Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0157 Acta Polytechnica 62(1):157–164, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague SWANSON HAMILTONIAN REVISITED THROUGH THE COMPLEX SCALING METHOD Marta Reboiroa, b, ∗, Romina Ramírezc, d, Viviano Fernándezc a University of La Plata, Faculty of Exact Science, Department of Physics, 49 & 115, 1900 La Plata, Argentine b CONICET, Institute of Physics of La Plata. 63 & diag. 113, 1900 La Plata, Argentine c University of La Plata, Faculty of Exact Science, Department of Mathematics, 50 & 115, 1900 La Plata, Argentine d CONICET, Institute Argentine of Mathematics. Saavedra 15 3º, C1083ACA Buenos Aires, Argentine ∗ corresponding author: reboiro@fisica.unlp.edu.ar Abstract. In this work, we study the non-hermitian PT-symmetry Swanson Hamiltonian in the framework of the Complex Scaling Method. We show that by applying this method we can work with eigenfunctions that are square-integrable both in the PT and in the non-PT symmetry phase. Keywords: PT-symmetric Hamiltonians, Swanson model, Complex Scaling Method. 1. Introduction The Swanson Model has been introduced in [1] as an example of a PT-symmetry hamiltonian [2–8]. Since then it has been extensively studied, allowing for several interesting extensions [9–25]. Among re- cent works, let us mention an extension of the Swan- son model with complex parameters [23, 25], this work introduces bicoherent-state path integration as a method to quantify non-Hermitian systems. Though the Swanson model is described by quadratic opera- tors, the underlying physics is nevertheless very rich. Depending on the region in the model parameter space, the Swanson model is similar to the hamiltonian of a parabolic barrier or the hamiltonian of a harmonic oscillator [26]. From the mathematical point of view, it is an example of a hamiltonian with eigenfunctions that do not belong to L2(R) in some regions of the space of parameters. Among the methods that are employed to describe the physics of resonances with complex energy, the Complex Scaling Method (CSM) [27–32] is one of the most powerful. It has been extensively used in the description of many-body resonant states and non-resonant continuum states observed in unstable nuclei [32]. In this work, we propose the use of the CSM to describe the dynamics of the Swanson model, particularly in the region of non-PT-symmetry. The work is organized as follows. In Section 2 we describe the application of the CSM to the Swanson Hamiltonian. We establish a similarity transforma- tion between the transformed hamiltonian and its adjoint operator. We discuss, according to the space of parameters of the model, the possibility of hav- ing square-integrable eigenfunctions. We present the mean values of some observables. In Section 3, we analyse with an example, the survival probability as a function of time for an initial coherent state. Con- clusions are drawn in Section 4. 