Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 36, pp. 1–22. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ENERGY-DEPENDENT HAMILTONIAN IN A NUCLEAR OPTICAL MODEL PIERRE CHAU HUU-TAI, BERNARD DUCOMET Abstract. We study the spectral properties of a 1D model of optical potential introduced by Morillon and Romain [21] in the context of nuclear reactions. We can localize the discrete spectrum and estimate the number of eigenvalues. We also study the continuous spectrum with possibly embedded spectral sin- gularities and give an expansion formula for an arbitrary function on a set of generalized eigenfunctions. We briefly study the resonances of a related model. 1. Introduction The optical model in [11, 12, 31] is an approximation scheme introduced in nuclear reactions [5] theory for describing the interaction between a nucleus and an incident nucleon. It aims at replacing the complicated initial N -body problem by a single Schrödinger equation, provided that a suitable potential is exhibited. It relies on the assumption that the main characteristics of the interaction can be described with good accuracy by a single particle potential called optical potential. The price to pay is that this phenomenological potential is generally non-local, complex and energy-dependent. Romain and Morillon [21] proposed a global spherical and dispersive optical potential for neutrons able to reproduce scattering data for a large mass range of nuclei. This phenomenogical potential is the sum of various contributions: a term of volume, a term of surface and the so-called spin-orbit term U(r, E) = UV (r, E) + US(r, E) + USO(r, E), where the potentials are given by the formulae UV (r, E) = −[VV (E) + iWV (E)]f(r,RV , aV ), US(r, E) = −[VS(E) + iWS(E)]g(r,RS , aS), and USO(r, E) = −[VSO(E) + iWSO(E)] ( h2 mπc )2 1 r g(r,RSO, aSO)~̀ · ~s, (1.1) where ~̀ is the orbital momentum, ~s the spin, j the angular momentum and po- tentials VV,S,SO, WV,S,SO (volume, surface and spin-orbit) are real functions and h,Mπ, c are positive constants of quantum origin. 2010 Mathematics Subject Classification. 35J10, 35P15, 47A55. Key words and phrases. Nonselfadjoint; complex potential; energy dependent potential; S-wave. c©2021 Texas State University. Submitted December 17, 2020. Published May 4, 2021. 1 2 P. CHAU HUU-TAI, B. DUCOMET EJDE-2021/36 The function f is the so-called Woods-Saxon form factor and g is the “derivative of Woods-Saxon” form factor. Both are given by f(r,R, a) = 1 1 + e r−R a , g(r,R, a) = −4a d dr f(r,R, a). The potential U(r, E) is complex valued and explicitely depends on energy E through its real and imaginary parts VV,S,SO et WV,S,SO related according [21] by subtracted (renormalized) dispersion relations. So the corresponding Schrödinger equation in the spherically symmetric case reads − d2 dr2 v + `(`+ 1) r2 v − { [VV (E) + iWV (E)]f(r,RV , aV ) + [VS(E) + iWS(E)]g(r,RS , aS) + λj,`,s[VSO(E) + iWSO(E)] ( h2 mπc )2 1 r g(r,RSO, aSO) } v = Ev, (1.2) for r > 0, where λj,`,s = j(j + 1) − `(` + 1) − s(s + 1), with j = ` ± 1/2 if ` 6= 0 and j = 1/2 if ` = 0 and s = 1/2. So we have j(j + 1) − `(` + 1) − s(s + 1) = j(j + 1)− `(`+ 1)− 3/4 which is zero for S-waves corresponding to j = 1/2, ` = 0. In the following we consider a simplified version of (1.2) restricted to S-waves (` = 0) leading to the 1D problem − d2 dr2 v + U(r, E)v = Ev E ∈ C, r > 0, u(0, E) = 0, (1.3) where the potential U has the form U(r, E) = N∑ j=1 Vj(r)Wj(E), (1.4) and N is a given integer. To comply with the model (1.1) we assume that Vj ∈ L2(0,∞;R) ∩ L1(0,∞;R), (1.5) and to take advantage of Jost theory [23] we assume the following analytic properties of Wj(z) for z ∈ C. (1) The functions z → Wj(z) for j = 1, . . . N are meromorphic in C but holo- morphic in the physical half-plane {=m(z) > 0}, with single poles defined by zj,m = EF − iBj,m for m = 1, . . . ,M with Bj,m > 0. (2) |Wj(z)| ≤W j when |z| → ∞, and we