The Inverse Problem for Perturbed Harmonic Oscillator on the Half-Line with a Dirichlet Boundary Condition

c

 2007 Birkh¨auser Verlag Basel/Switzerland

1424-0637/061115-36,published online September 7, 2007

DOI 10.1007/s00023-007-0330-z Annales Henri Poincar´e

The Inverse Problem for Perturbed Harmonic

Oscillator on the Half-Line with a Dirichlet

Boundary Condition

Dmitry Chelkak and Evgeny Korotyaev

Dedicated to Vladimir Buslaev on the occasion of his 70th birthday

Abstract. We consider the perturbed harmonic oscillator TDψ = −ψ+ x2_{ψ + q(x)ψ, ψ(0) = 0, in L}2_(R

+), where q ∈ H+ = {q, xq ∈ L2(R+)}

is a real-valued potential. We prove that the mappingq → spectral data = {eigenvalues of TD} ⊕ {norming constants} is one-to-one and onto. The com-plete characterization of the set of spectral data which corresponds toq ∈ H+

is given.

1. Introduction and main results

Consider the Schr¨odinger operator H =− ∂

2

∂x2+|x|

2_{+ q(|x|) x ∈ R}3_{, (1.1)}

acting in the space L2(R3). Let x = |x| and q be a real-valued bounded func-tion. The operator H has pure point spectrum. Using the standard transformation

u(x) → xu(x) and expansion in spherical harmonics, we obtain that H is unitary

equivalent to a direct sum of the Schr¨odinger operators acting on L2(R₊). The first operator from this sum is given by

T_Dψ =−ψ+ x2ψ + q(x)ψ , ψ(0) = 0 , x≥ 0 . (1.2) The second is−d2

dx2+ x2+x22+ q(x) etc. Below we consider the simplest case, i.e.,

the operator T_D. In our paper we assume that

q∈ H₊=q∈ L2(R₊) : q, xq∈ L2(R₊).

The similar class of potentials was used to solve the corresponding inverse problem on the real line [5]. Define the unperturbed operator T_D0ψ =−ψ+x2ψ , ψ(0) = 0 .

The spectrum σ(T_D) of T_D is the increasing sequence of simple eigenvalues σ_n =

σ_n0+ o(1) , where σ_n0 = 4n + 3, n≥ 0 , are the eigenvalues of T_D0. Note that σ(T_D) does not determine q uniquely, see Theorem 1.3. Then what does the isospectral set

Iso_D(q) =p∈ H₊: σ_n(p) = σ_n(q) for all n≥ 0 of all potentials p with the same Dirichlet spectrum as q look like?

The inverse problem consists of two parts:

1) to characterize the set of all sequences of real numbers which arise as the Dirichlet spectra of q∈ H₊.

2) to describe the set Iso_D(q).

We shall give the complete solution of these problems in Theorem 1.3. To describe the set Iso_D(q) we define the norming constants ν_n(q) by1

ν_n(q) = logϕ·, σ_n(q), q−2₊ = 2 log|ψ_n,D (0, q)| , n ≥ 0 , (1.3) where ϕ(x) = ϕ(x, λ, q) is the solution of the equation

−ϕ_{+ x}2_{ϕ + q(x)ϕ = λϕ , ϕ(0) = 0 , ϕ}_{(0) = 1 , (λ, q)∈ C × H}

+, (1.4)

and ψ_n,D is the n-th normalized (in L2(R₊)) eigenfunction of the operator T_D.

Remark. Let Ψ_{n,0(x) = Ψn,0}(|x|) be the n-th normalized (in L2(R3)) spherically-symmetric eigenfunction of the operator H given by (1.1). Then we obtain

Ψ_{n,0(x) =} ψn,D(|x|)

2√π| x| and |Ψn,0(0)|

2₌eνn(q)

4π , n≥ 0 .

We describe papers about the inverse problem for the perturbed harmonic oscillator, which are relevant to our paper. McKean and Trubowitz [11] considered the problem of reconstruction on the real line. They gave an algorithm for the reconstruction of q from norming constants for the class of real infinitely differ-entiable potentials, vanishing rapidly at ±∞ , for fixed eigenvalues λ_n(q) = λ0_n for all n and “norming constants” → 0 rapidly as n → ∞. Later on, Levi-tan [10] reproved some results of [11] without an exact definition of the class of potentials. Some uniqueness theorems were proved by Gesztesy, Simon [7] and Chelkak, Kargaev, Korotyaev [4]. Chelkak, Kargaev, Korotyaev [5] obtained the characterization and described the isospectral set for the case on the real line for

q ∈ H = {q ∈ L2(R) : q, xq ∈ L2(R)}. For uniqueness theorems we need some asymptotics of fundamental solutions and eigenvalues at high energy. For charac-terization we need “sharp” asymptotics of these values. Usually it is not simple. Note that recently the asymptotics λ_n(q) were determined for bounded potentials in [9]. Gesztesy and Simon [6] proved that the each Iso_D(q) is connected for various classes of potentials. Note that the inverse problem for harmonic oscillator on the half-line for the boundary conditions ψ(0) = bψ(0), b∈ R is solved in [3].

1_{Here and below we use the notations · += · }

Our approach is based on the methods from [5] and [13] (devoted to the inverse Dirichlet problem on [0, 1]). The main point in the inverse problem for the perturbed harmonic oscillator T_D on R₊ is the characterization of Iso_D(q). Note that, in contrast to the case of perturbed harmonic oscillator on the real-line [5], the characterization of{ν_n}∞_n=0, i.e., the parameterization of isospectral manifolds, is given in terms of the standard weighted 2- space. Thus there is a big difference between the case of the real line and the case of the half-line.

The present paper continues the series of papers [4, 5] devoted to the inverse spectral problem for the perturbed harmonic oscillator on the real line. Note that the set of spectra which correspond to potentials from H+ (see Section 3 for

details) is similar to the space of spectral data in [5] . In particular, the range of the linear operator f (z)→(1 − z)−1/2f (z) acting in some Hardy–Sobolev space in

the unit disc plays an important role. As a byproduct of our analysis, we give the simple proof of the equivalence between two definitions of the space of spectral data (Theorem 4.2 in [5]), which was established in [5] using a more complicated techniques.

We recall some basic results from [5]. Consider the operator

T ψ =−ψ+ x2ψ + q(x)ψ ,

q∈ H_even=q∈ L2(R) : q, xq∈ L2(R); q(x) = q(−x), x ∈ R,

acting in the space L2(R). The spectrum σ(T ) is an increasing sequence of simple eigenvalues given by

λ_n(q) = λ0_n+ μ_n(q) , where λ0_n = λ_n(0) = 2n + 1 , n≥ 0 ,

and μ_n(q)→ 0 as n → ∞ . Define the real weighted 2-space

_r2= ⎧ ⎨ ⎩c ={cn}∞n=0: cn∈ R , c2r2= n≥0 (1 + n)2r|c_n|2< +∞ ⎫ ⎬ ⎭, r≥ 0 , and the Hardy–Sobolev space of analytic functions in the unit discD={z : |z|<1}:

H_r2= H_r2(D) = ⎧ ⎨ ⎩f (z)≡ n≥0 f_nzn, z∈D : f_n∈R , |f_H2 r ={fn} ∞ n=0r2< +∞ ⎫ ⎬ ⎭, r≥ 0 . Introduce the space of spectral data from [5]

H = ⎧ ⎨ ⎩h ={hn}∞n=0: n≥0 h_nzn≡√f (z) 1− z, f ∈ H 2 3 4 ⎫ ⎬ ⎭, hH=fH_3/42 . (1.5)

Theorem 1.1 ([5]). The mapping q→ {λ_n(q)− λ0_n}∞_n=0is a real-analytic isomor-phism2 _{between the space of even potentials Heven} and the following open convex subset

S ={hn}∞n=0∈ H : λ00+ h0< λ01+ h1< λ02+ h2<· · ·



⊂ H .