2. Formalism The hamiltonian of Swanson [1] is given by H = ℏω ( a†a + 1 2 ) + ℏα a2 + ℏβ a† 2 , (1) with ω, α, β ∈ R. The hamiltonian of Eq. (1) can be written in terms of the coordinate operator, x̂, and the momentum operator, p̂, by implementing the following representation a = 1 √ 2 ( x̂ b0 + i b0 ℏ p̂ ) , a† = 1 √ 2 ( x̂ b0 − i b0 ℏ p̂ ) , (2) being b0 the characteristic length of the non- interacting system. The hamiltonian in Eq. (1) reads H(ω,α,β) = 1 2 ℏ(ω + α + β) ( x̂ b0 )2 + 1 2 ℏ(ω − α − β) ( b0 p̂ ℏ )2 +ℏ (α − β) 2 ( 2 x̂ i ℏ p̂ + 1 ) . (3) The adjoint hamiltonian of H(ω,α,β) is Hc = H(ω,β,α). As we showed in [26], some of the eigenfunctions of Eq. (3) do not belong to the usual Hilbert space, H = L2(R), so that we have to work in a Rigged Hilbert Space [33, 34]. An alternative approach to solve the eigenvalue problem of the hamiltonian of Eq. (1), is the use of the CSM method [27–32] . The aim of the CSM is to make a similarity transformation from the original hamiltonian to a hamiltonian which has eigenfunctions 157 https://doi.org/10.14311/AP.2022.62.0157 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en M. Reboiro, R. Ramírez, V. Fernández Acta Polytechnica that belong to L2(R). In the framework of the CSM, we shall introduce the transformation operator V̂ (θ) = e− θ 2ℏ (x̂p̂+p̂x̂) with a real scaling parameter θ: V̂ (θ)x̂V̂ −1(θ) = e iθ x̂, V̂ (θ)p̂V̂ −1(θ) = e−iθ p̂. (4) The hamiltonian of Eq. (3) is transformed as H(θ) = V̂ (θ)HV̂ −1(θ): H(θ) = H(θ,ω,α,β) = 1 2 ℏ(ω + α + β) ( e iθ x̂ b0 )2 + 1 2 ℏ(ω − α − β) ( b0 e −iθ p̂ ℏ )2 +ℏ (α − β) 2 ( 2 x̂ i ℏ p̂ + 1 ) . (5) It is straightforward to observe that H†(θ) = H(−θ,ω,β,α). (6) Notice that H(θ) is not invariant under the usual PT-symmetry given by x̂ → −x̂, and p̂ → p̂, and i → −i. We shall introduce the following similarity transfor- mation induced by the operator Υ(θ) = e − α−β ω−α−β e2iθ x2 2b20 . It reads Υ(θ) H(θ)Υ(θ)−1 = h(θ), (7) where h(θ) is given by h(θ) = 1 2m ( e−iθp̂ )2 + 1 2 k ( eiθx̂ )2 . (8) We have defined [26] k = m Ω2 and m = m(ω,α,β,b0) = ℏ (ω − α − β)b20 Ω = Ω(ω,α,β) = √ ω2 − 4αβ = |Ω|eiϕ. (9) Though h(θ) is a non-hermitian operator, h†(θ) = h(−θ) = V (−2θ)h(θ)V (−2θ)−1. Consequently Υ(−θ)−1H†(θ)Υ(−θ) = h(θ)∗, (Υ(−θ)V (−2θ))−1H†(θ)(Υ(−θ)V (−2θ)) = h(θ). (10) From Eqs. (7) and (10), it results H†(θ)S = SH(θ), with S = Υ(−θ)V (−2θ)Υ(θ) [35–37]. The eigenfunctions and eigenvalues of h(θ), ϕ(θ) and E(θ), are related to that of H and H† as follows. Given h(θ)ϕ(θ,x) = E(θ)ϕ(θ,x): H ϕ(θ,x) = Ẽ(θ) ϕ(θ,x), H† ψ(θ,x) = E(θ) ψ(θ,x), (11) with ϕ̃(θ,x) = Υ(θ)−1ϕ(θ,x), E(θ) = E(θ), ψ(θ,x) = Υ(−θ)(ϕ(θ,x))∗, E(θ) = E(θ)∗. (12) Thus, the eigenfunctions of H(θ) with eigenvalue Ẽν (θ) = Eν (θ) are given by ϕ̃ν (θ,x) = e α−β ω−α−β e2iθ x2 2b20 Nνϕν (θ,x) (13) with Nν a normalization constant. It can be shown that the eigenfunctions of H†(θ) are ψν (θ,x) = e − α−β ω−α−β e−2iθ x2 2b20 (Nνϕν (θ,x)) ∗ , (14) and the corresponding eigenvalue is given by Eν (θ) = Ẽν (θ)∗. A similar structure for Eqs. (10)-(14) can be found in [38, 39]. Moreover, the relation between the eigenvalues, Eq. (12), is a typical feature for operators which are self-adjoint in Krein spaces [39–41]. It should be mentioned that the Hamiltonian of Eq. (5), for α = β = 0 and ω = 1/ cos(2θ), reduces to the one introduced in [23–25]. Particularly, in [25] the dynamics under the action of this hamiltonian is described for values of θ ∈ (−π/4,π/4). For further results, the reader is kindly referred to [23–25]. In what follows, we aim to determine the range of values of θ for which ϕ(θ,x) belongs to the Hilbert space L2(R). 