Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0056 Acta Polytechnica 62(1):56–62, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague TIME-DEPENDENT STEP-LIKE POTENTIAL WITH A FREEZABLE BOUND STATE IN THE CONTINUUM Izamar Gutiérrez Altamiranoa, ∗, Alonso Contreras-Astorgab, Alfredo Raya Montañoa, c a Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edificio C-3, Ciudad Universitaria. Francisco J. Mújica s/n. Col. Felícitas del Río. 58040 Morelia, Michoacán, México b CONACyT–Physics Department, Cinvestav, P.O. Box. 14-740, 07000 Mexico City, Mexico c Centro de Ciencias Exactas, Universidad del Bío-Bío, Avda. Andrés Bello 720, Casilla 447, 3800708, Chillán, Chile ∗ corresponding author: izamar.gutierrez@umich.mx Abstract. In this work, we construct a time-dependent step-like potential supporting a normalizable state with energy embedded in the continuum. The potential is allowed to evolve until a stopping time ti, where it becomes static. The normalizable state also evolves but remains localized at every fixed time up to ti. After this time, the probability density of this state freezes becoming a Bound state In the Continuum. Closed expressions for the potential, the freezable bound state in the continuum, and scattering states are given. Keywords: Bound states in the continuum, supersymmetric quantum mechanics, time-dependent quantum systems. 1. Introduction The first discussion of Bound states In the Con- tinuum (BICs) in quantum mechanics dates back to von Neumann and Wigner [1] who constructed normalizable states corresponding to an energy em- bedded in the continuum in a periodic potential V (r) = E + ∇2ψ/ψ from a modulated free-particle wave function ψ(r) = (sin(r)/r)f(r), with twice the period of the potential. The localization of this state is interpreted as the result of its reflection in the Bragg mirror generated by the wrinkles of V (r) as r → ∞. The extended family of von-Neumann and Wigner potentials have been discussed and extended for many years [2–5] from different frameworks including the Gelfan-Levitan equation [6] also known as inverse scat- tering method [4, 7], Darboux transformations [8, 9] and supersymmetry (SUSY) [10–13], among others. Bound states In the Continuum are nowadays recog- nized as a general wave phenomenon and has been explored theoretically and experimentally in many different setups, see [14] for a recent review. Exact solutions to the time-dependent Schrödinger equation are known only in a few cases, including the potential wells with moving walls [15, 16], which has been explored from several approaches (see, for instance, Ref. [17] and references therein) including the adiabatic approximation [18] and perturbation theory [16] and through point transformations [19– 23], which combined with supersymmetry techniques allow to extend from the infinite potential well with a moving wall to the trigonometric Pöschl-Teller po- tential [24]. In this article, we present the construction of a time- dependent step-like potential. We depart from the standard stationary step potential and apply a second- order supersymmetric transformation to add a BIC. Then, by means of a point transformation, the poten- tial and the state become dynamic and we allow them to evolve. After a certain time, we assume that all the time-dependence of the potential is frozen, such