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EIGENVALUES OF -SYMMETRIC OSCILLATORS WITH PT POLYNOMIAL POTENTIALS KWANG C. SHIN Abstract. We study the eigenvalue problem u′′(z) [(iz)m +Pm−1(iz)]u(z) = λu(z) 5 − − with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays 0 0 argz =−π2 ± m2+π2, where Pm−1(z)=a1zm−1+a2zm−2+···+am−1z is a polynomial and 2 integers m 3. We provide an asymptotic expansion of the eigenvalues λn as n + , ≥ → ∞ n andprovethatforeachrealpolynomialPm−1,theeigenvaluesareallrealandpositive,with a only finitely many exceptions. J 3 2000 Mathematics subject classification: 34L40, 34L20 ] P S h. 1. Introduction t a m Forintegersm 3fixed,weareconsideringthe“non-standard”non-self-adjointeigenvalue ≥ [ problems 3 d2 v (1) Hu(z,λ) := (iz)m P (iz) u(z,λ) = λu(z,λ), for some λ C, 8 −dz2 − − m−1 ∈ 1 (cid:20) (cid:21) 0 with the boundary condition that 7 0 π 2π (2) u(z,λ) 0 exponentially, as z along the two rays arg(z) = , 4 → → ∞ −2 ± m+2 0 h/ where Pm−1 is a polynomial of degree at most m 1 of the form − t a m (3) P (z) = a zm−1 +a zm−2 + +a z, a C for 1 j m 1. m−1 1 2 m−1 j ··· ∈ ≤ ≤ − : v We let i X a := (a ,a ,...,a ) Cm−1 r 1 2 m−1 ∈ a be the coefficient vector of P (z). We are mainly interested in the case when P is m−1 m−1 real, that is, when a Rm−1. However, some interesting facts in this paper hold also for ∈ a Cm−1. So except for Theorem 4 below, we will use a Cm−1. ∈ ∈ If a nonconstant function u satisfies (1) with some λ C and the boundary condition ∈ (2), then we call λ an eigenvalue of H and u an eigenfunction of H associated with the eigenvalue λ. Also, the geometric multiplicity of an eigenvalue λ is the number of linearly independent eigenfunctions associated with the eigenvalue λ. The operator H in (1) with potential V(z) = (iz)m P (iz) is called -symmetric if V( z) = V(z), z C. Note m−1 − − PT − ∈ that V(z) = (iz)m P (iz) is a -symmetric potential if and only if a Rm−1. m−1 − − PT ∈ Date: January 02, 2005. 1 2 Before we state our main theorems, we first introduce some known facts by Sibuya [18] about the eigenvalues λ of H. Theorem 1. The eigenvalues λ of H have the following properties. n (I) The set of all eigenvalues is a discrete set in C. (II) The geometric multiplicity of every eigenvalue is one. (III) Infinitely many eigenvalues, accumulating at infinity, exist. (IV) The eigenvalues have the following asymptotic expansion 2m Γ 3 + 1 √π n 1 m+2 (4) λ = 2 m − 2 [1+o(1)] as n tends to infinity, n N, n sin π Γ 1+ 1 ∈ (cid:0) m (cid:1) (cid:0) m (cid:1)! where the err(cid:0)or(cid:1)ter(cid:0)m o(1)(cid:1)could be complex-valued. This paper is organized as follows. In Section 2, we will introduce work of Hille [12] and Sibuya [18], regarding properties of solutions of (1). We then improve on the asymptotics of a certain function in [18]. In Section 3, we introduce an entire function C(a,λ) whose zeros are the eigenvalues of H, due to Sibuya [18]. In Section 4, we then provide asymptotics of C(a,λ) as λ in the complex plane, improving the asymptotics of C(a,λ) in [18]. In → ∞ Section 5, we will improve the asymptotic expansion (4) of the eigenvalues. In particular, we will prove the following. Throughout this paper, we use that x is the largest integer ⌊ ⌋ that is less than or equal to x R. ∈ Theorem 2. Let a Cm−1 be fixed. Then there exist e (a) C, 1 ℓ m +1 such that ∈ ℓ ∈ ≤ ≤ 2 the eigenvalues λ of H have the asymptotic expansion n ⌊m+1⌋ 2 1−ℓ 1−1 (5) λ = λ + e (a)λ m +o λ2 m , n 0,n ℓ 0,n 0,n n→+∞ Xℓ=1 (cid:16) (cid:17) where 2m n+ 1 π m+2 ∞ λ = 2 with K = √1+tm √tm dt > 0. 