Remark. The inequalities in the definition ofS correspond to the monotonicity of

eigenvalues.

Recall the following sharp representation from [5]:

λ_n(q) = λ0_n+  Rq(t)dt πλ0_n +μn(q) ,  μn(q) ∞ 0 ∈ H0, q∈ H ,

where the subspaceH₀⊂ H of codimension 1 is given by

H0= ⎧ ⎨ ⎩h∈ H : √ 1− z n≥0 h_nzn    z=1 = f (1) = 0 ⎫ ⎬ ⎭. (1.6)

Remark that Lemma 3.4 yields μ_n(q) = O(n−3/4log1/2n) as n → ∞ . Also, we

need

Proposition 1.2 (Trace formula). For each q∈ H_even the following identity holds: q(0) = 2

n≥0



λ_2n(q)− λ_2n+1(q) + 2, (1.7)

where the sum converges absolutely.

We come to the inverse problem for the operator T_DonR₊. For each q∈ H₊ we set q(−x) = q(x) , x ≥ 0 . This gives a natural isomorphism between H₊ and Heven. Then

σ_n(q) = λ_2n+1(q) , n≥ 0 .

Let

SD=



{hn}∞n=0∈ H : σ00+ h0< σ10+ h1< σ20+ h2<· · ·. (1.8)

We formulate our main result.

Theorem 1.3.

(i) The sequence{σ_n(q)− σ0_n}∞_n=0belongs toS_D for each potential q∈ H₊.

(ii) For each q∈ H₊ the sequence {r_n(q)}∞_n=0∈ _3/42 , where r_n is given by ν_n(q) = ν_n0− q(0) 2(2n + 1) + rn(q) , and ν 0 n= νn(0) = log  4(2n + 1)! √ π 22n[n!]2  .

2_{By definition, the mapping of Hilbert spaces F : H1→ H2is a local real-analytic isomorphism iff}

for any y∈ H1it has an analytic continuation F into some complex neighborhood y∈ U ⊂ H1C

of y such that F is a bijection between U and some complex neighborhood F (y)∈ F (U )⊂ H_2C of F (y) and both F , F−1are analytic. The local isomorphism F is a (global) isomorphism iff it is a bijection.

(iii) The mapping q→ ({σ_n(q)− σ_n0}_n=0∞ , q(0) ,{r_n(q)}∞_n=0) is a real-analytic

iso-morphism between H+ andSD× R × _3/42 .

Remark. In particular, (p(0) ,{r_n(p)}∞_n=0) ∈ R × _3/42 are “independent coordi-nates” in Iso_D(q).

The ingredients of the proof of Theorem 1.3 are:

i) Uniqueness Theorem. We adopt the proof from [13] and [4]. This proof re-quires only some estimates of the fundamental solutions.

ii) Analysis of the Fr´echet derivative of the nonlinear spectral mapping

{potentials} → {spectral data} at the point q = 0 . We emphasize that this

linear operator is complicated (in particular, it is not the Fourier transform, as it was in [13]). Here we essentially use the technique of generating functions (from [5]), which are analytic in the unit disc.

iii) Asymptotic analysis of the difference between spectral data and its Fr´echet derivatives at q = 0 . Here the calculations and asymptotics from [5] play an important role.

iv) The proof that the spectral mapping is a surjection, i.e., the fact that each element of an appropriate Hilbert space can be obtained as spectral data of some potential q ∈ H₊. Here we use the standard Darboux transform of second-order differential equations.

The plan of the paper. Section 2 is devoted to the basic asymptotics of the

eigenval-ues σ_n(q) and the values log[(−1)n_ψ

+(0, σn(q), q)] . In Section 3 we introduce the

spaceH and obtain its equivalent definition (Corollary 3.6). Furthermore, we con-sider a kind of linear approximation of our spectral data and prove Theorem 3.10 that is, in a sense, the linear analogue of the main Theorem 1.3. Section 4 is de-voted to the asymptotics of the norming constants ν_n(q). Also, in this sect. we prove Proposition 1.2. In Section 5 we prove the main Theorem 1.3. All needed properties of fundamental solutions, gradients of spectral data and some technical lemmas are collected in Appendix.

2. Basic asymptotics

Let ψ0₊(x, λ) = Dλ−1

2 (

√

2x) be the decreasing near +∞ solution of the unperturbed equation

−ψ_{+ x}2_{ψ = λψ .}

We use the standard notation D_μ(x) for the Weber functions (or the parabolic cylinder functions), see [1]. Note that for each q∈ H₊ the perturbed equation

−ψ_{+ x}2_{ψ + q(x)ψ = λψ}

has the unique solution ψ₊(x, λ, q) such that ψ₊(x) = ψ0₊(x)(1 + o(1)) as x→ +∞ (see (A.11)).

Lemma 2.1. For each q∈ H₊ and n≥ 0 the following identities hold: ν_n(q) = 2 log  −ψ+  0, σ_n(q), q ˙ ψ₊0, σ_n(q), q  , (2.1) ˙ ψ₊0, σ_n(q), q= ψ 0 +  0, σ_n(q) σ_n(q)− σ_n0 ·  m:m=n σ_n(q)− σ_m(q) σ_n(q)− σ0_m , ∂ ∂λψ+= ˙ψ+. (2.2)

Remark. It is important that the values ˙ψ₊(0, σ_n(q), q) are uniquely determined by the spectrum σ(T_D) . In particular, ˙ψ₊(0, σ_n(p), p) = ˙ψ₊(0, σ_n(q), q) for all

p∈ Iso_D(q) and n≥ 0 .

Proof. The standard identity3 ψ₊2 ={ ˙ψ₊, ψ₊} yields  _+∞ 0 ψ₊2(x)dx =ψ˙₊, ψ₊+∞ 0 =− ˙ψ+(0)ψ  +(0) ,

where we omit σ_n(q) and q for short. Therefore,

eνn(q) ₌_ψ n,D(0, q) ₂ =  ψ₊0, σ_n(q), q ψ+·, σn(q), q  + ₂ =−ψ  +  0, σ_n(q), q ˙ ψ₊0, σ_n(q), q . Using the Hadamard Factorization Theorem, we obtain

ψ₊(0, σ, q) = ψ0₊(0, σ)· 

m≥0

σ− σ_m(q)

σ− σ0_m , σ∈ C .

The differentiation of ψ₊(0, σ, q) gives (2.2).  Let ψ_n0 be the normalized (in L2(R)) eigenfunctions of the unperturbed har-monic oscillator on R. Note that ψ0_n,D(·) = ψ_n,D(·, 0) = √2ψ0_2n+1(·) . It is well-known that ψ0_n(x) =n!√π−12_D n √ 2x=2nn!√π−12_H n(x)e− x2 2 , n≥ 0 ,

where H_n(x) are the Hermite polynomials. For each n≥ 0 we consider the second solution χ0_n(x) =  n!√π 2 1/2₍₋₁₎n 2 Im D_−n−1(i√2x) , n is even , (−1)n−12 Re D_−n−1(i√2x) , n is odd ,

of the equation−ψ+ x2ψ = λ0_nψ which is uniquely defined by the conditions {χ0

n, ψ0n} = 1 , (ψ0nχ0n)(−x) = −(ψ0nχ0n)(x) , x∈ R .

Note that (ψ_n0χ0_n)(x) = (−1)n+1x + O(x2) as x→ 0 , and (ψ_n0χ0_n)(x) =−x−1+

O(x−2) as x→ ∞, see [5]. Following [5], for q ∈ H₊, we introduce ∧ q_n=q, (ψ_n0)2 +, ∨ q_n=q, ψ0_nχ0_n + n≥ 0 . (2.3)

3_{Here and below we use the notations{f, g} = fg}_{− f}_{g , u}₌ ∂

Also, we introduce the constants

κ_n = ψ₊0(0, λ0_n) , κ_n = (ψ0₊)(0, λ0_n) , ˙κ_n= ˙ψ0₊(0, λ0_n) and so on .