that the potential becomes stationary again and explore the behavior of the normalizable state. Intriguingly, it is seen that the freezable BIC is not an eigenso- lution of the stationary Schrödinger equation in the frozen potential, but rather solves an equation that includes a vector potential that does not generate a magnetic field whatsoever. Thus, by an appropri- ate gauge transformation, we gauge away the vector potential and observe the BIC that remains frozen as an eigenstate of the stationary Hamiltonian after the potential ceases to evolve in time. In order to expose our results, we have organized the remaining of this article as follows: In Section 2 we describe the preliminaries of SUSY and a point transformation. Section 3 presents the construction of the time-dependent step-like potential and give ex- plicit expressions for the freezable BIC and scattering states. Final remarks are presented in Section 4. 2. Supersymmetry and a point transformation Point transformation is a successful technique to de- fine a time-dependent Schrödinger equation with a full time-dependent potential from a known stationary problem [19, 20, 24]. In this section, we use a trans- 56 https://doi.org/10.14311/AP.2022.62.0056 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 62 no. 1/2022 Step-like potential with a freezable bound state in the continuum formation of this kind in combination with a conflu- ent supersymmetry transformation to obtain a time- dependent step-like potential from the stationary case. 2.1. Confluent supersymmetry Darboux transformation, intertwining technique or supersymmetric quantum mechanics (SUSY) is a method to map solutions ψ of a Schrödinger equa- tion into solutions ψ̄ of another Schrödinger equation [25–29]. It is based on an intertwining relation where two Hamiltonians and a proposed operator L† must fulfill the relation H̄L† = L†H, (1) where H = − d2 dy2 + V0(y), H̄ = − d2 dy2 + V̄ (y). (2) The main ingredient of SUSY are the seed solutions, which correspond to solutions of the initial differential equation Hu = ϵu, where ϵ is a real constant called factorization energy. In this work we focus on the so called confluent supersymmetry, where L† is a second- order differential operator. Once a seed solution and a factorization energy are chosen, the next step is to construct the following auxiliary function v = 1 u ( ω + ∫ u2(y)dz ) , (3) where ω is a real constant to be fixed. Then, one way to fulfill (1) is by selecting L† = ( − d dy + v′ v ) ( − d dy + u′ u ) , (4) and the potential term in H̄ as V̄ (y) = V0(y) − 2 d2 dy2 ln ( ω + ∫ y y0 u2dz ) . (5) Then, solutions of the differential equation Hψ = Eψ, where E is energy, can be mapped using L† and the intertwining relation as follows: Hψ = Eψ, ⇓ times L† L†Hψ = EL†ψ, ⇓ using (1) H̄L†ψ = EL†ψ, ⇓ defining ψ̄ ∝ L†ψ H̄ψ̄ = Eψ̄. We define ψ̄ as ψ̄ = 1 E − ϵ L†ψ. (6) The factor (E − ϵ)−1 is introduced for normalization purposes. Moreover, H̄ could have an extra eigenstate that cannot be written in the form (6). This state is called missing state and plays an important role in this work. The missing state is obtained as follows: First we have seen that L† maps solutions of Hψ = Eψ into solutions of H̄ψ̄ = Eψ̄, by obtaining the adjoint equation of (1) HL = LH̄, where L = (L†)† we can construct the inverse mapping, but