0,n K sin 2π m − (cid:0)m (cid:1)m ! Z0 (cid:16) (cid:17) OnecancomputeK di(cid:0)rec(cid:1)tly(orseeequation(2.22)in[9]withtheidentity Γ(s)Γ(1 s) = m − πcsc(πs)) and obtains √πΓ 1+ 1 K = m . m 2cos π Γ 3 + 1 m(cid:0) 2 (cid:1)m In the last section, we prove the following theorem, regarding monotonicity of λ . (cid:0) (cid:1) (cid:0) (cid:1) | n| Theorem 3. For each a Cm−1 there exists M > 0 such that λ < λ if n M. n n+1 ∈ | | | | ≥ This is a consequence of (5). Finally, when H is -symmetric (i.e., a Rm−1), u(z,λ) is an eigenfunction associated PT ∈ with aneigenvalue λ if andonly if u( z,λ) is aneigenfunction associated with theeigenvalue − 3 λ. Thus, the eigenvalues either appear in complex conjugate pairs, or else are real. So Theorem 3 implies the following. Theorem 4. Suppose that a Rm−1. Then the eigenvalues λ of H are all real and positive, ∈ with only finitely many exceptions. For the rest of the Introduction, we will mention a brief history of problem (1). In recent years, these -symmetric operators have gathered considerable attention, be- PT cause ample numerical and asymptotic studies suggest that many of such operators have real eigenvalues only even though they are not self-adjoint. In particular, the differential operators H with some polynomial potential V and with the boundary condition (2) have been considered byBessis andZinn-Justin (not inprint), Bender andBoettcher [2]andmany other physicists [3, 4, 5, 9, 13, 14, 15, 17, 19]. Around 1992 Bessis and Zinn-Justin (not in print) conjectured that when V(z) = iz3 + βz2, β R, the eigenvalues are all real and positive, and in 1998, Bender and Boettcher ∈ [2] conjectured that when V(z) = (iz)m + βz2, β R, the eigenvalues are all real and ∈ positive. Many numerical, asymptotic and analytic studies support these conjectures (see, e. g., [3, 4, 5, 9, 13, 14, 15, 17, 19] and references therein and below). The first rigorous proof of reality and positivity of the eigenvalues of some non-self-adjoint H in (1) was given by Dorey, Dunning and Tateo [8] in 2001. They proved that the eigenvalues of H with the potential V(z) = (iz)2m α(iz)m−1 + ℓ(ℓ+1), m, α,ℓ R, are all real − − z2 ∈ if m > 1 and α < m+1+ 2ℓ+1 , and positive if m > 1 and α < m+1 2ℓ+1 . | | −| | Then in 2002 the present author [16] extended the polynomial potential results of Dorey, DunningandTateotomoregeneralpolynomialcases, byadaptingthemethodin[8]. Namely, when V(z) = (iz)m P (iz), the eigenvalues are all real and positive, provided that for m−1 − − some 1 j m the coefficients of the real polynomial P satisfy (j k)a 0 for all ≤ ≤ 2 m−1 − k ≥ 1 k m 1. ≤ ≤ − However, therearesome -symmetricpolynomialpotentialsthatproducenon-realeigen- PT values. Delabaere and Pham [6], and Delabaere and Trinh [7] studied the potential iz3+γiz and showed that a pair of non-real eigenvalues develops for large negative γ. Moreover, Handy [10], and Handy, Khan, Wang and Tymczak [11] showed that the same potential ad- mitsapairofnon-realeigenvaluesforsmallnegativevaluesofγ 3.0. Also, Bender, Berry, ≈ − Meisinger, Savage and Simsek [1] considered the problem with the potential V(z) = z4+iAz, A R, under decaying boundary conditions at both ends of the real axis, and their numeri- ∈ cal study showed that more and more non-real eigenvalues develop as A . So without | | → ∞ any restrictions on the coefficients a , Theorem 4 is the most general result one can expect k about reality of eigenvalues. 