Theorem 2.2. For each q∈ H₊ the following asymptotics4 hold: σ_n(q) = σ_n0+ 2q∧_2n+1+ 32 4+δ(n) , (2.4) logψ  +(0, σn(q), q) κ_2n+1 = ˙κ_2n+1 κ_2n+1  σ_n(q)− σ0_n−q∨_2n+1+ 32 4+δ(n) , (2.5)

uniformly on bounded subsets of H+, for some absolute constant δ > 0. Remark.

i) Proposition 3.2 immediately yields{q∧_n}∞

n=0∈ H . Using basic properties of

the spacesH, H₀(see Proposition 3.3 and Lemma 3.4), we obtain ∧ q_2n+1 = π−1_R +q(t)dt· (σ 0 n)− 1 2 + O  n−34log12 _{n .}

ii) In the proof we use some technical results from [5], formulated in Appendix A.1–A.4.

Proof. Let μ = σ_n(q)− σ0_n and m = 2n + 1. Recall that σ_n0= λ0_mand ψ₊(0, λ0_m+

μ, q) = 0. Lemma A.4 (i) yields μ = O(m−1/2). Due to Corollary A.3 and asymp-totics (A.13), we have

0 = ψ+(0, λ 0 m+ μ, q) ˙κ_m = ψ₊(1)(0, λ0_m, q) + ˙κ_m· μ ˙κ_m + O  m−1log m.

Hence, Lemma A.7 (ii) gives μ = 2q∧_m+ O(m−1log2m) . Using the similar

argu-ments, we deduce that 0 = 1 ˙κ_m  ψ(1)₊ + ˙ψ0₊·μ+ψ₊(2)+ ˙ψ₊(1)·2∧q_m+ ¨ ψ₊0 2 (2 ∧ q_m)2  (0, λ0_m, q)+O  m−32log3m .

Together with Lemmas A.7 (ii), A.6, this yields

μ 2 = ∧ q_m−q∨_mq∧_m+  ˙κ_m κ_m ∧ q_m+1 2 ∨ q_m  · 2q∧_m−κ¨m κ_m( ∧ q_m)2+ 32 4+δ(m) =q∧_m+  2 ˙κ_m κ_m − ¨ κ_m κ_m  · (q∧_m)2+ 32 4+δ(m) = ∧ q_m+ 23 4+δ(m) .

4_{Here and below an = bn+ }2

r(n) means that {an− bn}∞n=0 ∈ r2. We say that an(q) = bn(q) + _r2(n) holds true uniformly on some set iff norms{an(q)− bn(q)}∞_n=0_r2 are uniformly bounded on this set.

Furthermore, using Corollary A.3 and Lemma A.7 (ii), we obtain ψ₊0, σ_n(q), q κ_m =  ψ0₊+ ψ₊(1)+ ψ₊(2) (0, λ0_m+ μ, q) κ_m + O(m −3 2) =  ψ0₊+ ψ₊(1)+ ˙ψ₊0 · μ + ψ(2)₊ + ˙ψ₊(1)· 2q∧_m+1₂ψ¨₊0 · (2∧q_m)2 (0, λ0_m, q) κ_m + Om−32_log3_m = 1−q∨_m+ ˙κ  m κ_m · μ +  1 2( ∨ q_m)2−π 2 8 ( ∧ q_m)2  +  −˙κm κ_m ∨ q_m+π 2 8 ∧ q_m  · 2q∧_m +2¨κ  m κ_m ( ∧ q_m)2+ 23 4+δ(m) . Hence, logψ  +  0, σ_n(q), q κ_2n+1 =− ∨ q_m+ ˙κ  m κ_m · μ + 2  ¨ κ_m κ_m− ( ˙κ_m)2 (κ_m)2 + π2 16  · (q∧_m)2+ 23 4+δ(m) =−q∨_m+ ˙κ  m κ_m · μ + 2 3 4+δ(m) ,

where we have used Lemma A.6. 

3. Coefficients

q

,

q

and

q

Let



ψ0_n(x) = 21/4ψ0_n(√2x) , n≥ 0 .

Note that the mapping

q→q, ψ_n0∞

n=0, H →

2

1/2. (3.1)

is a linear isomorphism5. Moreover, since{ ψ0_2m}∞_m=0is the orthogonal basis of the space Heven, it is the orthogonal basis of H+. On the contrary, { ψ2m+10 }∞m=0 is

the orthogonal basis of the subspace

H0₊={q ∈ H₊: q(0) = 0}  H₊.

5_{We say that the linear operator is a linear isomorphism iff it is bounded and its inverse is}

Following [5], for each potential q∈ H₊ we define two (analytic in the unit discD) functions (F q)(z)≡ 1 (2π)1/4 k≥0  E_kq, ψ_2k0 ₊· zk, z∈ D , (Gq)(z)≡ −(2π) 1/4 2 k≥0 q, ψ0_2k+1+  (2k + 1)E_kz k_{, z∈ D ,} (3.2) where E_k = (2k)! 22k(k!)2 ∼ π −1 2_k−12 _{as k}→ ∞ . _(3.3) Lemma 3.1.

(i) The mapping q→ F q is a linear isomorphism between H₊ and H_3/42 .

(ii) The mapping q→ Gq is a linear isomorphism between H0₊ and H_3/42 .

(iii) (Gq)(·) ∈ C(T \ {−1}) and (Gq)(·) ∈ L1(T) for each q ∈ H₊. Proof. (i), (ii) Using (3.1) and (3.3), we deduce that the mappings

q→q, ψ_2m0 ₊∞ m=0→ F q , H+→ 2 1/2→ H3/42 q→q, ψ_2m+10 ₊∞ m=0→ Gq , H 0 +→ 1/22 → H3/42

are linear isomorphisms.

(iii) Let q∈ H₊. Since ψ0₀(0) = 21/4π−1/4 (see (3.8)), we obtain

q(x) = 2−14_π14_q(0)· _ψ0

0(x) + q0(x) , x≥ 0 ,

for some q₀∈ H0₊. Due to (ii), we have Gq₀∈ H_3/42 ⊂ C(T) ⊂ L1(T). Furthermore, (3.9) yields (G ψ0₀)(z) =−(2π) 1/4 2√2π m≥0 (−1)mzm 2m + 1 .

Hence, G ψ0₀∈ C(T \ {−1}), G ψ₀0∈ L1(T) and the same holds for Gq . 

Lemma 3.2 ([5]). Let q∈ H₊. Then the following identities6 hold:

n≥0 ∧ q_nzn≡(F q)(z)√ 1− z , n≥0 ∨ q_nzn≡ P₊  (Gq)(ζ)  1− ζ  , z∈ D , (3.4) (F q)(1) = (2π)−12  R+ q(t)dt , (F q)(−1) = 2−32q(0) , (3.5)

where the coefficients q∧_n and q∨_n, n≥ 0 , are defined by (2.3).

Remark.

i) Here and below we put (P₊f )(z)≡ _2πi1 _|ζ|=1 f (ζ)dζ_ζ−z for any f ∈ L1(T) and

z ∈ D . In particular, the identity (P₊!k_n=−kc_nζn)(z)≡!k_n=0c_nzn holds true for any c_n∈ C .

ii) Definition (1.5) of the space H is directly motivated by asymptotics (2.4) and (3.4)

Proof. Identities (3.4) were proved in [5] (Propositions 1.2 and 2.9). Also, in [5] it

was shown that 1 = k≥0  ψ_2k0 (x)  R  ψ0_2k(t)dt = (2π)14 k≥0  E_kψ0_2k(x) , in the sense of distributions, which gives (F q)(1) = (2π)−1/2_R

+q(t)dt . Further-more, δ(x) = k≥0  ψ0_2k(x)· ψ0_2k(0) = k≥0  ψ_2k0 (x)· 2 1/4_H 2k(0) √ π 22k(2k)!1/2 =  2 π 1 4 k≥0 (−1)kE_kψ_2k0 (x)

in the sense of distributions. Together with (3.2), this implies (F q)(−1) =

2−1/2· q(0)/2 . 