denote by ΓW the common analyticity domain of the family {Wj(z)}j≤N ΓW = C\ ( ∪Nj=1 ∪Mm=1zj,m ) . (1.6) Note that because the two previous potentials Wj are complex, the hamiltonian H := d2 dr2 + U is not self-adjoint and the standard spectral theory [8] has to be adapted. The 1D selfadjoint Schrödinger equation is a classical topic, and is the matter of a number of books (among them [4, 8, 23]). The 1D non-selfadjoint case goes EJDE-2021/36 ENERGY-DEPENDENT HAMILTONIAN 3 back to the pioneer work of Naimark [22] (and a large part of our study relies on this work). Among his followers, let us quote Lyantse [19, 20], Pavlov [24, 25, 26], Krall [17] and more recently Folland [13], Hruščev [16] and Brown [6]. Concerning energy-dependent potentials, let us mention studies by Friedman and Mishoe [14] and Bairamov and coauthors [1, 2, 3, 18], for specific types of energy dependence. This article is organized as follows: In Section 2 we consider special solutions of (1.3). In Section 3 we study the discrete spectrum of H. In Section 4 we analyze the continuous spectrum of H together with its scattering properties. Finally in Section 5 we study resonances of the model. 2. Some special solutions of (1.3) In analogy with classical studies of 1D selfadjoint operators [8, 23], we introduce several particular solutions of (1.3). 2.1. The solution φ(r, ρ). Taking benefit of the monodimensional geometry, for E = ρ2 we classically [4] introduce the problems Hφ = ρ2φ ρ ∈ C, r > 0, φ(0, ρ) = 0, φ′(0, ρ) = 1, (2.1) and Hψ = ρ2ψ ρ ∈ C, r > 0, ψ(0, ρ) = 1, ψ′(0, ρ) = 0, (2.2) where Hv = −v′′+Uv. It is well known that both these both of these two problem have a unique solution φ(r, E) and ψ(r, E), holomorphic in the positive complex half-plane {=m(E) > 0} for any r ≥ 0. Lemma 2.1. For any r ≥ 0 and ρ 6= 0 with =m(ρ) ≥ 0 one has |ρφ(r, ρ2)| ≤ exp ( r=m(ρ) + N∑ j=1 W j ∫ r 0 |Vj(r′)| dr′ ) . (2.3) Proof. Putting E = ρ2 for E > 0, φ satisfies the equation φ(r, ρ2) = sin ρr ρ + ∫ r 0 sin ρ(r − r′) ρ U(r′, ρ2)φ(r′, ρ2) dr′. (2.4) Defining z(r, ρ2) := ρeiρrφ(r, ρ2), z solves z(r, ρ) = eiρr sin ρr + ∫ r 0 sin(ρ(r − r′))eiρ(r−r ′)U(r′, ρ2)z(r′, ρ) dr′. (2.5) Supposing that =mρ ≥ 0, the first term in the right-hand side is less that 1; then |z(r, ρ)| ≤ 1 + ∫ r 0 |U(r′, ρ2)z(r′, ρ)| dr′. Multiplying (2.5) by |U(r, ρ2)| ( 1 + ∫ r 0 |U(r′, ρ2)z(r′, ρ)| dr′ )−1 we find the bound: log ( 1 + ∫ r 0 |U(r, ρ2)z(r′, ρ)| dr′ ) ≤ ∫ r 0 |U(r′, ρ2)| dr′ ≤ N∑ j=1 W j ∫ r 0 |Vj(r′)| dr′, 4 P. CHAU HUU-TAI, B. DUCOMET EJDE-2021/36 which gives |z(r, ρ)| ≤ exp ( N∑ j=1 W j ∫ r 0 |Vj(r′)| dr′ ) . Then estimate (2.3) follows. � 2.2. The solution e(r, ρ). Denoting by σj the well-defined functions σj(r) = ∫ ∞ r |Vj(r′)| dr′ for j = 1, . . . , N, (2.6) we have the following result. Proposition 2.2. (1) The equation − y′′ + U(r, ρ2)y = ρ2y, (2.7) admits a solution y = e(r, ρ) satisfying the integral equation e(r, ρ) = eiρr − ∫ ∞ r sin(ρ(r − r′)) ρ U(r′, ρ2)e(r′, ρ) dr′, (2.8) for ρ 6= 0 and =m(ρ) ≥ 0. (2) For any δ > 0 and for r →∞ e(r, ρ) = eiρr(1 + o(1)), ∂re(r, ρ) = eiρr(iρ+ o(1)), (2.9) uniformly with respect to ρ in the domain {=m(ρ) ≥ 0, |ρ| > δ}. Moreover for =m(ρ) ≥ 0 and |ρ| → ∞, e(r, ρ) = eiρr(1 + o(ρ−1)), ∂re(r, ρ) = ieiρr(1 + o(ρ−1)), (2.10) uniformly with respect to r in the domain {r ≥ 0}. (3) For any r ≥ 0 the function ρ→ e(r, ρ) is holomorphic in the open half-plane {=m(ρ) > 0}. Proof. (1) It is clear that e(r, ρ) given by (2.8) satisfies (2.7). Now noting ε(r, ρ) = e−iρre(r, ρ), we see that ε satisfies ε(r, ρ) = 1 + 1 2iρ ∫ ∞ r [e2iρ(r′−r) − 1]U(r′, ρ2)ε(r′, ρ) dr′. (2.11) We look for a solution of (2.11) in the form of the expansion ε(r, ρ) = ∞∑ n=0 εn(r, ρ), (2.12) for a sequence {εn}n≥0 given iteratively by εn+1(r, ρ) = 1 2iρ ∫ ∞ r [e2iρ(r′−r) − 1]U(r′, ρ2)εn(r′, ρ) dr′ for n ≥ 0, and ε0(r, ρ) = 1. Observing that |e2iρ(r′−r) − 1| ≤ 2 when r′ ≥ r and =m(ρ) ≥ 0, we deduce that |εn+1(r, ρ)| ≤ 1 |ρ| ∫ ∞ r |U(r′, ρ2)| |εn(r′, ρ)| dr′. EJDE-2021/36 ENERGY-DEPENDENT HAMILTONIAN 5 Supposing by induction that |εn(r′, ρ)| ≤ 1 n!