4 S 1 S 2 S 0 S-2 S-1 Figure 1. The Stokes sectors for m = 3. The dashed rays represent argz = π, 3π, π. ±5 ± 5 Also, the method used to prove Theorem 4 in this paper is new. The method used in [8, 16] is useful in proving reality of all eigenvalues, but I think that some critical arguments in proving reality of eigenvalues in [8, 16] cannot be applied to the cases when some non-real eigenvalues exist. The asymptotic expansion (5) itself is interesting, and also (5) implies Theorem 3. Note that (4) is not enough to conclude Theorem 3. Finally, Theorem 3 and -symmetry of H explained right before Theorem 4 above imply the partial reality of the PT eigenvalues in Theorem 4. 2. Properties of the solutions In this section, we introduce work of Hille [12] and Sibuya [18] about properties of the solutions of (1). First, we scale equation (1) because many facts that we need later are stated for the scaled equation. Let u be a solution of (1) and let v(z,λ) = u( iz,λ). Then v solves − (6) v′′(z,λ)+[zm +P (z)+λ]v(z,λ) = 0, m−1 − where m 3 and P is a polynomial (possibly, P 0) of the form (3). m−1 m−1 ≥ ≡ Since we scaled the argument of u, we must rotate the boundary conditions. We state them in a more general context by using the following definition. Definition. The Stokes sectors S of the equation (6) are k 2kπ π S = z C : arg(z) < for k Z. k ∈ − m+2 m+2 ∈ (cid:26) (cid:12) (cid:12) (cid:27) (cid:12) (cid:12) See Figure 1. It is known fr(cid:12)om Hille [12, 7(cid:12).4] that every nonconstant solution of (6) (cid:12) (cid:12) § either decays to zero or blows up exponentially, in each Stokes sector S . That is, one has k the following result. 5 Lemma 5 ([12, 7.4]). § (i) For each k Z, every solution v of (6) (with no boundary conditions imposed) is ∈ asymptotic to z (7) (const.)z−m4 exp [ξm +Pm−1(ξ)+λ]21 dξ ± (cid:20) Z (cid:21) as z in every closed subsector of S . k → ∞ (ii) If a nonconstant solution v of (6) decays in S , it must blow up in S S . k k−1 k+1 ∪ However, when v blows up in S , v need not be decaying in S or in S . k k−1 k+1 Lemma 5 (i) implies that if v decays along one ray in S , then it decays along all rays in k S . Also, if v blows up along one ray in S , then it blows up along all rays in S . Thus, k k k since the rotation z iz maps the two rays in (2) onto the center rays of S and S , −1 1 7→ the boundary conditions on u in (1) mean that v decays in S S . −1 1 ∪ Next we will introduce Sibuya’s results, but first we define a sequence of complex numbers b in terms of the a and λ, as follows. For λ C fixed, we expand j k ∈ (1+a z−1 +a z−2 + +a z1−m +λz−m)1/2 1 2 m−1 ··· ∞ 1 = 1+ 2 a z−1 +a z−2 + +a z1−m +λz−m k k 1 2 ··· m−1 k=1(cid:18) (cid:19) X (cid:0) (cid:1) ∞ b (a,λ) (8) = 1+ j , for large z . zj | | j=1 X Note that b , b , ..., b do not depend on λ, so we write b (a) = b (a,λ) for j = 1 2 m−1 j j 1, 2,..., m 