We need some results from [5] (see Lemmas 2.10, 2.11 [5]).

Proposition 3.3.

(i) For each {h_n}∞_n=0 ∈ H there exist unique v ∈ R and {h(0)_n }∞_n=0 ∈ H₀ such that h_n= v· (λ0_n)−1/2+ h(0)n . The mapping

h→v, h(0)

is a linear isomorphism betweenH and R × H₀. If h ={q∧_n}∞

n=0, q∈ H+, then v = 212π−12√1− z +∞ n=0 h_nzn z=1= π −1 R+ q(t)dt .

(ii) The set of finite sequences {(h₀, . . . , h_k, 0 , 0 , . . . ), k≥ 0 , h_j ∈ R} is dense inH₀.

(iii) The embeddings _3/42 ⊂ H₀⊂ _1/42 are fulfilled.

(iv) If{h_n}∞_n=0∈ H, then {h_n− h_n+1}∞_n=0∈ 2_3/4.

Remark. SinceH₀⊂ _1/42 , the sequence of leading terms{(λ0_n)−1/2v}∞_n=0doesn’t belong toH₀.

The next lemma gives the O-type estimate for sequences fromH₀.

Proof. The proof is similar to the proof of Lemma 2.1 in [2]. Definition (1.6) ofH₀ yields n≥0 h_nzn≡ ! k≥0fkzk √ 1− z , {fk} ∞ k=0∈ 23 4, k≥0 f_k= f (1) = 0 .

Recall that (1− z)−1/2≡!_m≥0E_mzm_{. Hence,}

h_n = n k=0 E_n−kf_k = n k=1 (E_n−k− E_n)f_k− E_n ∞ k=n+1 f_k.

It is easy to see that E_n = O(n−1/2) and E_n−k− E_n = O(kn−1(n− k + 1)−1/2) . Therefore,   En ∞ k=n+1 f_k   ≤ En " _+∞ k=n+1 k−32 #1 2" +∞ k=n+1 k32|f_k|2 #1 2 = O(n−34) ,    n k=1 (E_n−k− E_n)f_k   ≤ " _n k=1 O  k12 n2(n− k + 1)  #1/2"_n k=1 k32|f_k|2 #1/2 = O(n−34log12_{n) ,}

where the estimate!+∞_k=1k3/2|f_k|2< +∞ has been used. 

Recall that F q and Gq are defined by (3.2) and the system of functions

{ ψ_2k0 }∞_{k=0is a basis of H+}. Therefore, it is possible to rewrite Gq in terms of F q . Note that this situation differs from the case q∈ H (the perturbed harmonic oscil-lator on the whole real line [5]), where the functions F q and Gq are “independent coordinates” in the space of potentials.

For ζ = eiφ _{∈ T, φ ∈ (−π, π) , ζ = −1 , we define} √_{ζ = e}iφ2. We have the

identity 1 √ ζ = 2 π s∈Z (−1)s 2s + 1ζ s _{in L}2_{(T) . (3.6)}

Lemma 3.5. For each q∈ H₊ the following identity holds:

(Gq)(z)≡ −π 2P+  (F q)(ζ) √ ζ  , z∈ D . (3.7)

Proof. We determine the coefficients of the function ψ_2m+10 with respect to the basis{ψ_2k0 }∞_k=0. The standard identity {ψ_2k0 , ψ_2m+10 } = (λ0_2k− λ0_2m+1)ψ0_2kψ0_2m+1

yields (ψ_2m+10 , ψ0_2k)₊=  R+ ψ_2m+10 (x)ψ_2k0 (x)dx = {ψ 0 2k, ψ2m+10 }(0) λ0_2m+1− λ0_2k = ψ 0 2k(0)(ψ2m+10 )(0) 22(m− k) + 1 .

Note that ψ_2k0 (0) =_√ H2k(0) π 22k(2k)!1/2 = (−1) k₂k_{(2k− 1)!!} √_{π 22k(2k)!}1/2 = (−1)k π1/4  E_k, (3.8) (ψ_2m+10 )(0) = H  2m+1(0) √ π 22m+1(2m + 1)!1/2 = (−1) m₂m+1_{(2m + 1)!!} √ π 22m+1(2m + 1)!1/2 =(−1) m√₂ π1/4  (2m + 1)E_m. Therefore, ( ψ0_2m+1, ψ_2k0 )₊= (ψ0_2m+1, ψ0_2k)₊= √1 2π · (−1)m−k 2(m− k) + 1  (2m + 1)E_mE_k. (3.9) Since ψ_2k0 2₊= 1/2 , we obtain  ψ_2m+10  (2m + 1)E_m = $ 2 π +∞ k=0 (−1)m−k 2(m− k) + 1·  E_kψ0_2k. This gives P₊  (F q)(ζ) √ ζ  ≡ 23/4 π5/4P+  _+∞ l=−∞ (−1)l 2l + 1ζ l_· +∞ k=0  E_k(q, ψ0_2k)₊ζk  ≡ 21/4 π3/4 +∞ m=0 (q, ψ0_2m+1)₊  (2m + 1)E_mz m_{≡ −}2 π(Gq)(z) , z∈ D ,

where definition (3.2) of the functions F q and Gq has been used.  We introduce the formal linear operatorA by

(Af)(z) ≡ P₊  f (ζ) √ −ζ  ≡ P+ % 1− ζ ·√f (ζ) 1− ζ  , f ∈ L1(T) . (3.10) Let ◦ H2_3/4=f ∈ H2_3/4: f (1) = 0⊂ H2_3/4.

Using Lemma 3.5 we shall obtain the simple proof of Theorem 4.2 from [5] about the equivalent definition ofH₀.

Corollary 3.6.

(i) The operatorA :H◦2_3/4→ H2_3/4 and its inverse are bounded.

(ii) The following identity holds:

H0= ⎧ ⎨ ⎩{hn}∞n=0: n≥0 h_nzn≡ P₊  g(ζ)  1− ζ  , g∈ H_3/42 ⎫ ⎬ ⎭. (3.11)

The normsh_Handg_H2

3/4 are equivalent, i.e., C1gH3/42 ≤ hH≤ C2gH3/42

Remark. This equivalence was proved in [5] using different and complicated

argu-ments.

Proof. (i) Recall that the mapping q→ (Gq)(−z) is a linear isomorphism between

H0+ and H3/42 . Also, due to the identity (F q)(−1) = 2−3/2q(0) (see Lemma 3.2),

the mapping q→ (F q)(−z) is a linear isomorphism between H0₊andH◦2_3/4. There-fore, the mapping

f (z)≡ (F q)(−z) → q → (Gq)(−z) ≡ −π 2 P+  (F q)(−ζ) √ ζ  ≡ Af(z)

is a linear isomorphism betweenH◦2_3/4, H0+ and H23/4 respectively.

(ii) If g∈ H_3/42 , then g₀(z)≡ g(z)−g(1) ∈H◦2_3/4and so|g₀(ζ)| ≤ C|ζ −1|1/4,

|ζ| = 1 , for some constant C > 0 . Hence, the following equivalence is valid: c n≥0 h_nzn≡ P₊  g(ζ)  1− ζ  ⇔ n≥0 h_nzn≡ g(1) +√g0(z) 1− z − P−  g₀(ζ)  1− ζ  ⇔ P+ % 1− ζ n≥0 h_nζn  ≡ g(1) + g0(z)≡ g(z) , where P₋f ≡ f − P₊f is the projector to the subspace of antianalytic functions

inD . Therefore, the equation

f (z) √ 1− z ≡ n≥0 h_nzn≡ P₊  g(ζ)  1− ζ  , where f ∈H◦2_3/4, g∈ H2_3/4,

is equivalent to g(z)≡ (Af)(z) . Then, (3.11) follows from (i). 