|ρ|n qn(r), where q(r) := 1 |ρ| N∑ j=1 W jσj(r), we obtain |εn+1(r, ρ)| ≤ 1 |ρ|n+1 ∫ ∞ r dq(r′) dr′ qn(r′) dr′ ≤ 1 (n+ 1)!|ρ|n+1 qn+1(r). Then for ρ 6= 0, series (2.12) is uniformly convergent and its sum ε(r, ρ) is uniformly bounded on any domain {=m(ρ) ≥ 0, |ρ| > δ} by eq(r)/|ρ|. (2) Going back to e with e(r, ρ) = eiρr ( 1 + ∞∑ n=1 εn(r, ρ) ) , and using estimate on εn in (1), asymptotic statements (2.9) and (2.10) follow. (3) The result holds after applying analytic Fredholm theory to the integral equation (2.11) depending analytically on the parameter ρ in the open set {=mρ > 0} (see [28, 30]). � Proposition 2.3. If the integrals σ′j(r) = ∫ ∞ r r′|Vj(r′)| dr′ for j = 1, . . . , N, (2.13) are convergent, then the solution e of (2.7) is well defined for ρ = 0 and e(r, ρ) = eiρr + ∫ ∞ r K(r, r′)eiρr ′ dr′. (2.14) where the kernel K is continuous with respect to r and r′ and satisfies the bounds |K(r, r′)| ≤ 1 2 exp ( N∑ j=1 Wjσ ′ j(r) ) N∑ j=1 Wjσj (r + r′ 2 ) , |∂rK(r, r′)|, |∂r′K(r, r′)| ≤ 1 4 N∑ j=1 Wj ∣∣∣Vj(r + r′ 2 )∣∣∣+ exp ( N∑ j=1 Wjσj(r) ) , × N∑ j=1 Wjσj (r + r′ 2 ) N∑ j=1 Wjσj(r)), (2.15) Proof. Comparing (2.8) and (2.14) one has∫ ∞ r K(r, r′)eiρr ′ dr′ = − ∫ ∞ r sin(ρ(r − r′)) ρ U(r′, ρ2)e(r′, ρ) dr′. In the right-hand side, replacing e(r′, ρ) by its expression in (2.14) we obtain − ∫ ∞ r K(r, r′)eiρr ′ dr′ = ∫ ∞ r sin(ρ(r − r′)) ρ U(r′, ρ2) [ eiρr ′ + ∫ ∞ r′ K(r′, r′′)eiρr ′′ dr′′ ] dr′ 6 P. CHAU HUU-TAI, B. DUCOMET EJDE-2021/36 = ∫ ∞ r sin(ρ(r − r′)) ρ U(r′, ρ2)eiρr ′ dr′ + ∫ ∞ r sin(ρ(r − r′)) ρ U(r′, ρ2) ∫ ∞ r′ K(r′, r′′)eiρr ′′ dr′′ dr′ =: J1 + J2. Using the identity sin(ρ(r−r′)) ρ eiρr ′ = 1 2 ∫ 2r′−r r eiρr ′′ dr′′ and applying Fubini’s theo- rem (σ′j(r) <∞), we obtain J1 = ∫ ∞ r eiρr ′′ dr′′ 1 2 ∫ r+r′′ 2 Vj(r ′)dr′. In the same stroke, using identity sin(ρ(r−r′)) ρ eiρr ′′ = 1 2 ∫ r′−r+r′′ r−r′+r′′ e iρτdτ we obtain J2 = ∫ ∞ r eiρr ′′ {1 2 ∫ r+r′′ 2 r U(ξ, ρ2) ∫ r′′+r−ξ r′′+ξ−r K(ξ, η) dη dξ + 1 2 ∫ ∞ r+r′′ 2 U(ξ, ρ2) ∫ r′′+ξ−r ξ K(ξ, η) dη dξ } dr′′. Then K satisfies the integral equation K(r, r′) = 1 2 ∫ ∞ r+r′ 2 U(ξ, ρ2)dξ + 1 2 ∫ r+r′ 2 r U(ξ, ρ2)dξ ∫ r′+ξ−r r′+r−ξ K(ξ, η) dη + 1 2 ∫ ∞ r+r′ 2 U(ξ, ρ2)dξ ∫ r′+ξ−r ξ K(ξ, η) dη, (2.16) for any 0 ≤ r ≤ r′. Similarly as in Proposition 2.2, we look for a solution in the form of the expansion K(r, r′) = ∞∑ n=0 Kn(r, r′), (2.17) for a sequence {Kn}n≥0 given iteratively by Kn+1(r, r′) = 1 2 ∫ r+r′ 2 r U(ξ, ρ2) ∫ r′+ξ−r r′+r−ξ Kn(ξ, η) dη dξ + 1 2 ∫ ∞ r+r′ 2 U(ξ, ρ2) ∫ r′+ξ−r ξ Kn(ξ, η) dη dξ, for n ≥ 0, and K0(r, r′) = 1 2 ∫ ∞ r+r′ 2 U(ξ, ρ2) dξ. We conclude by induction that |Kn(r, r′)| ≤ 1 2 N∑ j=1 Wjσj (r + r′ 2 ) 1 n! [ N∑ j=1 Wjσ ′ j(r) ]n for n ≥ 0, which gives the first bound. Bounds on derivatives are obtained in the same way by derivating equation (2.16) and using the first bound (2.15). � EJDE-2021/36 ENERGY-DEPENDENT HAMILTONIAN 7 2.3. Solution ẽ(r, ρ). By Proposition 2.2 we know that r → e(r, ρ) is asymptotic to the free solution eiρr. Let us show that a similar asymptotic result holds for another solution ẽ(r, ρ) which will be asymptotic to e−iρr. Proposition 2.4. For any δ > 0 there exists a positive number a = aδ large enough such that equation − y′′ + U(r, ρ2)y = ρ2y, (2.18) admits a solution y = ẽ(r, ρ) satisfying the integral equation ẽ(r, ρ) = e−iρr + 1 2iρ ∫ r a eiρ(r−r ′))U(r′, ρ2)ẽ(r′, ρ) dr′ + 1 2iρ ∫ ∞ r e−iρ(r−r ′))U(r′, ρ2)ẽ(r′, ρ) dr′, (2.19) for 0 < a ≤ r <∞, in the domain {ρ ≥ δ; =m(ρ) ≥ 0}. There exists Cδ large enough such that in this domain |ẽ(r, ρ)| ≤ Cδer=m(ρ). Moreover, for any α > 0 and for r →∞, ẽ(r, ρ) = e−iρr(1 + o(1)), ∂r ẽ(r, ρ) = e−iρr(−iρ+ o(1)), (2.20) uniformly with respect to ρ in the domain {=m(ρ) ≥ α, |ρ| ≥ δ}. Moreover ρ → ẽ(r, ρ) is holomorphic in the domain {=m(ρ) ≥ 0 : |ρ| ≥ δ} and for |ρ| → ∞, ẽ(r, ρ) = eiρr(1 + o(ρ−1)), ∂r ẽ(r, ρ) = ieiρr(1 + o(ρ−1)), (2.21) uniformly with respect to r in the domain {r ≥ a}. Proof. As the proof is similar to that of Proposition 2.2, we only sketch it, leaving the details to the reader. It is clear that ẽ(r, ρ) given by (2.19) satisfies (2.18). Now noting ε̃(r, ρ) = eiρr ẽ(r, ρ), we see that ε satisfies ε̃(r, ρ) = 1 + 1 2iρ ∫ r a e2iρ(r−r′))U(r′, ρ2)ε̃(r′, ρ) dr′ + 1 2iρ ∫ ∞ r U(r′, ρ2)ε̃(r′, ρ) dr′, Looking for a solution of (2.11) in the form ε̃(r, ρ) = ∞∑ n=0 ε̃n(r, ρ), (2.22) for a sequence {ε̃n}n≥0 given iteratively by ε̃n+1(r, ρ) = 1 2iρ ∫ r a e2iρ(r−r′))U(r′, ρ2)ε̃n(r′, ρ) dr′ + 1 2iρ ∫ ∞ r U(r′, ρ2)ε̃n(r′, ρ) dr′ for n ≥ 1. Then for =m(ρ) ≥ 0 we deduce that |ε̃n(r, ρ)| ≤ [ 1 |ρ| N∑ j=1 Wjσj(a) ]n for n ≥ 1. 8 P. CHAU HUU-TAI, B. DUCOMET EJDE-2021/36 Then as σj(r) → 0 when r → ∞, for any δ > 0 there is a number a large enough such that N∑ j=1 Wjσj(a) < δ, so series (2.22) is uniformly convergent and its sum ε̃(r, ρ) is uniformly bounded on any domain {=m(ρ) ≥ 0 : |ρ| > δ, r ≥ a}. � Finally we have the following consequence. Proposition 2.5. If the integrals σ′j(r) = ∫∞ r r′|Vj(r′)| dr′ are convergent, then the solution ẽ(r) := ẽ(r, 0) of (2.7) is well defined as the solution of the integral equation ẽ(r) = r − r ∫ ∞ r U(r′, 0)ẽ(r′) dr′ − ∫ r a r′U(r′, 0)ẽ(r′) dr′, (2.23) and ẽ rewrites ẽ(r) = rε̃(r), where ε̃ is bounded as r →∞. Moreover if for some γ > 0, the integrals ∫∞ r ξ1+γ |Vj(ξ)| dξ are convergent, then as r →∞, one has ẽ(r, 0) = r(1 + o(r−γ)), ∂r ẽ(r, ρ) = 1 + o(r−γ). (2.24) We omit the proof of the above proposition. 2.4. The Wronskian W ( e(ρ, r), ẽ(ρ, r) ) . From the asymptotic behavior of solu- tions e and ẽ for r →∞ we observe that their Wronskian W (e, ẽ) := e′ẽ− e, ẽ′, has the asymptotic behavior W (e, ẽ) = −2iρ+ o(1). (2.25) As first order derivative are absent from equation (2.7), W (r) does not depend on r and in fact W (e, ẽ) = −2iρ in the domain {=m(ρ) ≥ 0 : |ρ| > δ}. (2.26) 3. Discrete spectrum σd(H) For =m(ρ) ≤ 0, we put E(r, ρ) = e(r,−ρ), where e is the solution studied in Proposition 2.2. Using the previous information about solutions of (2.7) we have the following result. EJDE-2021/36 ENERGY-DEPENDENT HAMILTONIAN 9 Theorem 3.1. (1) H has no positive eigenvalue. (2) If for a γ > 0 ∫ ∞ r ξ1+γ |Vj(ξ)| dξ <∞, for j = 1, . . . , N , then 0 is not an eigenvalue of H. (3) A necessary and sufficient condition for λ 6= 0 to be an eigenvalue of H is λ = ρ2, =m(ρ) > 0, e(0, ρ) = 0. (4) The set of eigenvalues of H is at most countable and its limit points belong to the half line {λ ≥ 0}. Proof. (1) The general solution y of (2.7) reads y(r, ρ) = C1e(r, ρ) + C2E(r, ρ) with ρ = √ λ. By Proposition 2.2, we have y(r, ρ) = C1e iρr + C2e −iρr + o(1). As the leading part is periodic, it cannot belong to L2(0,∞). (2) In the same stroke, after Propositions 2.3 and 2.5, the general solution Y of −Y ′′ + U(r, 0)Y = 0 satisfies Y (r) = C1 + C2r [ 1 + o ( 1 rγ )] , which cannot belong to L2(0,∞). (3) We consider now solutions of (2.7) when ρ is non-positive or complex. As the general solution of (1.3) is given by y(r) = Cφ(r, ρ), it is clear that λ is an eigenvalue of H if and only if φ(·, λ) ∈ L2(0,∞). From (2.26) and the definition (2.1) of φ, φ(r, ρ2) = ẽ(ρ)e(r, ρ)− e(ρ)ẽ(r, ρ) 2iρ , (3.1) where ẽ(ρ)) = ẽ(0, ρ) and e(ρ) = e(0, ρ), for ρ ∈ ΓW , =m(ρ) > 0 and |ρ| ≥ δ, the result follows. Indeed, from Proposition 2.2 e(·, ρ) ∈ L2(0,∞) but by Proposition 2.4, ẽ(·, ρ)) /∈ L2(0,∞) so s(·, ρ) ∈ L2(0,∞) if and only if e(ρ) = 0. (4) We know (by Proposition 2.2) that for =m(ρ) ≥ 0 and |ρ| → ∞ e(ρ) = 1 + o ( ρ−1 ) . Therefore ρ → e(ρ) is bounded and holomorphic in the half-plane =m(ρ) > 0, therefore the set of zeros of e(ρ) = 0 is at most countable and can have limit points only on the axis � Now we are in position to give an approximate localization of the discrete spec- trum, following Davies [7]. 