2.1. Eigenfunctions and eigenvectors For ω − (α + β) ̸= 0, Eq. (8) can be also written as − d2ϕ(y) dy2 + ( 1 4 y2 − ϵ ) ϕ(y) = 0, (15) with ϵ = E ℏΩ = E ℏ|Ω| eiϕ (16) and y = √ 2 |σ|ei(θ+γ) x b0 , (17) where we have defined σ = ( mΩ ℏ )1/2 b0 = eiγ |σ|. (18) 158 vol. 62 no. 1/2022 Swanson Hamiltonian revisited through the CSM (a) 0 π 2 π 3 π 2 2 π -1 0 1 θ u (x )/ |u (x )| (b) 0 π 2 π 3 π 2 2 π -1 0 1 θ u (x )/ |u (x )| (c) 0 π 2 π 3 π 2 2 π -1 0 1 θ u (x )/ |u (x )| (d) 0 π 2 π 3 π 2 2 π -1 0 1 θ u (x )/ |u (x )| Figure 1. Effective potential of Eq. (19), u(θ,x)|u(θ,x)| , for a fixed x in the regions determined by the signs of the parameters m(ω, α, β, b0) and Ω2(ω, α, β, b0). In Panel (a), (sg(m), sg(Ω2)) = (+, +), Region I. For Panel (b),(sg(m), sg(Ω2)) = (+, −), Region II. While, (sg(m), sg(Ω2)) = (−, +) in Panel (c), Region III. In Region IV (sg(m), sg(Ω2)) = (−, −), for Panel (d). The real part of the effective potential, Re ( u(θ,x) |u(θ,x)| ) , is displayed in solid lines, while the imaginary part of the effective potential, Im ( u(θ,x) |u(θ,x)| ) , is drawn with dashed lines. Eq. (15) is the Schrödinger equation corresponding to the effective potential u(θ,x) = U(θ,x) ℏΩ = e2i(θ+γ) 1 2 |σ|2 x2 b20 . (19) Solutions corresponding to Eq. (15) represent dif- ferent physical systems according to the signs of m(ω,α,β,b0) and Ω2(ω,α,β,b0) [26]. In what follows, we shall refer to Region I when (sg(m),sg(Ω2)) = (+, +), Region II for the case (sg(m),sg(Ω2)) = (+, −), Region III for (sg(m),sg(Ω2)) = (−, +), and Region IV for (sg(m),sg(Ω2)) = (−, −), respectively. In Figure 1 we present the behaviour of the effective potential of Eq. (19), u(x)|u(x)| , as a function of θ, for x, |σ| and |Ω| fixed, in the different regions of the parameter model-space. The real part of the effec- tive potential, Re ( u(x) |u(x)| ) , is displayed in solid lines, while the imaginary part of the effective potential, Im ( u(x) |u(x)| ) , is drawn with dashed lines. 2.1.1. Discrete spectrum For the discrete sector of the spectrum, eigenvalues and the eigenfunctions are given by En = ℏ Ω [n] = ℏ |Ω|eiϕ [n], ϕ̃m(θ,x) = e α−β ω−α−β e 2iθ x2 2b20 ϕm(θ,x), (20) ψm(θ,x) = e − α−β ω−α−β e −2iθ x2 2b20 (ϕm(θ,x))∗, (21) where ϕn(θ,x) can be written as (a) 0 π 2 π 3 π 2 2 π -1 0 1 θ R e (u (x )/ |u (x )| ) (b) 0 π 2 π 3 π 2 2 π -1 0 1 θ R e (u (x )/ |u (x )| ) (c) 0 π 2 π 3 π 2 2 π -1 0 1 θ R e (u (x )/ |u (x )| ) (d) 0 π 2 π 3 π 2 2 π -1 0 1 θ R e (u (x )/ |u (x )| ) Figure 2. Real part of the effective potential of Eq. (19), u(θ,x)|u(θ,x)| . The shadowed sectors correspond to the values of θ for which the solutions of Eq. (15) are square-integrable. In Panels (a), (b), c) and (d) we present the results for Regions I, II, III and IV, respectively. ϕn(θ,x) = Nne −e2i(θ+γ) x 2 2b20 |σ|2 Hn ( ei(θ+γ) x b0 |σ| ) . Nn2 = ei(θ+γ) √ πn!2n |σ| b0 , (22) being Hn(z) the Hermite Polynomial of order n, and [n] = n + 1/2. Eigenfunctions ϕn(θ,x) are square-integrable for θ- intervals where Re(u(θ,x)) takes positive values. In Figure 2, we plot Re(u(θ,x)/|u(θ,x)|) for every region, the gray regions correspond to the intervals for which the eigenfunctions are square integrable. In Table (1), we summarize the sign of the