Lemma 3.7. For each q∈ H₊ the following identity holds:

n≥0 ∨ q_2n+1zn≡ π 2P+  1 √ −ζ n≥0 ∧ q_2nζn  , z∈ D . (3.12)

Proof. Due to identities (3.2) and Lemma 3.5, we have

n≥0 ∨ q_nzn ≡ P₊  (Gq)(ζ)  1− ζ  ≡ −π 2P+  (F q)(ζ) √ ζ1− ζ  ≡ −π 2P+  √ 1− ζ √ ζ1− ζ n≥0 ∧ q_nζn  ≡ −π 2P+ √ −ζ √ ζ n≥0 ∧ q_nζn  .

Therefore, n≥0 ∨ q_2n+1z2n+1≡ −π 4P+ √_−ζ √ ζ n≥0 ∧ q_nζn− √ ζ √ −ζ n≥0 ∧ q_n(−ζ)n  ≡ π 4P+  √ ζ √ −ζ n≥0 ∧ q_n(ζn+ (−ζ)n)  ≡ π 2P+  √ ζ √ −ζ n≥0 ∧ q_2nζ2n  . This yields n≥0 ∨ q_2n+1z2n ≡π 2P+  1 ζ · √ ζ √ −ζ n≥0 ∧ q_2nζ2n  ≡π 2P+  1  −ζ2 n≥0 ∧ q_2nζ2n  , since√−ζ ·√ζ =−ζ2 for ζ∈ T , ζ = ±1 .  We consider linear terms{q∧_2n+1}∞

n=0and{ ∨ q_2n+1}∞ n=0 in asymptotics (2.4), (2.5). Proposition 3.8.

(i) For each q∈ H₊ the following identity is fulfilled:

n≥0 ∧ q_2n+1zn ≡(F√Dq)(z) 1− z , z∈ D , (3.13) where (F_Dq)(z2)≡ 1 2z  (F q)(z)√1 + z− (F q)(−z)√1− z.

(ii) For each q∈ H₊ the following identity is fulfilled:

n≥0 ∨ q_2n+1zn≡π 2P+  1 √ −ζ  (G_Dq)(ζ) + n≥0 ∧ q_2n+1ζn  , z∈ D , (3.14) where (G_Dq)(z2)≡ 1 2z  (F q)(−z)√1 + z− (F q)(z)√1− z.

(iii) The mapping

q→ (F_Dq ; G_Dq)

is a linear isomorphism between H+ and H_3/42 × H_3/42 .

Proof. (i) Due to (3.2), we have

n≥0 ∧ q_2n+1z2n≡ 1 2z  (F q)(z) √ 1− z − (F q)(−z) √ 1 + z  ≡ (F_√Dq)(z2) 1− z2 . This gives (3.13).

(ii) Recall that Lemma 3.7 yields n≥0 ∨ q_2n+1zn≡π 2 P+  1 √ −ζ n≥0 ∧ q_2nζn  . Using (3.2), we obtain n≥0 (q∧_2n−q∧_2n+1)z2n≡1 2  (F q)(z) √ 1− z + (F q)(−z) √ 1 + z  −(F_√Dq)(z2) 1− z2 ≡ (GDq)(z 2_{) .} This gives (3.14).

(iii) Recall that the mapping q → F q is a linear isomorphism between H₊ and H_3/42 . Therefore, we need to prove that F q → (F_Dq ; G_Dq) is a linear

iso-morphism between H_3/42 and H_3/42 × H_3/42 . Due to definitions of F_Dand G_D, the direct mapping is bounded. Since

(F q)(z)≡ (F_Dq)(z2)√1 + z + (G_Dq)(z2)√1− z ,

the inverse mapping is bounded too. 

Definition 3.9. For q∈ H₊ define coefficients ∼q_n, n≥ 0 , by n≥0 ∼ q_nzn≡π 2P+  1 √ −ζ  (G_Dq)(ζ)− (G_Dq)(1)  , z∈ D .

Remark. Due to Proposition 3.8, we have G_Dq∈ H2_3/4. Hence, (G_Dq)(·)−(G_Dq)(1) ∈H◦2_3/4 and Corollary 3.6 gives{q∼_n}∞

n=0∈ 3/42 . Theorem 3.10.

(i) For each q∈ H₊ the following identities hold:

∨ q_2n+1 = q(0) 4(2n + 1) + ∼ q_n+ m≥0 ∧ q_2m+1 2(n− m) + 1, n≥ 0 . (3.15) (ii) The mapping

q→{q∧_2n+1}∞ n=0; q(0) ;{ ∼ q_n}∞ n=0 

is a linear isomorphism between H+ andH × R × _3/42 . Proof. (i) Due to identity (3.14) and Definition 3.9, we have

n≥0 ∨ q_2n+1zn≡ n≥0 ∼ q_nzn+π 2P+  1 √ −ζ  (G_Dq)(1) + n≥0 ∧ q_2n+1zn  . (3.16)

Note that (G_Dq)(1) = 2−1/2(F q)(−1) . Then, identity (3.5) yields (G_Dq)(1) =

1

4q(0) . Substituting the identity 1/

√ −ζ = 2 π ! s∈Z ζs 2s+1 in L2(T) (see (3.6)) into (3.16), we obtain (3.15).

(ii) Due to Proposition 3.8 and identity (3.13), the mappings

q → (F_Dq ; G_Dq) → {q∧_2n+1}∞_n=0; G_Dq,

H+ → H_3/42 × H_3/42 → H × H_3/42 ,

are linear isomorphisms. Using Corollary 3.6, we deduce that the mapping

G_Dq→(G_Dq)(1) ;{∼q_n}∞

n=0



, H_3/42 → R × _3/42 ,

is a linear isomorphism too. The identity (G_Dq)(1) = 1₄q(0) completes the proof.



4. Asymptotics of

ν

(

q) and proof of Proposition 1.2

Lemma 4.1. For each q∈ H₊ the following identity holds:

n≥0



σ_n(q)− σ0_n− 2q∧_2n+1= 0 . (4.1)

where the series converges absolutely.

Proof. By asymptotics (2.4), the series !_n≥0(σ_n(q)− σ_n0 − 2∧q_2n+1) converges absolutely. Due to Lemma A.47, ∂σn(q)

∂q(x) = ψ2n,D(x, q) , where ψn,D is the n-th

normalized (in L2(R₊)) eigenfunction of the operator T_D. Therefore,

σ_n(q)− σ_n0 =  ₁ 0 d dsσn(sq)ds =  ₁ 0  ψ2_n,D(x, sq), q(x)₊ds .

Recall that ψ2_n,D(x, 0) = 2(ψ0_2n+1)2(x) . Then,

σ_n(q)− σ0_n− 2q∧_2n+1 =  ₁ 0  ψ2_n,D(x, sq)− ψ2_n,D(x, 0), q(x) L2(R+, dx)ds .