1. Sotheaboveexpansionwithouttheλz−m termstillgivesb for1 j m 1. j − ≤ ≤ − We further define rm = −m4 if m is odd, and rm = −m4 −bm2+1(a) if m is even. The following theorem is a special case of Theorems 6.1, 7.2, 19.1 and 20.1 of Sibuya [18] that is the main ingredient of the proofs of the main results in this paper. Theorem 6. Equation (6), with a Cm−1, admits a solution f(z,a,λ) with the following ∈ properties. (i) f(z,a,λ) is an entire function of z,a and λ. (ii) f(z,a,λ) and f′(z,a,λ) = ∂ f(z,a,λ) admit the following asymptotic expansions. ∂z Let ε > 0. Then f(z,a,λ) = zrm(1+O(z−1/2))exp[ F(z,a,λ)], − f′(z,a,λ) = zrm+m2 (1+O(z−1/2))exp[ F(z,a,λ)], − − 6 as z tends to infinity in the sector argz 3π ε, uniformly on each compact set | | ≤ m+2 − of (a,λ)-values . Here 2 2 F(z,a,λ) = zm2+1 + bj(a)z12(m+2−2j). m+2 m+2 2j 1≤j<m+1 − X2 (iii) Properties (i) and (ii) uniquely determine the solution f(z,a,λ) of (6). (iv) For each fixed a Cm−1 and δ > 0, f and f′ also admit the asymptotic expansions, ∈ (9) f(0,a,λ) =[1+o(1)]λ−1/4exp[L(a,λ)], (10) f′(0,a,λ) = [1+o(1)]λ1/4exp[L(a,λ)], − as λ in the sector arg(λ) π δ, where → ∞ | | ≤ − L(a,λ) = 0+∞ tm +Pm−1(t)+λ−tm2 − jm=2+11 bj(a)tm2−j dt if m is odd,  0+∞ tRm +P(cid:16)mp−1(t)+λ−tm2 − jm2=1bj(aP)tm2−j − bm2t++11(a)(cid:17)dt if m is even. (v) The eRntire(cid:16)fupnctions λ f(0,a,λ) andPλ f′(0,a,λ) have order(cid:17)s 1 + 1.  7→ 7→ 2 m Proof. In Sibuya’s book [18], see Theorem 6.1 for a proof of (i) and (ii); Theorem 7.2 for a proof of (iii); and Theorem 19.1 for a proof of (iv). Moreover, (v) is a consequence of (iv) along with Theorem 20.1. Note that properties (i), (ii) and (iv) are summarized on pages (cid:3) 112–113 of Sibuya [18]. Using this theorem, Sibuya [18, Theorem 19.1] also showed the following corollary that will be useful later on. Corollary 7. Let a Cm−1 be fixed. Then L(a,λ) = Kmλ21+m1 (1 + o(1)) as λ tends to ∈ infinity in the sector argλ π δ, and hence | | ≤ − m+2 (11) Re (L(a,λ)) = Kmcos 2m arg(λ) |λ|21+m1 (1+o(1)) (cid:18) (cid:19) as λ in the sector arg(λ) π δ. → ∞ | | ≤ − In particular, Re (L(a,λ)) + as λ in any closed subsector of the sector → ∞ → ∞ arg(λ) < mπ . In addition, Re (L(a,λ)) as λ in any closed subsector of | | m+2 → −∞ → ∞ the sectors mπ < arg(λ) < π δ. m+2 | | − Proof. This asymptotic expansion will be clear from Lemma 8 below, or alternatively, see (cid:3) [18, Theorem 19.1] for a proof. Based on the above Corollary, Sibuya [18, Theorem 29.1] also proved the following asymptotic expansion of the eigenvalues. 2m ( 2n+1)π m+2 (12) λ = ωm − [1+o(1)], as n , n 2K sin 2π → ∞ m m ! (cid:0) (cid:1) 7 where 2πi ω = exp . m+2 (cid:20) (cid:21) Notice that in this paper we consider the boundary conditions of the scaled equation (6) where v decays in S S , while Sibuya studies equation (6) with boundary conditions such −1 1 ∪ that v decays in S S . The factor ωm in our formula (12) is due to this scaling of the 0 2 ∪ problem. Remark. Throughout this paper, we will deal with numbers like (ωνλ)s for some s R, ∈ and ν C. As usual, we will use ∈ 2πi ων = exp ν m+2 (cid:20) (cid:21) and if arg(λ) is specified, then 2π arg((ωνλ)s) = s[arg(ων)+arg(λ)] = s Re(ν) +arg(λ) , s R. m+2 ∈ (cid:20) (cid:21) If s Z then the branch of λs is chosen to be the negative real axis. 6∈ Next, we provide animproved asymptotic expansion ofL. We willuse this new asymptotic expansion of L to improve the asymptotic expansion (12) of the eigenvalues. Lemma 8. Let m 3 and a Cm−1 be fixed. Then there exist constants K (a) C, m,j ≥ ∈ ∈ 0 j m +1, such that ≤ ≤ 2 L(a,λ) = jm=2+01 Km,j(a)λ12+1m−j +O |λ|−21m if m is odd,  Pjm2=+01Km,j(a)λ21+1m−j − bm2(cid:16)+m1(a) ln((cid:17)λ)+O |λ|−m1 if m is even, (cid:16) (cid:17) as λ in the sectPor arg(λ) π δ.  → ∞ | | ≤ − Proof. The function L(a,λ) is defined as an integral over 0 t < + in Theorem 6. We ≤ ∞ will rotate the contour of integration using Cauchy’s integral formula. In doing so, we need to justify that the integrand in the definition of L(a,λ) is analytic in some domain in the complex plane. Let 0 < δ < π be a fixed number. Suppose that 0 arg(λ) π δ. Then if m+2 ≤ ≤ − 0 arg(t) 1 arg(λ), there exists M > 0 such that ≤ ≤ m 0 δ δ δ π < arg(tm +P (t)) arg(λ)+ π , m−1 − −2 ≤ ≤ 2 ≤ − 2 provided that t M . Since tm + P (t) lies in a large disk centered at the origin for 0 m−1 | | ≥ t M , we see that forallλ with λ large, we have that δ < arg(tm+P (t)+λ) < π δ | | ≤ 0 | | −2 m−1 −2 and tm + P (t) + λ > 0 for all t in the sector 0 arg(t) 1 arg(λ), and hence | m−1 | ≤ ≤ m tm +P (t)+λ is analytic in the sector 0 arg(t) 1 arg(λ) if λ lies outside a large m−1 ≤ ≤ m disk and in the sector 0 arg(λ) π δ. p ≤ ≤ − 8 Let Q(t,a,λ) = tm +Pm−1(t)+λ−tm2 − jm=2+11 bj(a)tm2−j if m is odd, ( ptm +Pm−1(t)+λ−tm2 −Pjm2=1bj(a)tm2−j − bm2t++11(a) if m is even. Then, since Q(t,a,λp) = O t −m as t tends toPinfinity in the sector 0 arg(t) 1 arg(λ), 2 | | | | ≤ ≤ m we have by Cauchy’s integr(cid:0)al form(cid:1)ula, upon substituting t = λm1 τ for all λ with λ large | | enough, +∞ +∞ 1 1 (13) L(a,λ) = Q(t,a,λ)dt = λm Q(λmτ,a,λ)dτ, Z0 Z0 where 1 Q(λmτ,a,λ) λ12 τm +1+ Pm−1λ(λm1 τ) −τm2 − jm=2+11 bj(a)τλm2mj−j if m is odd, = (cid:18)q (cid:19)  λ21 τm +1+ Pm−1λ(λm1 τ) −τm2 −Pjm2=1bj(a)τλm2mj−j − λ−λ12mb1m2τ++11(a) if m is even. (cid:18)q (cid:19)  P Similarly, (13) holds for π +δ arg(λ) 0. − ≤ ≤ Next, we examine the following square root in Q(λm1 τ,a,λ): τm +1+ Pm−1(λm1 τ) = √τm +1 1+ Pm−1(λm1 τ) s λ s λ(τm +1) = √τm +1 1+ ∞ 12 Pm−1(λm1 τ) k  k λ(τm +1)  ! k=1(cid:18) (cid:19) X  ∞  g (τ) l=et √τm +1+ j , j λ m j=1 X whereg (τ)arefunctionssuchthatg (τ) areallintegrableon[0, R]foranyR > 0. Moreover, j j by the definition of b in (8), we see that for 1 j m 1, j ≤ ≤ − j b (a)τmk−j j g (τ) = j,k for some constants b (a) such that b (a) = b (a). j (τm +1)k−21 j,k j,k j k=1 k=1 X X Thus, j τmk−j gj(τ)−bj(a)τm2−j = bj,k(a) (τm +1)k−21 −τm2−j! k=1 X j 1 = bj,k(a)τm2−jO τ→∞ τm k=1 (cid:18) (cid:19) X 1 m+1 = O for all 1 j . τ→∞ τm2+j ≤ ≤ 2 (cid:18) (cid:19) 9 So ∞ g (τ) b (a)τm−j dτ < + for all 1 j m+1. Next, when m is even and 0 j − j 2 ∞ ≤ ≤ 2 j = m +1, we write 2R (cid:12) (cid:12) (cid:12) (cid:12) ∞ bm+1(a) Z0 (cid:18)gm2+1(τ)− τ +2 λ−m1 (cid:19) dτ ∞ bm+1(a) ∞ 1 1 = Z0 (cid:18)gm2+1(τ)− τ2 +1 (cid:19) dτ +bm2+1(a)Z0 (cid:18)τ +1 − τ +λ−m1 (cid:19) dτ let bm+1(a) = Km,m2+1(a)− 2 m ln(λ), where we take Im(ln(λ)) = arg(λ) ( π,π). ∈ − Thus, we have that L(a,λ) = jm=2+01 Km,j(a)λ21+1m−j +O |λ|−21m if m is odd,  Pjm2=+01Km,j(a)λ21+1m−j − bm2(cid:16)+m1(a) ln((cid:17)λ)+O |λ|−m1 if m is even, (cid:16) (cid:17) as λ in the sectPor arg(λ) π δ, where  → ∞ | | ≤ − ∞ K (a) = K = √1+tm √tm dt > 0 for all m 3, m,0 m − ≥ ∞ Z0 (cid:16) (cid:17) m+1 (14) Km,j(a) = gj(t) bj(a)tm2−j dt for all 1 j , − ≤ ≤ 2 Z0 ∞(cid:0) bm+1((cid:1)a) Km,m2+1(a) = gm2+1(t)− 2t+1 dt when m is even. Z0 (cid:18) (cid:19) (cid:3) This completes the proof. 