there is a solution ψ̄ϵ such that Lψ̄ϵ = 0. This solution is explicitly: ψ̄ϵ = Cϵ 1 v = Cϵ u ω + ∫ u2(y)dy , (7) where Cϵ is a normalization constant if ψ̄ϵ is square integrable. This state fulfills H̄ψ̄ϵ = ϵψ̄ϵ. Notice that the selection of u, ϵ and ω is very relevant, we must choose these carefully to avoid the introduction of sin- gularities in the potential V̄ that lead to singularities also in ψ̄. The function ω + ∫ u2dy must be node- less. We can satisfy this if either limy→∞ u(y) = 0 or limy→−∞ u(y) = 0 and if ω is appropriately chosen. 2.2. Point transformation Given that we know the solution of the time indepen- dent Schrödinger equation d2 dy2 ψ̄(y) + [ E − V̄ (y) ] ψ̄(y) = 0 (8) with a potential defined in y ∈ (−∞, ∞), let us con- sider the following change of variable y(x,t) = x 4t + 1 , (9) where x ∈ (−∞, ∞) is considered as a spatial variable and t ∈ [0, ∞) a temporal one. Then, the wavefunc- tion ϕ(x,t) = 1 √ 4t + 1 exp { i(x2 + E4 ) 4t + 1 } ψ̄ ( x 4t + 1 ) , (10) solves the time-dependent Schrödinger equation i ∂ ∂t ϕ(x,t) + ∂2 ∂x2 ϕ(x,t) − V (x,t)ϕ(x,t) = 0, (11) where the potential term is V (x,t) = 1 (4t + 1)2 V̄ ( x 4t + 1 ) . (12) In other words, the change of variable (9) together with the replacements V̄ → V and ψ̄ → ϕ trans- form a stationary Schrödinger equation into a time dependent solvable one. 3. Time dependent step-like potential with a freezable bound state in the continuum In this section, we depart from the well-known step potential V (y) = V̂ Θ(−y) as time independent sys- tem. Then, using confluent supersymmetry we will 57 I. Gutiérrez, A. Contreras-Astorga, A. Raya Acta Polytechnica add a single BIC. Furthermore, with the point trans- formation previously introduced we transform the stationary system into a time-dependent system with an explicitly time-dependent potential. We will choose a stopping time or freezing time ti after which the potential no longer evolves: VF (x,t) = { V (x,t) 0 ≤ t < ti, V (x,ti) t ≥ ti. (13) Finally, the solutions of the Schrödinger equation will be presented. Let us commence our discussion by considering the Step-Potential V0(y) = { V̂ y ≤ 0, 0 y > 0, (14) defined along the axis y ∈ (−∞, ∞) and V̂ is a positive constant. The solutions of this system are well known in the literature (see [30, 31]). Restricting ourselves to the case 0 < Eq < V̂ , the solutions are: ψ(y) = { exp(ρy) y ≤ 0, cos(qy) + κ k sin(qy) y > 0, (15) with energy Eq = q2 and ρ = √ V̂ − Eq. Next, to perform the confluent supersymmetric transformation we choose a factorization energy such that 0 < ϵ < V̂ and the corresponding seed solution u(y) as u(y) = { exp(κy) y ≤ 0, cos(ky) + κ k sin(ky) y > 0, (16) with k2 = ϵ and κ2 = V̂ − ϵ. Note that u(y) → 0 when y → −∞. Then, from (5) we obtain explicitly the SUSY partner V̄ : V̄ (y) =   V̂ − 16 exp(2κy)κ 3 ω (exp(2κy)+2κω)2 y ≤ 0 32k2 ( k cos(ky) + κ sin(ky) ṽ(y) v̂(y) ) y > 0, (17) where the functions ṽ(y) and v̂(y) are ṽ(y) = [ (k2 + κ2)(k2x + κ) + 2k4ω ] sin(ky) − k [ (k2 + κ2)(κy + 1) + 2k2κω ] , v̂(y) = [ 2ky(k2 + κ2) + 4k3ω − 2kκ cos(2ky) +(k2 − κ2) sin(2ky) ]2 . We can calculate directly from (7) the missing state associated to the factorization energy ϵ: ψ̄ϵ(y) = Cϵ   2κ exp(κy) 2κω+exp(2κy) y ≤ 0, 4k3(cos(ky)+ κ k sin(ky)) ψ̂ϵ(y) y > 0, (18) where ψ̂ϵ(y) = (k2 − κ2) sin(2ky) − 2κk cos(2ky) + 4ωk3 + 2ky ( κ2 + k2 ) . ������ 0 5 10 15 20 0.00 0.02 0.04 0.06 0.08 0.10 0.12 y |ψ (y ) 2 | ψϵ(y) 2 Fit |A(y) 2 Figure 1. |ψ̄ϵ(y)|2 and an envelop function of the form A(y) = a b+y , with a = 2k(κ 2 + k2)−1/2, b = 2ωk2(κ2 + k2)−1. The scale of the graph is fixed with V̂ = 5, k = 1, κ = 2 and Cϵ = 1, in the appropri- ate units. In order to confirm that ψ̄ϵ is square integrable, we proceed in the following way. First, we separate the integral ||ψ̄ϵ||2 = ∫ ∞ −∞ |ψ̄ϵ| 2dy = ∫ 0 −∞ |ψ̄ϵ| 2dy +∫ ∞ 0 |ψ̄ϵ| 2dy. The first integral can be directly calcu- lated:∫ 0 −∞ |ψ̄ϵ|2dy = |Cϵ|2 √ 2 κω tan−1 ( 1 √ 2κω ) . For the second integral, we can show that it is bounded by a square integrable function: ∫ ∞ 0 |ψ̄ϵ| 2dy |Cϵ|2 = ∫ ∞ 0 ∣∣∣∣∣ 4k 3(cos(ky) + κ k sin(ky)) ψ̂ϵ(y) ∣∣∣∣∣ 2 dy ≤ ∫ ∞ 0 ∣∣∣∣∣ 4k 2 √ k2 + κ2 4ωk3 + 2ky (κ2 + k2) ∣∣∣∣∣ 2 dy = ∫ ∞ 0 ∣∣∣∣ ab + y ∣∣∣∣2 dy = a2b , (19) where a = 2k√ κ2+k2 , b = 2ωk 2 κ2+k2 . Figure 1 shows a fair fit to the squared modulus of eq. (18) for y > 0. For an energy E = q2 ̸= ϵ, the wavefunction solving H̄ψ̄ = Eψ̄ is constructed using (6), and (15). It reads ψ̄(y) =   [ (κ−ρ) exp(ρy) (q2−k2) ] ψ̄−(y) y ≤ 0, ψ̄+(y)−q2 cos(qy)−qρ sin(qy) q2−k2 y > 0, (20) where we abbreviated ψ̄−(y) = 2κω0(κ + ρ) + (ρ − κ) exp(2κy) 2κω + exp(2κy) , ψ̄+(y) = k2(ρ sin(qy) + q cos(qy)) q + 4k(κ sin(ky) + k cos(ky)) ψ̂ϵ(y) × [ k q ( κ cos(ky) − k sin(ky) ) ( ρ sin(qy) + q cos(qy) ) + ( κ sin(ky) + k cos(ky) )( q sin(qy) − ρ cos(qy) )] . 58 vol. 62 no. 1/2022 Step-like potential with a freezable bound state in the continuum In Figure 2 the potential V̄ (y), along with the proba- bility densities of the missing state |ψ̄ϵ(y)|2 and a scat- tering state |ψ̄(y)|2 are shown. We observe that the wavefunction of the BIC has an envelop function which tends to zero as |y| → ∞, whereas the state ψ̄(y) is not localized. The next step is to construct a time dependent potential from (17) using the point transformation presented in (9-12). Notice that x = y at t = 0. Then V̄ transforms as the piecewise potential: V (x,t) = 1 (4t + 1)2 { V̂ − 16κ3ω exp( 2κx4t+1 )[ 2κω + exp( 2κx4t+1 ) ]2 } (21) if x ≤ 0, otherwise V (x,t) = 32k2 (4t + 1)2 × [ k cos ( kx 4t + 1 ) + κ sin ( kx 4t + 1 ) ṽ(y(x,t)) v̂(y(x,t)) ] . (22) In Figure 3 (top) we show the potential V (x,t) at t = 0, t = 0.1 and t = 0.2. Its shape changes in time and its spatial profile oscillates as expected, vanishing as x → ∞. Analogously, for the time-dependent BIC, the associated wavefunction for energy ϵ is explicitly ϕϵ(x,t) = 1 √ 4t + 1 exp { i(x2 + k 2 4 ) 4t + 1 } ψ̄ϵ ( x 4t + 1 ) , (23) This function solves the time-dependent Schrödinger equation i∂tϕϵ + ∂xxϕϵ − V ϕϵ = 0 and its square integrability is guaranteed since ψ̄ϵ(y) is a square integrable function: ||ϕϵ||2 = ∫ ∞ −∞ |ϕϵ(x,t)|2dx = 1 4t + 1 ∫ ∞ −∞ ∣∣ψ̄ϵ ( x4t + 1 ) ∣∣2dx = ∫ ∞ −∞ ∣∣ψ̄ϵ(y)∣∣2dy = ||ψ̄ϵ||2. (24) where we used the change of variable (9). Its proba- bility density is shown in Figure 3 (center) at different times. This state is localized and the first peak in the probability density broadens and diminishes height as time increases. For states with energy Eq = q2 ̸= ϵ, the correspond- ing time-dependent wavefunction has the explicit form ϕ(x,t) = 1 √ 4t + 1 exp { i(x2 + q 2 4 ) 4t + 1 } ψ̄ ( x 4t + 1 ) , (25) The behavior of the probability density |ϕ(x,t)|2, for E = 2 at different times