The standard perturbation theory (e.g., see [8]) yields

∂ψ_n,D(x, q) ∂q(y) = m:m=n ψ_n,D(y, q)ψ_m,D(y, q) σ_n(q)− σ_m(q) ψm,D(x, q) . Hence,  ψ2_n,D(x, sq)− (ψ_n,D0 )2(x), q(x) L2(R+, dx) =  s 0 & d dtψ 2 n,D(x, tq), q(x) ' L2(R+, dx) dt

7_{Here and below ∂ξ(q)}(_{∂q = ζ(q) means that for any v∈ L}2 _{the equation (dqξ)(v) =v, ζ}

L2

=  s 0 ) 2ψ_n,D(x, tq) ×& m:m=n ψ_n,D(y, tq)ψ_m,D(y, tq) σ_n(q)− σ_m(q) ψm,D(x, tq), q(y) ' L2(dy) q(x) * L2(dx) dt . This gives n≥0  σ_n(q)− σ0_n− 2q∧_2n+1= 2  ₁ 0 ds  s 0 n≥0 m:m=n  (ψ_n,Dψ_m,D)(tq), q2₊ σ_n(tq)− σ_m(tq) dt . Let S_k= k n=0 m:m=n  (ψ_n,Dψ_m,D)(tq), q2 + σ_n(tq)− σ_m(tq) = k n=0 +∞ m=k+1  (ψ_n,Dψ_m,D)(tq), q2 + σ_n(tq)− σ_m(tq) . Due to Lemma A.8, for some absolute constant ε > 0 we have

 (ψ_n,Dψ_m,D)(tq), q2 +=  On−12m−12 for all n, m≥ 0 , On−12−ε2m−12, if m≥ n + n12+ε.

Using the simple estimate |σ_n(tq)− σ_m(tq)|−1 = O(|n − m|−1) and technical Lemma A.9, we obtain S_k → 0 as k → ∞ , i.e.,!_n≥0(σ_n(q)−σ_n0−2q∧_2n+1) = 0 . 

Proof of Proposition 1.2. Repeating the proof of Lemma 4.1, we obtain

n≥0



λ_n(q)− λ0_n− 2q∧_n= 0 ,

for q∈ H_even. Recall that λ_2n+1(q) = σ_n(q) and λ0_2n+1 = σ_n0. Using (4.1), we get n≥0 (−1)nλ_n(q)− λ0_n− 2q∧_n= n≥0  λ_n(q)− λ0_n− 2∧q_n − 2 n≥0  σ_n(q)− σ_n0− 2q∧_2n+1= 0 . Due to Lemma 3.2, Propsition 3.3 (iv) this yields

n≥0 (−1)nλ_n(q)− λ0_n= 2 n≥0 (−1)nq∧_n= 2·(F q)(z)√ 1− z   z=−1 =q(0) 2 . Hence, q(0) = 2!_n≥0(−1)n_(λ n(q)− λ0n), which gives (1.2). 

Recall that each sequence{σ_n(q)− σ_n0}, q ∈ H₊ belongs to the set S_D⊂ H given by (1.8).

Theorem 4.2.

(i) Each function r_n(q) = ν_n(q)− ν_n0+_2(2n+1)q(0) , q∈ H₊ satisfies r_n(q) =−2q∼_n+ R_n(μ) + 32 4+δ(n) , μ ={μm} ∞ m=0=  σ_m(q)− σ₀∞ m=0 (4.2)

uniformly on bounded subsets of H+,where δ > 0 is some absolute constant and R_n(μ) =−2 log  ψ₊0(0, σ0_n+ μ_n) ˙κ_2n+1· μ_n  m:m=n (σ_n0+ μ_n)− (σ_m0 + μ_m) (σ_n0+ μ_n)− σ_m0  +2 ˙κ  2n+1 κ_2n+1 · μn− m≥0 μ_m 2(n− m) + 1, n≥ 0 . (4.3)

(ii) For each {μ_m}∞_m=0 ∈ S_D the sequence {R_n}∞_n=0 belongs to the space _3/42 . Moreover, the mapping R : S_D → _3/42 given by {μ_m}∞_m=0 → {R_n}∞_n=0, is locally bounded.

Remark. Note thatq∼_n= _3/42 (n) due to Theorem 3.10 (ii). Therefore,{r_n(q)}∞_n=0∈

_3/42 .

Proof. (i) Let σ_n= σ_n(q) and μ_n= σ_n(q)− σ_n0, n≥ 0 . Lemma 2.1 yields

ν_n(q)− ν_n0 2 = log  ψ₊ (0, σ_n, q) ˙ ψ₊(0, σ_n, q)· ˙κ_2n+1 κ_2n+1  .

Using Theorem 2.2 (ii) and Theorem 3.10 (i), we obtain logψ  +(0, σn, q) κ_2n+1 = ˙κ_2n+1 κ_2n+1 μn− q(0) 4(2n + 1)− ∼ q_n− m≥0 ∧ q_2m+1 2(n− m) + 1+ 2 3 4+δ(n) .

Furthermore, identity (2.2) gives log ˙ ψ₊(0, σ_n, q) ˙κ_2n+1 = log  ψ₊0(0, σ_n) ˙κ_2n+1· μ_n  m:m=n σ_n− σ_m σ_n− σ0_m  Hence, r_n(q) = ν_n(q)− ν_n0+ q(0) 2(2n + 1) =−2 ∼ q_n+ R_n(μ) + 23 4+δ(n) + hn, where h_n= m≥0 μ_m− 2q∧_2m+1 2(n− m) + 1 , n≥ 0 .

In order to prove that h_n= _3/4+δ2 (n), we note that identity (3.6) yields

h(z)≡ n≥0 h_nzn≡ π 2P+  g(ζ) √ −ζ  , where g(z)≡ m≥0 (μ_m− 2∧q_2m+1)zm

Due to asymptotics (2.4) and identity (4.1), we have g ∈ H_3/4+δ2 and g(1) = 0 . Hence8, g(ζ) √ −ζ ∈ W234+δ(T) , and so P+  g(ζ) √ −ζ  ∈ H2 3 4+δ. Thus,!_n≥0h_nzn_{∈ H}2 3/4+δ, i.e.,{hn}∞n=0∈ 3/4+δ2 .

(ii) Let{μ_m}∞_m=0∈ S_D. We rewrite (4.3) in the form R_n= R(1)_n +R(2)_n +R_n(3), where R(1)_n =−2 log  ψ₊0(0, σ0_n+ μ_n) ˙κ_2n+1· μ_n  +2 ˙κ  2n+1 κ_2n+1 · μn, R(2)_n =− m:m=n " 2 log  1− μm 4(n− m) + μ_n  + μm 2(n− m) # , R(3)_n =1 2 m:m=n μ_m n− m− m≥0 μ_m 2(n− m) + 1.

In the following Lemmas 4.3–4.5 we will analyze these terms separately. Recall that Proposition 3.3 and Lemma 3.4 give μ_n= O(n−1/2) as n→ ∞ and

μ_n= v· (n + 1)−12 ₊ 2

1/4(n) , (4.4)

where v∈ R is some constant.

Lemma 4.3. The asymptotics R(1)_n = π₄₈2v2(n + 1)−1+ _3/42 (n) hold true.

Proof. Due to ψ₊0(0, σ_n0) = 0, μ_n = O(n−1/2) and the estimates from Corol-lary A.3, we have

ψ0₊(0, λ0_n+ μ_n) ˙κ_2n+1· μ_n = 1 + ¨ κ_2n+1 2 ˙κ_2n+1 μn+ ... κ_2n+1 6 ˙κ_2n+1μ 2 n+ O(n− 3 2 log4n) . Therefore, R(1)_n =  2 ˙κ_2n+1 κ_2n+1 − ¨ κ_2n+1 ˙κ_2n+1  μ_n− 2 ... κ_2n+1 6 ˙κ_2n+1 − (¨κ_2n+1)2 8( ˙κ_2n+1)2  μ2_n+ O(n−32log4n) = π 2 48μ 2 n+ O(n− 3 2_log4_{n) =}π 2_v2 48 (n + 1) −1₊ 2 3/4(n) ,

where we have used Lemma A.6 and (4.4). 

Lemma 4.4. The asymptotics R(2)n =−π 2_v2

48 (n + 1)−1+ 3/42 (n) hold true.