3. Eigenvalues are zeros of an entire function In this section, we will prove that the eigenvalues are zeros of an entire function. First, we let Gk(a) := (ω−ka ,ω−2ka ,...,ω−(m−1)ka ) for k Z. 1 2 m−1 ∈ Then recall that the function f(z,a,λ) in Theorem 6 solves (6) and decays to zero exponentially as z in S , and blows up in S S . Next, one can check that the function 0 −1 1 → ∞ ∪ f (z,a,λ) := f(ω−kz,Gk(a),ω−mkλ), k which is obtained by scaling f(z,Gk(a),ω−mkλ) in the z-variable, also solves (6). It is clear that f (z,a,λ) = f(z,a,λ). Also, f (z,a,λ) decays in S and blows up in S S since 0 k k k−1 k+1 ∪ f(z,Gk(a),ω−mkλ) decays in S . Since no nonconstant solution decays in two consecutive 0 Stokessectors(seeLemma5(ii)),f andf arelinearlyindependent andhenceanysolution k k+1 of (6) can be expressed as a linear combination of these two. Especially, there exist some coefficients C(a,λ) and C(a,λ) such that (15) f (z,a,λ) = C(a,λ)f (z,a,λ)+C(a,λ)f (z,a,λ). −1 e 0 1 e 10 We then see that W (a,λ) W (a,λ) (16) C(a,λ) = −1,1 and C(a,λ) = −1,0 , W (a,λ) − W (a,λ) 0,1 0,1 where Wj,k = fjfk′ −fj′fk is the Wronskian of fj aend fk. Since both fj and fk are solutions of the same linear equation (6), we know that the Wronskians are constant functions of z. Also, since f and f are linearly independent, W = 0 for all k Z. Moreover, we k k+1 k,k+1 6 ∈ have the following lemma that is useful later on. Lemma 9. Suppose k, j Z. Then ∈ (17) W (a,λ) = ω−1W (G(a),ω2λ), k+1,j+1 k,j and W (a,λ) = 2ωµ(a), where 0,1 m if m is odd, µ(a) = 4 (cid:26) m4 −bm2+1(a) if m is even. Moreover, W (a,λ) W (G−1(a),ω−2λ) C(a,λ) = −1,0 = ω 0,1 = ω1+2ν(a), − W (a,λ) − W (a,λ) − 0,1 0,1 where e 0 if m is odd, (18) ν(a) = bm+1(a) if m is even. (cid:26) 2 Proof. See Sibuya [18, pages 116-118] for proof. Here, we mention that by (8), we have bm+1(G−1(a)) = bm+1(a) and hence ν(G−1(a)) = ν(a). (cid:3) 2 − 2 − Now we can identify the eigenvalues of H as the zeros of the entire function λ C(a,λ). 7→ Theorem 10. For each fixed a Cm−1, the function λ C(a,λ) is entire. Moreover, λ is ∈ 7→ an eigenvalue of H if and only if C(a,λ) = 0. Proof. Since W (a,λ) = 0 and since W (a,λ) is a Wronskian of two entire functions, it 0,1 −1,1 6 is clear from (16) that C(a,λ) is an entire function of λ for each fixed a Cm−1. ∈ Next, suppose that λ is an eigenvalue of H with a corresponding eigenfunction u, then the scaled eigenfunction v(z,λ) = u( iz,λ) solves (6) and decays in S S . Hence, v is −1 1 − ∪ a (nonzero) constant multiple of f since both decays in S . Similarly, v is also a constant 1 1 multiple of f . Thus, f is a constant multiple of f , implying C(a,λ) = 0. −1 −1 1 Conversely, if C(a,λ) = 0, then f is a constant multiple of f , and hence f also decays −1 1 1 in S . Thus, f decays in S S and is a scaled eigenfunction with the eigenvalue λ. (cid:3) −1 1 −1 1 ∪ Moreover, the following is an easy consequence of (15): For each k Z we have ∈ (19) W (a,λ) = C(a,λ)W (a,λ)+C(a)W (a,λ), −1,k 0,k 1,k where we use C(a) for C(a,λ) since it is independent of λ. e e e