB(a)

2σa ,

u′(a) = σC1 − σC2 +ψB(a)

2σ(1 − σa) ,

which yields

2σC1 = u′(a) + σu(a)− ψB(a)

2σ.

We will show that this is nonzero. Using (E.1) together with the bound-state conditionψ′B(a) + σψB(a) = 0, we obtain

2σC1 = γ2(a)(ψ′2(a) + σψ2(a)

)− ψB(a)

2σ.

The condition C1 = 0 together with the bound-state condition give the pair⎧⎨⎩γ2(a)

(ψ′2(a) + σψ2(a)

)− ψB(a)

2σ= 0 ,

ψ′B(a) + σψB(a) = 0 .

Multiplying the first of these equations by ψB(a) and the second by γ2(a)ψ2(a) andsubtracting yields

γ2(a)(ψB(a)ψ

′2(a)− ψ2(a)ψ

′B(a)

)− ψ2B(a)

2σ= 0 .

Using the fact that theW [ψB, ψ2](x) ≡ 1 and the expression (E.2) for γ2, this equalitybecomes

−∫ a

0

ψB(y)2 dy − ψ2

B(a)

2σ= 0 ,

which is untenable. This means that C1 must be nonzero.

Appendix F. WKB analysis. The formal WKB solution of the initial-valueproblem (2.1), (2.2) is

(F.1) ψWKB(x) = cos

(kx− 1

2k

∫ x

0

V (y) dy

)+O(|k|−2)e|Imk|x,

in which the O(|k|−2) factor is uniform in x. This can be obtained rigorously asfollows. The solution ψ satisfies the integral equation ψ = cos(k·) + k−1Tψ, wherethe operator T is defined by

(Tφ)(x) :=

∫ x

0

V (y)φ(y) sin(k(x − y)) dy.

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This operator is bounded in the weighted uniform norm ‖φ‖|Imk| =

sup0<x<L∣∣φ(x)e−|Im k|x∣∣. Thus, for large enough k, we have an explicit expansion

of the solution,

(F.2) ψ = (1 + k−1T + k−2T 2 + · · · ) cos(k ·),

which yields ψ = ψWKB.The radiation condition (2.3) applied to the WKB solution,

ik cos

(kx− 1

2k

∫ x

0

V (y) dy

)+

(k − V0

2k

)sin

(kx− 1

2k

∫ x

0

V (y) dy

)= 0,

yields the resonance equation

(F.3) exp

(2ikL− i

k

∫ L

0

V (y) dy

)= 1− 4k2

V0.

Excluding a term of order O(|k|−2) that comes from the constant 1, we obtain

(F.4) k =π(m+ 1/2 + ic)

L− i

Llog k +

1

2kL

∫ L

0

V (y)dy.

Observe that if V (x) ≡ V0, then (4.2) is recovered up to order O(|k|−1). As we havelearned from the case V (x) = 0 for 0 < x < a, in which solutions of this equationapproximate the resonances well only for large enough L, the WKB result should betreated with caution. This subtlety comes from the exponential factor e|Imk|x in theasymptotics.

Acknowledgments. The authors wish to thank the Banff International Re-search Station (BIRS) for its hospitality. This research was initiated during the au-thors’ participation in the BIRS Research in Teams event, Research in Photonics:Modeling, Analysis, and Optimization, in September 2010.

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[2] A.-S. Bonnet-BenDhia and F. Starling, Guided waves by electromagnetic gratings andnonuniqueness examples for the diffraction problem, Math. Methods Appl. Sci., 17 (1994),pp. 305–338.

[3] K. Datchev, Computing Resonances by Generalized Complex Scaling, preprint,http://math.mit.edu/ datchev/res.ps (2006).

[4] R. Froese, Asymptotic distribution of resonances in one dimension, J. Differential Equations,137 (1997), pp. 251–272.

[5] B. Helffer and J. Sjostrand, Resonances en limite semi-classique, Mem. Soc. Math. Fr.,24–25, (1986), p. 228.

[6] A. Iantchenko, Resonance spectrum for one-dimensional layered media, Appl. Anal., 85(2006), pp. 1383–1410.

[7] T. Regge, Analytic properties of the scattering matrix, Nuovo Cimento, 8 (1958), pp. 671–679.[8] B. Simon, The definition of molecular resonance curves by the method of exterior complex

scaling, Phys. Lett. A, 71 (1979), pp. 211–214.[9] B. Simon, Resonances in one dimension and Fredholm determinants, J. Funct. Anal., 178

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[11] M. Wei, G. Majda, and W. Strauss, Numerical computation of the scattering frequenciesfor acoustic wave equations, J. Comput. Phys., 75 (1988), pp. 345–358.

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Documents

Resonances of a Potential Well with a Thick Barrier