10 P. CHAU HUU-TAI, B. DUCOMET EJDE-2021/36 Proposition 3.2. Suppose that Vj ∈ L1(0,∞) ∩ L2(0,∞) then every eigenvalue λ of H satisfies |λ| ≤ 1 2 ( N∑ j=1 W j‖Vj‖1 )2 . (3.2) Proof. Let λ = −z2 be an eigenvalue of H with 0 and let ψ be the corresponding eigenfunction. Then ψ ∈ H2(R+) ⊂ C0(R+). We have (−∂2 r + z2)ψ = −Uψ, so −ψ = (−∂2 r + z2)−1Uψ. Noting X := |U |1/2, W := U/X and χ := Wψ ∈ L2(0,∞), we obtain −χ = W (−∂2 r + z2)−1Xχ, so −1 ∈ Spec ( W (−∂2 r + z2)−1X ) . We estimate the Hilbert-Schmidt norm of the operator W (−∂2 r+z2)−1X with kernel W (r) [e−z(r−r′) 2z − e−z(r+r ′) 2z ] X(r′). We have 1 ≤ ‖W (−∂2 r + z2)−1X‖22 = 1 4|z|2 ∫ R+×R+ |W (r)|2 [ e−2 0. Then the following properties hold (1) The number of eigenvalues of H is finite. (2) Denoting by E the modulus of the largest eigenvalue EN of H, suppose that A an arbitrary positive number such that A > E 2 α − α 4 . (3.4) Then the number N(H) of eigenvalues of H in the disk D(0,Λ), with Λ = 1 2 ( N∑ j=1 W j‖Vj‖1 )2 , EJDE-2021/36 ENERGY-DEPENDENT HAMILTONIAN 11 satisfies the bound N(H) ≤ ( log A+ α 2√ A2 + Λ2 )−1{ N∑ j=1 W j‖(1 + eαr ′ )Vj‖1 − log ( 2− exp ( 1 A log 2 √ Λ 2 ))} . (3.5) Proof. (1) By (3.3) the discrete spectrum σd(H) of H which we know not to have any accumulation point on the open positive semi-axis cannot include 0. So in fact σd(H) ⊂ D(0,Λ) has no accumulation point, and therefore is finite. (2) As E is the modulus of the largest eigenvalue EN of H, we have E ≤ Λ. Supposing A satisfy (3.4), pick A arbitrary in the interval (√ A2 + E 2 , A+ α 2 ) and consider the translated function Φ(z) := e(z + iA). The number N(H) is at most the number of zeroes of Φ in D(0, √ A2 + E 2 ). Applying Jensen’s Theorem to the analytic function Φ(z) we obtain N(H) log A√ A2 + E 2 ≤ ∑ |zk|≤ √ A2+E 2 log A |zk| = 1 2π ∫ 2π 0 log |Φ(Aeiθ)| dθ − log |Φ(0)|. We observe now that |Φ(Aeiθ)| ≤ exp (∫ ∞ r ( 1 + e−2(A+A sin θ)r′ ) |U(r′, ρ2)| dr′ ) ≤ exp (∫ ∞ 0 ( 1 + e2(A−A)r′ ) |U(r′, ρ2)| dr′ ) , and |Φ(0)‖ ≥ 2− exp ( 1 2A ∫ ∞ r ( 1 + e−2Ar′ ) |U(r′, ρ2)| dr′ ) ≥ 2− exp ( 1 2A ∫ ∞ 0 |U(r′, ρ2)| dr′ ) ≥ 2− exp ( 1 2A N∑ j=1 W j‖Vj‖1 ) . Then for any A ∈ (√ A2 + E 2 , A+ α 2 ) we obtain N(H) log A√ A2 + E 2 ≤ ∫ ∞ 0 ( 1 + e2(A−A)r′ ) |U(r′, ρ2)| dr′ − log ( 2− exp ( 1 A log 2 √ Λ 2 )) , which gives (3.5) if we choose A = A+ α 2 . � 12 P. CHAU HUU-TAI, B. DUCOMET EJDE-2021/36 Remark 3.4. Using the bound given by Franck, Laptev, Lieb and Seiringer [15] we can give another estimate for N(H). Let κ > 0. Then for the eigenvalues outside the cone {|=m(z)| < κ 0}, that σess(H) = σc(H) (no positive eigenvalues) and moreover that σac(H) = (0,+∞)\Ωs where Ωs is finite. 4.1. Continuous spectrum and resolvent of H. Proposition 4.1. (1) Each real number λ = ρ2 such that e(ρ) 6= 0 and =m(ρ) > 0 belongs to the resolvent set %(H) of H. (2) The resolvent R(λ) := (H − λ)−1 of H is given by the formula R(λ)f(r) = ∫ ∞ 0 R(r, r′;λ)f(r′) dr′, where R(r, r′;λ) =  e(r,ρ)φ(r′,λ) e(ρ) if 0 < r′ < r, φ(r,λ)e(r′,ρ) e(ρ) if 0 < r < r′. (4.1) (3) For any δ > 0 there exists a constant Cδ > 0 such that ‖R(λ)‖ ≤ Cδ |e(ρ)|=m(ρ) . Proof. Rewriting F (r) := R(λ)f(r) = (R1 +R2)f(r) with R1f(r) = e(r, ρ) e(ρ) ∫ r 0 φ(r′, ρ2) dr′ and R2f(r) = φ(r, λ) e(ρ) ∫ r 0 e(r′, ρ) dr′, one checks that F solves equation HF = ρ2F + f and the boundary condition F (0) = 0. From Lemma 2.1 and Proposition 2.2 we obtain the estimates |φ(r, ρ2)| ≤ Cδeτr, |e(r, ρ)| ≤ Cδe−τr, for =mρ = τ > 0 and |ρ| ≥ δ. We deduce that ‖R1‖ ≤ Cδ |e(ρ)|τ and ‖R2‖ ≤ Cδ |e(ρ)|τ , and the proof is complete. � EJDE-2021/36 ENERGY-DEPENDENT HAMILTONIAN 13 Proposition 4.2. For any R > 0, there exists Cδ > 0 such that ‖R(ρ2)‖ ≥ Cδ |e(ρ) √ =m(ρ) , for any ρ ∈ {=m(ρ) > 0 : |ρ| ≥ R}. In particular ‖R(λ)‖ → ∞ as long as λ ∈ Res(H) → λ0 ∈ R+. In other words R+ ⊂ σc(H). Proof. Consider for a R > 0 the truncated function ΦR(r) ΦR(r) = { φ(r, λ) if 0 < r < R, 0 if R < r. From Proposition 4.1, ΦR ∈ L2 and R(λ)ΦR(r) = ‖ΦR‖22 e(r, ρ) e(ρ) for r > R. Then ‖R(λ)ΦR‖2 ≥ ∫ ∞ R |R(λ)ΦR(r)|2dr = ‖ΦR‖42 |e(ρ)|2 ∫ ∞ R |e(r, ρ)|2dr. Choosing R = Rδ so large that for r > R, =m(ρ) ≥ 0 and |ρ| ≥ δ one has |e(r, ρ)| > 1 2 e−τr, τ = =m(ρ), we obtain ∫ ∞ R |e(r, ρ)|2dr ≥ e−τR 8τ , ‖R(λ)ΦR‖2 ≥ ‖ΦR‖2e−τR |2e(ρ)| √ 2τ , which completes the proof. � We are now in a position to characterize the continuous spectrum of H. Theorem 4.3. Every number λ > 0 belongs to the continuous spectrum of H. Moreover, under Condition 2 in Theorem 3.1, σc(H) = R+. Proof. It is sufficient to check that for any λ > 0, the range Rλ of H − λ is dense in L2(R+) or equivalently that its orthogonal R⊥λ is reduced to 0. As R⊥λ is the set of solutions of H∗f = λf , where H∗ is the adjoint of H defined by − d2 dr2 v + U(r, E)v, v(0) = 0, one sees from Theorem 3.1 that λ cannot be eigenvalue of H∗. � 14 P. CHAU HUU-TAI, B. DUCOMET EJDE-2021/36 4.2. Spectral expansion. We show that the set of generalized eigenfunctions of H is complete (see below) and we give a convergent expansion in term of these eigenfunction for any L2 function. In the following we assume an additional constraint on the potential U ,∫ ∞ 0 eεr|U(r, E) dr <∞, (4.2) which is clearly satisfied by (1.1). This condition implies that∫ ∞ r |U(r, E) dr < Cεe −εr and ∫ ∞ r r|U(r, E) dr < Cε′e −ε′r, for 0 < ε′ < ε, then estimates (2.15) can be improved as follows |K(r, r′)| ≤ Ce−ε r+r′ 2 , |∂rK(r, r′)|, |∂r′K(r, r′)| ≤ 1 4 N∑ j=1 Wj ∣∣Vj(r + r′ 2 )∣∣+ Ce−ε( 3 2 r+r ′), (4.3) for 0 ≤ r ≤ r′ and C > 0. Proposition 4.4. Under Condition 4.2 and for any r ≥ 0, the function e(r, ρ) = eiρr + ∫ ∞ r K(r, r′)e−iρr ′ dr′ (4.4) is holomorphic in the half-plane {=m(ρ) > − ε 2}. The same holds for e(ρ) = e(0, ρ). Proof. As =m(ρ) > −ε/2, using (4.3), the integral at the right hand side of (4.4) converges, and so do all the derivatives with respect to ρ � We have the following simple Corollary of Proposition 4.4 Proposition 4.5. On the strip {|=m(ρ)| < ε 2}, the second-order ode Hψ = ρ2ψ admits the fundamental system of solutions e1 = e(r, ρ), e2 = ẽ(r, ρ) = e(r,−ρ), with Wronskian W (e1, e2) = −2iρ, |=m(ρ)| < ε 2 . Proof. By analytic continuation, e(r, ρ) satisfies the integral equation (2.8) on the half-plane {=m(ρ)| > −ε/2} then it is also solution of He = ρ2e in the same region. � Observe that from (3.1) the relation φ(r, ρ2) = ẽ(ρ)e(r, ρ)− e(ρ)ẽ(r, ρ) 2iρ = e(−ρ)e(r, ρ)− e(ρ)e(r,−ρ) 2iρ , (4.5) holds on the strip {|=m(ρ)| < ε/2}. As Naimark, we call singular value of H any root ρk of the equation e(ρ) = 0 such that ρk 6= 0 and =m(ρk) ≥ 0. Proposition 4.6. Under condition (4.2), the set Ωs of singular values of H is finite. EJDE-2021/36 ENERGY-DEPENDENT HAMILTONIAN 15 Proof. This result is a direct consequence of analytic properties of e in a neigh- borhood of the real given in Proposition 4.4 and asymptotic properties of e given in Proposition 2.2: as e is analytic and bounded on the positive half-line, Ωs is actually finite. � The non-real singular values of H are denoted by ρ1, ρ2, . . . ρK , and its real singular values by ρK+1, ρK+2, . . . ρL, with multiplicities mk. Then =mρk > 0 for k = 1, . . . ,K and =mρk = 0 for k = K + 1, . . . , L. From assertion 3 in Theorem 3.1 we know that the λk = ρ2 k for k = 1, . . . ,K are the eigenvalues of H and from Schwartz [27] we call spectral singularities of H the remaining λk = ρ2 k for k = K + 1, . . . , L. From Theorem 4.3 we see that spectral singularities of H are embedded into its continuous spectrum. 4.3. Principal functions. Under Condition (4.2), for any integer m ∈ N, e(m)(·, ρ) := ∂mρ e(·, ρ) ∈ L2(R+) for =m(ρ) > 0, and from (4.5), taking into account that e(m)(ρk) = 0 for k = 1, . . . ,K, m = 0, . . . ,mk − 1, we see that φ(m)(·, ρk) := ∂mρ φ(·, ρk) ∈ L2(R+) for k = 1, . . . ,K, m = 0, . . . ,mk − 1, where mk is the multiplicity of the singular value ρk and λk = ρ2 k. Functions φ(m)(·, ρk), k = 1, . . . ,K, m = 0, . . . ,mk − 1 are called principal functions of the point spectrum. Functions φ(m)(·, λ), λ > 0 are called principal functions of the continuous spectrum (note that φ(m)(·, λ) /∈ L2). In the same stroke: functions φ(m)(·, ρk), k = K + 1, . . . , L, m = 0, . . . ,mk − 1 are called principal functions for the spectral singularities. Lemma 4.7. The following estimates hold sup r≥0 |∂mρ e(r, ρ)| (1 + r)m <∞ for =m(ρ) > 0, m = 0, 1, . . . , sup r≥0 |∂mλ φ(r, λ)| (1 + r)m <∞ for λ > 0, m = 0, 1, . . . . The above lemma is a consequence of (4.3) and the formula in formula (4.4) in Proposition 4.4. 