8_{Here and below W}2

Proof. For m= n we have 2 log  1− μm 4(n− m) + μ_n  + μm 2(n− m) = μ_mμ_n 2(n− m)4(n− m) + μ_n −_ μ2m 4(n− m) + μ_n2 + O  m−3/2 (n− m)3  = μmμn 8(n− m)2 − μ2_m 16(n− m)2 + O  m−3/2+ m−1/2n−1 (n− m)3  . Therefore, 16R(2)_n =−2μ_n· m:m=n μ_m (n− m)2 + m:m=n μ2_m (n− m)2 + O(n −3 2) .

Recall that μ_n = v(n + 1)−1/2+ _1/42 (n) and μ_n2 = v2(n + 1)−1+ _3/42 (n) . Using simple technical Lemma A.10, we deduce that

16R_n(2)=−2  v (n + 1)1/2 + 2 1 4(n)  π2v 3(n + 1)1/2 + 2 1 4(n)  +  π2v2 3(n + 1)+ 2 3 4(n)  + O(n−32_{) .} This gives 48R(2)_n =−π2v2(n + 1)−1+ _3/42 (n) . 

Lemma 4.5. The asymptotics R(3)_n = _3/42 (n) hold true.

Proof. Note that the following identities are fulfilled in L2(T): 2 +∞ l=−∞ ζl 2l + 1 = π √ −ζ , l:l=0 ζl l =− log(−ζ) ,

where the branches of√−ζ and log(−ζ), ζ ∈ T \ {1} are such that √1 = 1 and log 1 = 0 . Then, n≥0 R(3)_n zn≡ −1 2P+  π √ −ζ + log(−ζ)  · n≥0 μ_nζn  . Since{μ_n}∞_n=0∈ H , we have n≥0 μ_nzn≡ √F (z) 1− z, where F ∈ H 2 3/4.

Introduce the function

γ(ζ) = π

(√

−ζ + log(−ζ) √

It is clear that γ ∈ C∞(T \ {1}) . Note that γ(ζ) → 0 as ζ → 1±i0. This yields

γ∈ W_3/42 (T), γF ∈ W_3/42 (T) and P₊[γF ]∈ H_3/42 . The last statement is equivalent

to{R(3)_n }∞_n=0∈ _3/42 . 

Lemmas 4.3–4.5 give R_n = R_n(1)+ R_n(2)+ R(3)_n = _3/42 (n) . Note that all estimates are uniform on bounded subsets of S_D. The proof of Theorem 4.2 is

finished. 

5. Proof of Theorem 1.3

Introduce the mapping

Φ : q→μ_n(q)∞

n=0; q(0) ;



r_n(q)∞

n=0 ,

where μ_n(q) = σ_n(q)− σ_n0. Due to Theorems 1.1, 4.2, we have Φ : H+→ SD× R × 23

4.

Theorem 1.3 claims that Φ is a real-analytic isomorphism. The proof given below consists of five steps: Φ is injective (Section 5.1); Φ is real-analytic (Section 5.2); the Fr´echet derivative d_qΦ is a Fredholm operator for each q∈ H₊ (Section 5.3);

d_qΦ is invertible for each q ∈ H₊, i.e., Φ is a local real-analytic isomorphism (Section 5.4); Φ is surjective (Section 5.5).

5.1. Uniqueness theorem

Let ({μ_n(p)}∞_n=0; p(0) ;{r_n(p)}∞_n=0) = ({μ_n(q)}∞_n=0; q(0) ;{r_n(q)}∞_n=0) for some

p, q∈ H₊. By definitions of μ_n and r_n, it is equivalent to

σ_n= σ_n(p) = σ_n(q) and ν_n = ν_n(p) = ν_n(q) for all n≥ 0 . Using Lemma 2.1, we obtain

ψ₊(0, σ_n, p) = ψ₊(0, σ_n, q) for all n≥ 0 .

The rest of the proof is standard (see also [4]). Recall that ϕ(x, λ, q) is the solution of (1.4) such that ϕ(0, λ, q) = 0 , ϕ(0, λ, q) = 1 . Introduce the functions

f₁(λ; x, q, p) = F1(λ; x, q, p) ψ₊(0, λ, q) , F₁(λ; x, q, p) = ψ₊(x, λ, p)ϕ(x, λ, q)− ϕ(x, λ, p)ψ₊ (x, λ, q) , f₂(λ; x, q, p) = F2(λ; x, q, p) ψ₊(0, λ, q) , F₂(λ; x, q, p) = ψ₊(x, λ, p)ϕ(x, λ, q)− ϕ(x, λ, p)ψ₊(x, λ, q) .

Both f₁and f₂are entire with respect to λ for each x∈ R₊. Indeed, all roots

σ_n, n≥ 0 , of the denominator ψ₊(0,·, q) are simple and all these values are roots of the numerators F₁, F₂, since

ψ₊(x, σ_n, p) ϕ(x, σ_n, p) = ψ  +(0, σn, p) = ψ+ (0, σn, q) = ψ₊(x, σ_n, q) ϕ(x, σ_n, q)

for all x ∈ R₊ and n ≥ 0 . Standard estimates (see Lemma A.2 and asymp-totics (A.6)) of ϕ and ψ₊ give

f₁(λ; x, p, q) = 1+O  |λ|−1 2 , f₂(λ; x, p, q) = O  |λ|−1 2 , |λ|=λ0_2k, k→ ∞ .

Then, the maximum principle implies

f₁(λ; x, p, q) = 1 , f₂(λ; x, p, q) = 0 , λ∈ C .

This yields ϕ(x, λ, p) = ϕ(x, λ, q) and ψ₊(x, λ, p) = ψ₊(x, λ, q), i.e., p = q . 

5.2. Φ is a real-analytic mapping

Recall that for some δ > 0 the following asymptotics are fulfilled (see (2.4) and (4.2)): μ_n(q) = 2q∧_2n+1+ 32 4+δ(n) , rn(q) =−2 ∼ q_n+R_nμ_m(q)∞ m=0 + 2 3 4+δ(n) , (5.1) where R : SD→ 32 4 , R : {μm} ∞ m=0→ {Rn}∞n=0,

is a locally bounded mapping given by (4.3). Let H+C be the complexification of

H+. Due to Lemma A.4 (ii), for each q∈H₊all functions σ_n(q) and ψ₊(0, σ_n(q), q),

n≥ 0 , have analytic continuations into some complex neighborhood of q .

More-over, due to Lemma 2.1, all functions ν_n(q) have analytic continuations into some complex neighborhood of q. Therefore, for each real potential q∈ H₊ all “coordi-nate functions” μ_n(q) , q(0) , r_n(q) of the mapping Φ have an analytic continuation into some small complex neighborhood Q of q.

Repeating the proof of (5.1) we obtain that these asymptotics hold true uniformly on bounded subsets of Q . Let

Φ(0): q→{2q∧_2n+1}∞ n=0; q(0) ;{−2 ∼ q_n}∞ n=0  .

Due to Theorem 3.10 (i), Φ(0)_{is a linear isomorphism between H+}andH×R× _3/42 . In particular, Φ(0) is a real-analytic mapping. Consider the difference

Φ− Φ(0): q→μ_n(q)− 2q∧_2n+1∞ n=0; 0 ;  r_n(q) + 2q∼_n∞ n=0 , Φ− Φ(0)_{: H+}→ 23 4+δ× R × 2 3 4.

All “coordinate functions” μ_n(q)− 2q∧_2n+1, r_n(q) + 2∼q_nare analytic and Φ− Φ(0) is correctly defined and bounded in some small complex neighborhood of each real potential (since (5.1) holds true uniformly on bounded subsets). Then, Φ− Φ(0) is

a real-analytic mapping from H+ into _3/4+δ2 ×R× _3/42 and Φ is real-analytic too,

since _3/4+δ2 ⊂ _3/42 ⊂ H . 