parame- ter m and Ω2, which characterize the different regions of the model, and for each region we present the val- ues of phases γ and ϕ, and the interval where the eigenfunctions are square-integrable. In Regions (I) and (III), we can define two well- defined θ−domains: I1 = [−π, −3π/4)∪(−π/4,π/4)∪ (3π/4,π] and I2 = (−3π/4, −π/4) ∪ (π/4, 3π/4). While, in Regions (II) and (IV), the θ-domains are: I3 = (−π, −π/2) ∪ (0,π/2) and I4 = (−π/2, 0) ∪ (π/2,π). The intervals repeat themselves periodically, with period π. In the domains summarized in Table 1, eigenfunc- tions {ψν (θ,x), ϕ̃ν (θ,x)} form a biorthogonal com- plete set. ∫ ∞ −∞ (ψm(θ,x)) ∗ϕ̃n(θ,x) dx =∫ ∞ −∞ ϕm(θ,x)ϕn(θ,x) dx = δmn. (23) It should be noticed that in all regions, the θ−domains of positive spectrum are different from the domains with negative spectrum. They represent different physical boundary conditions. 159 M. Reboiro, R. Ramírez, V. Fernández Acta Polytechnica sg(m) sg(Ω2) γ ϕ I θc I + + 0 0 I1 ±π/4π/2 π I2 III - + π/2 0 I2 0 π I1 II + - π/4 π/2 I4 0 ±π/2 π −π/4 −π/2 I3 IV - - −π/4 π/2 I3 π/4 −π/2 I4 Table 1. Values of the characteristic parameters for the different model-space regions. In columns 2 and 3 we give the sign of m and Ω2, respec- tively. Phases γ and ϕ, for the different regions, are given in columns 4 and 5, respectively. In col- umn 6 we present the θ-interval for which the dif- ferent eigenfunctions are square-integrable. In the Table I1 = [−π, −3π/4) ∪ (−π/4, π/4) ∪ (3π/4, π], I2 = (−3π/4, −π/4) ∪ (π/4, 3π/4), I3 = (−π, −π/2) ∪ (0, π/2) and I4 = (−π/2, 0) ∪ (π/2, π). In the last col- umn, we give the values of θc for which the eigenfunc- tions of the continuous spectrum are square-integrable. The intervals repeat themselves periodically, with pe- riod π. 2.1.2. Continuous spectrum The eigenfunctions associated to the continuous spec- trum [26, 42–45] are given, in terms of the eigenfunc- tions of h(θ) of Eq. (8), ϕE±(θ,x), by ϕ̃E±(θ,x) = e α−β ω−α−β e 2iθ x2 2b20 ϕE±(θ,x), (24) ψ E ±(θ,x) = e − α−β ω−α−β e −2iθ x2 2b20 (ϕE±(θ,x)) ∗, (25) with ϕ E ±(θ, x) = C Γ(ν + 1)D−ν−1 ( ∓ √ −2ei(θ+γ)|σ| x b0 ) . (26) being D−ν−1(y) the parabolic cylinder functions and ν = ϵ− 12 . The normalization constant takes the value C = e iπ/8iν/2( |σ| b0 ei(θ+γ) )1/2 π23/4 . The biorthogonality and the completeness relation can be written as ∫ ∞ −∞ (ψ E ±(θ,x)) ∗ϕ̃E ′ ± (θ,x)dx = δ(E − E ′), ∑ s=± ∫ ∞ −∞ (ψ E s (θ,x)) ∗ϕ̃Es (θ,x)dE = δ(x − x ′). (27) The possible values that the parameter θ can take to fulfill the requirements of biorthogonality and com- pleteness of Eq. (27), θc, are presented in the last column of Table 1. In the framework of the CSM, the continuous spec- trum lies along the line 2θ. In Regions II and IV, the 2θc = ± π so that E ∈ (−∞, +∞) Meanwhile, in Re- gion I and III, 2θc = ± π2 , so that E takes imaginary values. Consequently, the parameter ν associated to the order of the eigenfunctions of Eq. (26) takes the value ν = −i|ϵ| − 12 . If we look at the effective potential u(θ,x), the values of θc correspond to the values of θ for which Re(u(θ,x)) = 0. 