4.4. Eigenfunction expansion of the resolvent in term of principal func- tions. Theorem 4.8. The integral kernel R(r, r′;λ) of the resolvent of H has the expan- sion R(r, r′; z) = 1 π Pf ∫ ∞ 0 φ(r, λ)φ(r′, λ) λ− z √ λ e( √ λ)e(− √ λ) dλ + L∑ k=1 ∂mk−1 λ M(λ) φ(r, λ)φ(r′, λ) λ− z ∣∣ λ=λk , where φ is defined by (2.1) and Pf is the finite part (in the Hadamard’s sense) of the corresponding singular integral in formula (4.7) below. 16 P. CHAU HUU-TAI, B. DUCOMET EJDE-2021/36 Proof. Denoting the two disks D(0, R) = {λ ∈ C : |λ| < R} and D(0, η) = {λ ∈ C : |λ| < η}, where R >> 1 and η << 1, and the half-strip Sη = {λ ∈ C : |=m(λ) < η, 0}, we define the domain ΩR,η := D(0, R)\(Sη ∪D(0, η)), and we define the closed contour ΓR,η as the boundary of ΩR,η, ΓR,η = ∂ΩR,η, where the direction on C(0, R) is clockwise. Let z ∈ %(H), choose R large enough and η small enough to ensure that {z, λ1, . . . , λK} ∈ ΩR,η and consider the Cauchy integral JR,η = ∫ ΓR,η R(r, r′;λ) λ− z dλ, where R(r, r′;λ) is the integral kernel of the resolvent of H. From the previous results λ → R(r, r′;λ) is meromorphic in ΩR,η with poles λ1, . . . , λK with multiplicities m1, . . . ,mK . So, applying Cauchy’s residue theorem, we obtain JR,η = R(r, r′; z) + K∑ k=1 Res ∣∣ λ=λk {R(r, r′;λ) λ− z } . (4.6) From (2.1) and (2.2) we see that e(r, ρ) = ∂re(0, ρ)φ(r, λ) + e(0, ρ)ψ(r, λ). As λ→ ψ(r, ρ) is an entire function in λ, with ρ = √ λ, the resolvent kernel reads R(r, r′;λ) = ∂re(0, ρ) e(0, ρ) φ(r, λ)φ(r′, λ) + Ψ(r, r′;λ), for =m(ρ) > 0, where λ→ Ψ(r, r′;λ) is entire. Plugging this into (4.6) we obtain R(r, r′; z) = JR,η + K∑ k=1 ∂mk−1 λ M(λ) φ(r, λ)φ(r′, λ) λ− z ∣∣ λ=λk , where M(λ) = (λ− λk)mk (mk − 1)! ∂re(0, ρ) e(0, ρ) , for k = 1, . . . ,K. Observing that for |ρ| large, |R(r, r′;λ)| < C ρ and checking that the multiplicity of 0, as a root of e(ρ), cannot be greater than unity, we conclude that we can pass to the limit R→∞, η → 0 in the contributions on the circles C(0, R) and C(0, η) of JR,η). To evaluate the contribution of spectral singularities on the real positive axis, we consider the indented domains ω+,ε := {=λ > 0}\ ∪Lk=K+1 D(λk, ε), ω−,ε := {=λ < 0}\ ∪Lk=K+1 D(λk, ε), for ε > 0 small enough, and their boundaries γ+,ε = ∂ω+,ε and γ−,ε = ∂ω−,ε. Now, after the previous steps, we can clearly replace the limit limR→∞,η→0 JR,η by limε→0 ∫ γ+,ε∪γ+,ε with suitable orientations. EJDE-2021/36 ENERGY-DEPENDENT HAMILTONIAN 17 Applying once more Cauchy’s residue theorem and plugging into (4.6), we obtain finally lim R→∞,η→0 JR,η = 1 π Pf ∫ ∞ 0 φ(r, λ)φ(r′, λ) λ− z √ λ e( √ λ)e(− √ λ) dλ + K∑ k=1 ∂mk−1 λ M(λ) φ(r, λ)φ(r′, λ) λ− z ∣∣ λ=λk , (4.7) where Pf is the (“Hadamard’s) finite part of the corresponding singular integral, which completes the proof. � Note that in the case where real spectral singularities are simple poles we can simply erase the Pf in front of the integral. Denoting φ(f, λ) := ∫ ∞ 0 φ(r, λ)f(r) dr, we have the following Fourier-type property. Lemma 4.9. (1) For any function f ∈ L2(R+) there exists C > 0 such that∫ ∞ 0 |φ(f, λ)|2 √ λ dλ ≤ C‖f‖2L2(R+). (4.8) (2) For any f ∈ L2(R+) one has∫ ∞ 0 ∣∣∣φ(f, λ)− ∫ r 0 f(r′)φ(r, λ) dr′ ∣∣∣2√λ dλ→ 0, when r →∞. Proof. (1) Supposing first that f is compactly supported and defining the operator K by Kf(r) = ∫ ∞ 0 K(r, r′)f(r′) dr′, where the kernel K(r, r′) satisfies (4.5) one checks |Kf(r)| ≤ Ce− ε2 r ∫ r 0 e− ε 2 r ′ |f(r′)| dr′, (4.9) then I +K is bounded in L2(R+). From (4.4) we see that