5.3. The Fr´_{echet derivative dqΦ is a Fredholm operator for each q ∈ H+}

In other words, we will prove that d_qΦ is the sum of invertible and compact operators. Let Φ(1): q→  0 ; 0 ;Rμ_m(q)∞ m=0  and Φ(2)= Φ− Φ(0)− Φ(1).

Using the same arguments as above, we obtain that R_D : S_D → _3/42 is a real-analytic mapping (since it is locally bounded in some small complex neighborhood of each real point μ∈ S_D and all “coordinate function” R_n are analytic). Then, Φ(1)is real-analytic as a composition of real-analytic mappings. Theorems 2.2, 4.2 yield Φ(2)_{: H+}→ 23 4+δ× R × 2 3 4+δ.

Repeating above arguments again, we obtain that Φ(2) is a real-analytic mapping too.

Fix some q∈ H₊. The Fr´echet derivatives d_qΦ , d_qΦ(j) of the analytic map-pings Φ, Φ(j)at the point q are bounded linear operators and

d_qΦ = d_qΦ(0)+ d_qΦ(1)+ d_qΦ(2)=Φ(0)+ d_qΦ(1)+ d_qΦ(2).

Note that the operator

d_qΦ(2)_{: H+}→ H × R × 23 4

is compact since it maps H+into _3/4+δ2 × R × _3/4+δ2 and the embedding _3/4+δ2 ⊂

_3/42 is compact. In order to prove that Φ(0)+ d_qΦ(1) is invertible, we introduce two linear operators

A : H+→ H , p → Ap = {2p∧2n+1}∞n=0, B : H+→ 32 4, p→ Bp = {−2 ∼ p_n}∞ n=0.

Recall that Φ(0)p = (Ap ; p(0) ; Bp) . The chain rule implies

d_qΦ(1) p =0 ; 0 ; (d_μ(q)R)Ap,

where d_μ(q)R is the Fr´echet derivative of the mapping R at the point μ(q) =

{μm(q)}∞m=0 ∈ SD. Hence, Φ(0)+ dqΦ(1) = CΦ(0), where both operators C and

C−1 given by

C±1 :H × R × _3/42 → H × R × _3/42 , C : (h; t; r)→h; t; r±(d_μ(q)R_D)h

are bounded. Recall that (Φ(0)_D )−1is bounded due to Theorem 3.10 (ii). Therefore, the operator (Φ(0)+ d_qΦ(1))−1 is bounded too. 

5.4. Φ is a local real-analytic isomorphism

By Fredholm’s Theory, in order to prove that (d_qΦ)−1 is bounded, it is sufficient to check that the range Ran d_qΦ is dense:

H × R × 2

3/4= Ran dqΦ . (5.2)

Note that Lemma A.4 (ii) gives

∂σ_n(q) ∂q(t) = ψ 2 n,D(t, q) , ∂ log + (−1)n_ψ +  0, σ_n(q), q, ∂q(t) =−(ψn,Dχn,D)(t, q) , (5.3)

where ψ_n,D(·, q) is the n-th normalized eigenfunction of T_D and χ_n,D(·, q) is some special solution of (1.4) for λ = σ_n(q) such that{χ_n,D, ψ_n,D} = 1 . In particular,

(ψ_n,Dχ_n,D)(t, q)∼ t , t → 0 , (ψ_n,Dχ_n,D)(t, q)∼ −t−1, t→ +∞ .

Due to Lemma A.5 (i), for each q∈ H₊the following standard identities are fulfilled:  (ψ_n,D2 )(q), ψ_m,D2 (q)₊= 0 ,  (ψ_n,D2 )(q), ψ_m,Dχ_m,D)(q) += 12δmn,  (ψ_n,Dχ_n,D)(q), (ψ_m,Dχ_m,D)(q) += 0 , n, m≥ 0 . (5.4)

Note that (ψ2_m,D)(·, q) ∈ H₊. Using (5.3), (5.4), we obtain  ∂μ_n(q) ∂q , (ψ 2 m,D)(q)  + = 0 and _ ∂ log(−1)n_ψ +(0, σn(q), q)  ∂q , (ψ 2 m,D)(q)  + =δnm 2 for all n, m≥ 0 . Due to Lemma 2.1, this implies

 ∂ ˙ψ₊(0, σ_n(q), q) ∂q , (ψ 2 m,D)(q)  + = 0 and  ∂ν_n(q) ∂q , (ψ 2 m,D)(q)  + = δ_nm. The identity _ ∂q(0) ∂q , (ψ 2 m,D)(q)  + = (ψ_m,D2 )(0) = 0 gives _ ∂r_n(q) ∂q , (ψ 2 m,D)(q)  + =  ∂ν_n(q) ∂q , (ψ 2 m,D)(q)  + = δ_nm. Thus, (d_qΦ)(ψ2_m,D)(q)_{= (0 ; 0 ; em}) ,

where 0 = (0, 0, 0, . . . ) , e0= (1, 0, 0, . . . ) , e1= (0, 1, 0, . . . ) and so on. Therefore,  (0; 0)× 3/42 =  (0; 0; c) : c ∈ 3/42  ⊂ Ran dqΦ . (5.5)

We come to the second component of (d_qΦ)ξ , i.e., to the value ξ(0) . We consider the lowest eigenvalue λ₀(q) of the operator T_N (with the same potential q and the Neumann boundary condition ψ(0) = 0) and the function

ξ(t) = (ϕϑ)t, λ₀(q), q, t∈ R ,

where ϑ(t) is the solution of−ψ+ x2ψ + q(x)ψ = λψ such that ϑ(0) = 1 and ϑ(0) = 1. Note that ξ(0) = 1. Asymptotics (A.11), (A.12) give ξ ∈ H₊ since

ϑ(·, λ₀(q), q) is proportional to ψ₊(·, λ₀(q), q). Moreover, using ϕ(0, λ₀(q), q) =

ψ_n,D(0, q) = 0, we obtain  ∂μ_n(q) ∂q , ξ  + =ψ2_n,D(q) , ξ += −{ψn,D, ϕ}{ψn,D, ϑ}  0, λ₀(q), q 2σ_n(q)− λ₀(q) = 0 . Hence, (d_qΦ)ξ =_{0 ; 1 ; (dq}r)ξ, where (d_qr)ξ =  ∂r_n(q) ∂q , ξ  + -∞ n=0 ∈ 2 3/4.

Together with (5.5) this implies

{0} × R × 2

3/4⊂ Ran dqΦ .

Furthermore, we consider the functions−2(ψ_m,Dχ_m,D)(q)∈ H₊(see asymp-totics (A.11), (A.12)). Identities (5.3), (5.4) and (ψ_m,Dχ_m,D)(0, q) = 1 give

(d_qΦ)− 2(ψ_m,Dχ_m,D)(q)=  em;−2 ; (dqr)  − 2(ψm,Dχm,D)(q)  .

Due to Proposition 3.3 (ii), the set of finite sequences is dense inH₀. Therefore,

H0× R × 3/42 ⊂ Ran dqΦ . (5.6)

In conclusion, we consider an arbitrary function ζ∈ H₊ such that_R

+ζ(t)dt= 0.

Proposition 3.3 (i) implies

(d_qΦ)ζ /∈ H₀× R × _3/42 .

Together with (5.6) this yields (5.2), since the codimension of H₀ in H is equal

to 1. 

5.5. Φ is surjective

Lemma 5.1. Let q∈ H₊, n≥ 0 and t ∈ R . Denote q_nt(x) = q(x)− 2 d 2 dx2log η t n(x, q) , η_nt(x, q) = 1 + (et− 1)  _+∞ x ψ_n,D2 (s, q)ds . Then qt n ∈ H+ and σ_m(q_nt) = σ_m(q) , ν_m(q_nt) = ν_m(q) + tδ_mn

for all m≥ 0. Moreover, qt