is shown in Figure 3 (bottom). This state is unlocalized at any time. Finally, we choose the freezing or stopping time ti. Then, we can consider a charge particle in a potential: VF (x,t) = { V (x,t) 0 ≤ t < ti, V (x,ti) t ≥ ti. (26) Figure 2. Potential V̄ (y), along with the probability densities of the missing state |ψ̄ϵ(y)|2 and a scattering state |ψ̄(y)|2 are shown. The scale of the graph is fixed with V̂ = 5, k = 1, κ = 2, q = √ 2 and ω = 4. Figure 3. Behavior of the potential V (x,t) (top), the BIC ϕϵ(x,t) (center) and the scattering state ϕ(x,t) (bottom) at the times t = 0, t = 0.1, and t = 0.2. The scale of the graphs is fixed by V̂ = 5, k = 1, κ = 2, q = √ 2 and ω = 4. 59 I. Gutiérrez, A. Contreras-Astorga, A. Raya Acta Polytechnica where V (x,t) is given by (21,22). Notice that when t ∈ [0, ti) the potential is changing in time, and when t ≥ ti the potential is frozen. This potential is in fact a family, parametrized by ω > 0, recall that ω was introduced by the confluent SUSY transformation. Neither ϕ(x,t) nor ϕϵ(x,t) are stationary states, they evolve in time, and they are not eigenfunctions of the operator −∂xx + V . At any time t ≥ ti, the functions ϕ(x,ti) and ϕϵ(x,ti) satisfy the eigenvalue equation:[( − ∂ ∂x + iAx(x) )2 + V (x,ti) ] ϕ(x,ti) = E (4ti + 1)2 ϕ(x,ti), t ≥ ti, (27) where Ax(x) = −∂xθ(x) and θ(x) = i 4ti + 1 ( x2 + E 4 ) . (28) Equation (27) is the Schrödinger equation for a charged particle under the influence of a vector potential A = (Ax, 0, 0) that, nevertheless, does not generate magnetic field since B = ∇ × A = 0. Let us recall that the Schrödinger equation for a charged particle of charge q immersed in an external electro- magnetic field is better written in terms of the scalar φ and vector potentials A through the Hamiltonian H = (p̂ + qA)2 + qφ. (29) These electromagnetic potentials allow us to define the electric and magnetic fields as E = −∇φ − ∂A ∂t , B = ∇ × A, (30) definition that does not change if the following trans- formations are performed simultaneously, A → A′ = A + ∇λ, φ → φ′ = φ − ∂λ ∂t , (31) where λ = λ(x,t) is a scalar function. This is a state- ment of gauge invariance of Maxwell’s equations. In quantum mechanics, the time-dependent Schrödinger equation i ∂ψ ∂t = Hψ (32) retains this feature if along the transformations in Eq. (31) in the Hamiltonian (29), the wavefunction changes according to the local phase transformation ψ → ψ′ = eiλψ. (33) In our example at hand, this freedom allows us to select λ in such a way that if at certain instant of time ti the vector potential A ̸= 0 but before we had A = 0, one can still have a Schrödinger equation without vector potential by tuning appropriately the scalar potential. In particular, by selecting λ(x,t) = ℓ(x)Θ(t − ti), (34) we can shift the scalar potential such that the time- dependent equation governing this state never de- velops a vector potential to begin with. Then, by choosing a vector potential A(x,t) = (Ax(x,t), 0, 0) where Ax(x,t) = −Θ(t − ti)∂xθ(x), we observe that the piecewise function ϕF (x,t) = { ϕ(x,t) 0 ≤ t < ti, ψ̄ ( x 4ti+1 ) t ≥ ti. (35) becomes a solution of i∂tϕF (x,t) = [−∂xx + VF (x,t)] ϕF (x,t) = HϕF (x,t). In particular, the function ϕFϵ(x,t) = { ϕϵ(x,t) 0 ≤ t < ti, ψ̄ϵ ( x 4ti+1 ) t ≥ ti, (36) before the freezing time ti is just a time dependent wave packet but for t > ti it becomes a Frozen Bound state In the Continuum satisfying the eigenvalue equa- tion HϕFϵ = εϕFϵ, where ε = ϵ/(4ti + 1)2. In Figure 4 we plot the potential VF (top), the Freezable Bound State in the Continuum ϕFϵ (center) and a scattering state ϕF (bottom) at t = 0.8, t = 1 and t = 1.8, the freezing time is ti = 1, note that after t = 1 neither the potential nor the wavefunctions evolve. 