2.1.3. Particular cases Case (a): Ω = 0. When Ω = 0 and ω − (α + β) ̸= 0, the problem reduces to that of a free particle of energy E = ε e−2iθ . Eq. (8) reduces to − ℏ2 2m e2iθ d2f(x) dx2 = E f(x), (28) the wave function can be written as f(x) = Aeikx + Ae−ikx, with k = √ 2ε ℏ(ω−α−β)b20 . Case (b): ω − (α + β) = 0, α ̸= β. To study this case we have to look at Eq. (5). If ω − (α + β) = 0, it reads H(θ) = ℏ(α + β) ( e iθ x̂ b0 )2 +ℏ (α − β) 2 ( 2 x̂ i ℏ p̂ + 1 ) , (29) f(x) = e −e2iθ x̂ 2 4b20 α+β α−β x − 12 + εe−2iθ ℏ(α−β) . (30) In Table 2 we present the values of E for which the wavefunction f(x) is square-integrable. (α + β)/(α − β) (α − β) cos(2θ) ε + + I1 ε| cos(2θ)| ℏ|α−β| < 1 2 - - I2 + - I1 ε| cos(2θ)| ℏ|α−β| > 1 2 - + I2 Table 2. Regions for which the wave function of Eq. (30) is square-integrable. 2.2. Mean values of observables To compute the mean values, we use operators P̂ and X̂ defined as [19, 46, 47] P̂ = Υ−1V (θ + γ)p̂V (θ + γ)−1Υ = e−i(θ+γ)p̂ + iℏei(θ+γ) α − β (ω − α − β)b20 x̂, X̂ = Υ−1V (θ + γ)x̂V (θ + γ)−1Υ = ei(θ+γ)x̂, (31) 160 vol. 62 no. 1/2022 Swanson Hamiltonian revisited through the CSM that satisfy [X̂, P̂ ] = iℏ. (32) For the discrete spectrum of H, it can be proved that ⟨m|P̂ |n⟩ = ∫ ∞ −∞ (ψm(θ,x)) ∗ P̂ ϕ̃n(θ,x)dx = ∫ ∞ −∞ ϕm(θ,x) e−i(θ+γ)p̂ ϕn(θ,x)dx = iℏ √ 2b0r (√ n + 1δm,n+1 − √ nδm,n−1 ) , ⟨m|P̂ 2|n⟩ = ∫ ∞ −∞ (ψm(θ,x)) ∗ P̂ 2 ϕ̃n(θ,x)dx = ∫ ∞ −∞ ϕm(θ,x) e−2i(θ+γ)p̂2 ϕn(θ,x)dx = iℏ √ 2b0r (√ n + 1δm,n+1 − √ nδm,n−1 ) = − ℏ2 2b20r (√ (n + 2)(n + 1)δm,n+2 −(2n + 1)δm,n + √ n(n − 1)δm,n−2 ) , (33) and ⟨m|X̂|n⟩ = ∫ ∞ −∞ (ψm(θ,x)) ∗ X̂ ϕ̃n(θ,x)dx = ∫ ∞ −∞ ϕ±m(θ,x) e i(θ+γ)x̂ ϕ±n (θ,x)dx = b0r√ 2 (√ n + 1δm,n+1 + √ nδm,n−1 ) , ⟨m|X̂2|n⟩ = ∫ ∞ −∞ (ψm(θ,x)) ∗ X̂2 ϕ̃n(θ,x)dx = ∫ ∞ −∞ ϕm(θ,x) e2i(θ+γ)x̂2 ϕn(θ,x)dx = b20r 2 (√ (n + 2)(n + 1)δm,n+2 +(2n + 1)δm,n + √ n(n − 1)δm,n−2 ) , (34) with b0r = b0/|σ|. 2.3. Time dependent mean values From the Schrödinger equation iℏ ∂ ∂t Φ̃n(θ,x,t) = H(θ)Φ̃n(θ,x,t), (35) it results Φ̃n(θ,x,t) = e−iẼn t ℏ ϕ̃n(θ,x). (36) In the same way iℏ ∂ ∂t Ψn(θ,x,t) = H(θ)†ψn(θ,x,t), (37) it results Ψn(θ,x,t) = e−iEn t ℏ ψn(θ,x). (38) 2.3.1. Reigions I and III: Real spectrum In Regions I and III, the discrete eigenvalues of H(θ) take the values E±n = ±ℏ|Ω|[n], with eigenfunctions ϕ̃±n (θ,x). In Region I, the eigenfunctions of the posi- tive (negative) are square integrable in interval I1 (I2), see Table 1. Meanwhile, in Region III, tthe eigenfunc- tions of the positive (negative) are square integrable in interval I2 (I1). Consequently the time evolution of the states is given by Φ̃±n (θ,x,t) = e −i Ẽn tℏ ϕ̃n(θ,x), = e∓i(n+ 1 2 )|Ω|tϕ̃±n (θ,x). Ψ ± n (θ,x,t) = e −i En tℏ ψn(θ,x), = e∓i(n+ 1 2 )|Ω|tψ ± n (θ,x), (39) and then ⟨m|Ô|n⟩ = e∓i(n−m)|Ω|t ∫ ∞ −∞ (ψ ± m(θ,x)) ∗Ôϕ̃±n (θ,x)dx. (40) 2.3.2. Region II and IV: Complex spectrum In Regions II and IV, the discrete eigenvalues of H(θ) take the values E±n = ±iℏ|Ω|[n], with eigenfunctions ϕ̃±n (θ,x). In Region II, the eigenfunctions of the posi- tive (negative) are square integrable in interval I4 (I3). Meanwhile, in Region III, the eigenfunctions of the positive (negative) are square integrable in interval I3 (I4). So that the time evolution of the eigenfunctions are given by Φ̃±n (θ,x,t) = e −i En tℏ ϕ̃±n (θ,x), = e±(n+ 1 2 )|Ω|tϕ̃n(θ,x). (41) and Ψ ± n (θ,x,t) = e −i E∗n t ℏ ψn(θ,x), = e∓(n+ 1 2 )|Ω|tψ ± n (θ,x). (42) As a result ⟨m|Ô|n⟩ = e±(n−m)|Ω|t ∫ ∞ −∞ (ψ ± m(θ,x)) ∗Ôϕ̃±n (θ,x)dx. (43) 