φ(f, λ) = ∫ ∞ 0 φ(r, λ)f(r) dr = ∫ ∞ 0 (I +K)f eiρr dr, (4.10) and from (4.5) we have 2iρφ(f, λ) = e(−ρ)φ(f, ρ)− e(ρ)φ(f,−ρ); so using (4.10) and Parseval’s Theorem we obtain∫ ∞ 0 |φ(f, λ)|2 √ λ dλ ≤ C ∫ ∞ 0 |(I +K)f(r)|2 dr, which is (4.8). The general case holds for any f ∈ L2(R+) by denseness. Statement (2) follows from (1). � At this point, one can show a spectral expansion for an arbitrary L2 function. The following expansion holds 18 P. CHAU HUU-TAI, B. DUCOMET EJDE-2021/36 Theorem 4.10. One has the following expansion for an arbitrary function f ∈ L2(R+) in the spirit of [10], f(r) = 1 π ∫ ∞ 0 φ(f, λ)φ(r, λ) √ λ e( √ λ)e(− √ λ) dλ + L∑ k=1 ∂mk−1 λ M(λ)φ(f, λ)φ(r, λ) ∣∣ λ=λk . (4.11) Proof. (1) We first assume that H has no singular value. Supposing that f ∈ C2 0 (R+), we define g(r) := −f ′′(r) + U(r, z)f(r). (4.12) From (4.1) (see Proposition 4.1), the kernel R(r, r′; z) satisfies the equation −R′′(r, r′; z) + U(r, z)R(r, r′; z) = zR(r, r′; z) + δ(r − r′). (4.13) Multiplying (4.12) by R(r, r′; z) and (4.13) by f(r), subtracting and integrating by parts, we obtain∫ ∞ 0 R(r, r′; z)f(r′) dr′ = −1 z f(r) + 1 z ∫ ∞ 0 R(r, r′; z)f(r′) dr′. Integrating this relation on the circle γR = {z ∈ C : |z| = R} and using the asymptotic properties of z → R(r, r′; z), we obtain the identity f(r) = − lim R→∞ ∫ γR [ ∫ ∞ 0 R(r, r′; z)f(r′) dr′ ] dz. Plugging (4.1) into this formula and deforming γR exactly as in the proof of Theo- rem 3.1, we obtain (4.11) for f smooth. Using the denseness of C2 0 (R+) in L2(R+), the general case follows. (2) If H has spectral singularities, formula (4.11) still holds provided that the function f satisfy the conditions djk dλjk φ(f, λk) = 0, λk = ρ2 k, k = K + 1, . . . , L. As one checks easily that this set of functions is dense in L2, formula (4.11) holds for any f ∈ L2(R+). � 5. Resonances for a simplified model We are interested now in the resonances of the (simplified) truncated Hamilton- ian Hb = − d2 dr2 + U bV (r, E), where U bV (r, E) = −[VV (E) + iWV (E)]f b(r,RV , aV ), and f b is the truncated Woods-Saxon form factor f b(r,R, a) = { 1 1+e r−R a if r < b, 0 if r > b, where b > R and we discarded surface and spin-orbit contributions. EJDE-2021/36 ENERGY-DEPENDENT HAMILTONIAN 19 Following the 1D presentation of Dyatlov and Zworski in [9], resonance is a λ ∈ C\{0} for which there exists a pure outgoing state: a solution φ to (Hb − λ2)φ ≡ [ − d2 dr2 + U bV (r, λ2)− λ2 ] φ = 0, satisfying φ(r) = eiλr for r > b. Let g a non decreasing C1 function with g(0) = 0 and supported outside the interval (0, b), we consider curves Γ ⊂ C which are the graphs of such g. Let γ(t) a parametrization of Γ and let f ∈ C1(Γ) in the sense: f ◦ γ ∈ C1(R). We define ∂Γ z f(z0) = γ′(t0)−1∂t(f ◦ γ)(t0), where z = x + iy and γ(t0) = z0 and put Dz = −i∂Γ z . By the chain rule, if f is differentiable near Γ, this is independent of the parametrization and if γ(t) := γr(t) + iγi(t) one gets γ′(t0)−1∂t(f ◦ γ)(t0) = γ′(t0)−1 (∂xf(z0)γ′r(t0) + i∂yf(z0)γ′i(t0)) . If f is holomorphic near Γ one checks that ∂Γ z f(z0) reduces to the holomorphic differential ∂z. The space L2(Γ) is L2(Γ) = { f : Γ→ C : ∫ γ |f |2dz := ∫ |f ◦ γ)(t)|2γ′(t)| dt <∞ } . As U bV (z, λ2) is well defined, so is HΓ := ( DΓ z )2 + U bV (z, λ2). Our point is that some resonances of HV coincide with eigenvalues of HΓ considered as an operator in L2(Γ). Let us introduce two fixed angles 0 ≤ θ1 ≤ θ2 ≤ π/2 and ε > 0 such that for any z ∈ Γ outside a given compact set K, one has θ1 + ε ≤ arg z ≤ θ2 − ε. Theorem 5.1. Each λ such that −θ1 ≤ arg λ ≤ π − θ2 is a resonance of HV of multiplicity m if and only if it is an eigenvalue of HΓ of multiplicity m. Proof. Suppose that λ is a resonance of multiplicity m of HV . Then there is a function φ : R→ C such that (Hb − λ2)m−1φ(r) = Aeiλr if r ≥ b. This means φ(r) = P (r)eiλr, for r ≥ b, for a suitable polynomial P (r). We defining now Φ : R→ C such that Φ(r) = { P (r)eiλr if