4. Final remarks In this article, we apply a confluent supersymmetric transformation to the standard Step-Potential defined in the whole real axis. The seed solution that we use makes it possible to embed a localized squared integrable state in the continuum spectrum, a BIC. We have provided the system, potential, and states, with time evolution through a point transformation. Nevertheless, we notice that the wrinkles in the po- tential as x → ∞ still localize a BIC at every fixed time. Next, we allow the evolution of the system continue and at a given stopping time ti, we freeze the potential and fix it stationary. Upon exploring the behavior of the BIC with this static potential after the freeze-out time, we surprisingly observe that it does not cor- respond to a solution of the stationary Schrödinger equation, but instead it develops a geometric phase encoded in a vector potential which does not gener- ate any magnetic field. Thus, by gauging out this geometric phase, the resulting state becomes indeed an eigenstate of the frozen Hamiltonian. We call this state a Freezable Bound state In the Continuum. Further examples are being examined under the strategy presented in this work, including vector po- tentials which might be relevant for pseudo-relativistic systems. 60 vol. 62 no. 1/2022 Step-like potential with a freezable bound state in the continuum Figure 4. Behavior of the potential VF (x,t) (top), the FBIC ϕF ϵ(x,t) (center) and the scattering state ϕF (x,t) (bottom) at the times t = 0.8, t = 1, and t = 1.8. The freezing time is ti = 1. The scale of the graph is fixed by V̂ = 5, k = 1, κ = 2, q = √ 2 and ω = 4. Acknowledgements The authors acknowledge Consejo Nacional de Cien- cia y Tecnología (CONACyT-México) under grant FORDECYT-PRONACES/61533/2020. IG and AR ac- knowledge valuable discussions with Juan Angel Casimiro Olivares. References [1] J. von Neuman, E. Wigner. Uber merkwürdige diskrete Eigenwerte. Uber das Verhalten von Eigenwerten bei adiabatischen Prozessen. 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Plenum, New York, NY, 1980. https://cds.cern.ch/record/102017. 62 http://arxiv.org/abs/1703.05282 https://doi.org/10.1098/rspa.1984.0023 https://doi.org/10.1103/PhysRevA.26.729 https://doi.org/10.1137/0143084 https://doi.org/10.1063/1.4903257 https://doi.org/10.1088/1402-4896/ab5cbf https://doi.org/10.1007/978-3-030-55777-5_28 https://doi.org/10.1007/978-3-030-20087-9_11 https://books.google.com.mx/books?id=pJDjvwEACAAJ https://books.google.com.mx/books?id=pJDjvwEACAAJ https://doi.org/10.1016/0370-1573(94)00080-M https://doi.org/10.1063/1.1853203 https://cds.cern.ch/record/101367 https://cds.cern.ch/record/102017 Acta Polytechnica 62(1):56–62, 2022 1 Introduction 2 Supersymmetry and a point transformation 2.1 Confluent supersymmetry 2.2 Point transformation 3 Time dependent step-like potential with a freezable bound state in the continuum 4 Final remarks Acknowledgements References