161 M. Reboiro, R. Ramírez, V. Fernández Acta Polytechnica 3. Results and discussion In order to evaluate the benefits of the present ap- proach, let us consider the time evolution of a given initial state when the parameters of the model corre- spond to Region II. In [26] we have analysed the Swanson model by solving its eigenvalue problem in the Rigged Hilbert Space. We have found that in Region II the hamilto- nian was similar to the one of a particle in a parabolic barrier. In the framework of the CSM, we model the effective interaction by a complex potential. This fact resembles the spirit of the Optical Potential in Nuclear Physics [48, 49], as the potential seen by an incident nucleon on a nucleus is modeled by a complex effective potential accounting for the loss of flux due to the interaction of an incident particle with the nucleons of the nucleus. We shall consider the solutions with eigenvalues En = −iℏ|Ω|(n + 1/2), which evolve in time as e−|Ω|(n+1/2)t. They correspond to the boundary prob- lem for 0 < t < ∞. In this case γ = −π/4 and θ ∈ I3. For simplicity, let us assume that the initial state is a coherent state of the form ϕI (z,θ,x) = e−|z| 2/2 ∞∑ k=0 zk √ k! ϕk(θ,x), (44) where ϕ̃k(θ,x) is the k-eigenfunction of H(θ). The survival probability of the state can be computed as p(t) = ∣∣∣∣ ∫ ∞ −∞ (ψI (z,θ,x)) ∗e−iH(θ)t/ℏϕI(z,θ, x)dx ∣∣∣∣2 = ∣∣∣∣∣e−|z|2−|Ω|t/2 ∞∑ k=0 (|z|2e−|Ω|t)k k! ∣∣∣∣∣ 2 = e−|Ω|t+2|z| 2(e−|Ω|t−1). (45) Notice that, in this particular case, p(t) is independent of the parameter θ. 4. Conclusions In this work we analyse the advantages of the CSM for describing the dynamics of a non-hermitian sys- tem when the eigenfunctions of the problem do not belong to L2(R). We have shown that we can cast the original problem into a complex potential, which includes absorption and dissipation effects according to the sign of its imaginary component. We have shown that for a range of values of θ in the different regions of the model, the resulting eigenfunctions are square-integrable. This feature facilitates the study of the dynamics of the system from the computa- tional point of view. The price we have to pay is the lack of PT-symmetry invariance of the transformed hamiltonian. Work is in progress concerning the application of the CSM to a more involved problem as the one presented in [50]. 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Physics Letters A 380(11-12):1117–1124, 2016. https://doi.org/10.1016/j.physleta.2016.01.027. 164 https://doi.org/10.1063/1.1599074 https://doi.org/10.1063/1.1644751 https://doi.org/10.1103/PhysRevA.94.052136 https://doi.org/10.1016/j.aop.2013.02.015 https://doi.org/10.1103/PhysRevD.86.064038 https://doi.org/10.1088/1751-8113/46/33/335304 https://doi.org/10.1016/0003-4916(69)90161-4 https://doi.org/10.1103/PhysRevC.95.024315 https://doi.org/10.1016/j.physleta.2016.01.027 Acta Polytechnica 62(1):157–164, 2022 1 Introduction 2 Formalism 2.1 Eigenfunctions and eigenvectors 2.1.1 Discrete spectrum 2.1.2 Continuous spectrum 2.1.3 Particular cases 2.2 Mean values of observables 2.3 Time dependent mean values 2.3.1 Reigions I and III: Real spectrum 2.3.2 Region II and IV: Complex spectrum 3 Results and discussion 4 Conclusions Acknowledgements References