1.Introduction

Bound states in the continuum (BICs) are a wave phenomenon that refers to states whose wave functions remain localized among a continuum of radiating waves with positive energies. This phenomenon of localized waves has been identified in atomic and molecular physics, optics, acoustics and other fields ^1-5. One review concerning this type of bound states was recently published ⁶.

BICs were first proposed by von Neumann and Wigner in 1929 ⁷, since then they have stimulated many experimental and theoretical research in various branches of physics. The first experimental evidence in this direction was reported by F. Capasso et al., they found an electronic bound state with energy greater than the barrier height in a semiconductor superlattice ⁸. More recently, BICs have been observed in optical waveguide arrays ^9-11. M. Koirala et al. found a class of critical states embedded in the continuum in a one-dimensional arrangement of waveguides ¹², and M. I. Molina et al. observed surface bound states in the continuum in a linear optical band of a discrete lattice ¹³.

In quantum physics, there exists a particular type of potentials that support BICs and are solely defined by their oscillatory asymptotic behaviour. Typically, a radial potential with an oscillatory asymptotic behaviour that falls to zero slowly,

supports a BIC with energy E=b2/4, if the parameters a and b satisfy the relation |a|>|b| ¹⁴. Potentials of the form (1) are known as von Neumann-Wigner potentials ⁷, and have been studied with methods of supersymmetric quantum mechanics (SUSY QM) ¹⁵, Darboux transformation ¹⁶ and the inverse scattering method (Gel’fand-Levitan equation) ¹⁷. Further development in the techniques for this purpose was made in the following years. A. A. Andrianov and A. V. Sokolov studied a non-Hermitian Hamiltonian in the whole axis with complex von Neumann-Wigner type potential obtained with SUSY methods, and found the normalized eigenfunction and associated eigenfunction and their orthogonal relations; as the Hamiltonian is non-Hermitian and self-orthogonality of the normalized eigenfunction occurs, they related the BIC to an exceptional point ¹⁸. A. Khelashvili and N. Kiknadze established a one to one correspondence between the decay law in von Neumann-Wigner type potentials and the asymptotic behaviour of the wave functions representing the bound states ¹⁹, while T. A. Weber and D. L. Pursey showed that by truncating a von Neumann-Wigner potential the BIC manifest itself as a resonance . E. Hernández et al. studied a particular spectral singularity produced by the coalescence of two BICs ²¹.

In this paper, in an attempt to develop further results in this field, particularly concerning the nature of the eigenfunction corresponding to a BIC, we study the Jost solutions of a radial Hamiltonian with a real von Neumann-Wigner type potential and obtain the Jordan cycle of length two of generalized eigenfunctions, one of which is the BIC, and their respective time evolution. Although the exact model presented in this paper may not be readily applied to a real physical system, it may prove useful in determining further properties of more realistic Hamiltonians. The paper is organized as follows: In Sec. 2, by performing a two times iterated and completely degenerated Darboux transformation with initial Hamiltonian the free particle, a von Neumann-Wigner type potential is obtained supporting a BIC with energy E = q ² . In Sec. 3, the set of Jost solutions normalized to unit flux are explicitly constructed and it is shown that the unnormalized Jost solutions coalesce at k = q to give rise to a Jordan chain of rank two. In Sec. 4, the generalized eigenfunctions are found from the pole decomposition of the normalized Jost solutions and the corresponding Jordan chain is established. The time evolution of the generalized eigenfunctions is described in Sec. 5. In Sec. 6, the scattering matrix is obtained as a function of k and the cross section is calculated. In Sec. 7, a summary of the results and conclusions is presented.

2.Von Neumann-Wigner potentials and Darboux transformations

A particular class of completely degenerated Darboux transformations, with initial Hamiltonian the free particle, generates new potentials of von Neumann-Wigner type supporting any number of bound states embedded in the continuum (16). These transformations are obtained from Crum’s theorem, a generalization of the Darboux transformation, which states that the function

is an eigenfunction of the Hamiltonian

with eigenvalue E, and the potential U _n is given by

The auxiliary or transformation functions ϕi are eigenfunctions of the initial Hamiltonian H ₀ with eigenvalues E _i , i=1,…,n, and ϕE is also an eigenfunction of H ₀ with eigenvalue E. In the above expressions W(ϕ1,ϕ2,…,ϕn) is the Wronskian of the eigenfunctions ϕi ^22,23.

The completely degenerated case occurs when we take all E _i energies close to each other, i.e.Ei→E¯+ϵi with ϵi≪1, and later taking the limit ϵi→0. After this procedure, all eigenfunctions ϕi coalesce in ϕ and the respective eigenvalues E 𝑖 in E¯. From Crum’s theorem it follows that the function

is an eigenfunction of the Hamiltonian H _n , with eigenvalue E, and the potential U _n is substituted by V _n given as

The partial derivative is with respect to the energy E¯.

From here on, we consider as the initial Hamiltonian the free particle in spherical coordinates with potential V ₀ = 0, and auxiliary eigenfunction

with eigenvalue E¯=q2 and δ(q) an arbitrary phase shift which sets the parameters of the system. In this case, differentiation with respect to the energy E¯ is equivalent to differentiation with respect to wave number q in expressions (5) and (6), and in what follows we derive with respect to q.

Let us consider now the simplest case that produces a von Neumann-Wigner potential: n = 2. The Hamiltonian H ₂ is given in (3), with the potential V ₂ obtained from (6) as

and calculating the Wronskian W2≡W(ϕ,∂ϕ) with (7) we obtain:

where

and as a convenient notation the prime in δ'(q) indicates the first derivative of δ(q).

Using (9) in (8) the potential obtained is:

with asymptotic behaviour given by:

and comparing it with (1) we see that it is a potential of von Neumann-Wigner type and, given that a=-4q and b=2q, supports a bound state in the continuum with energy E=q2.

A requirement for the validity of the Darboux transformation is the absence of singularities in the new potential not present in the initial potential. From (12) we see that the singularities of V ₂ occur at the zeros of W ₂ . The Wronskian W ₂ as a function of r grows linearly with a negative slope, and has only one real zero.

As the Hamiltonian H ₂ is defined in the positive semi-axis, we set the condition

which locates the real zero for negative r and therefore setting the potential V ₂ an analytical function of r in the physical space. Condition (14) provides a differential relation for the phase shift δ(q) as a function of q. Evaluating (9) in r = 0 we get,

W2(0)=12sin2δ(q)-2qdδ(q)dq,

and defining t(q)=tanδ(q) we can write condition (14) as

This relation is readily solved by considering a positive constant β to eliminate the inequality, and solving the resulting differential equation yields the result

with α and β real constants, and β > 0.

The potential V ₂ in (12) is now determined by the parameters α and β. Figure 1 shows the potential V ₂ as a function of r for a given choice of parameters.

FIGURE 1 Potential V₂ (continuous line) and density probability of the eigenfunction of the bound state embedded in the continuum (dashed line) for parameters values α = 2, β = 3 and q = 1. 

3. Jost solutions of the Hamiltonian H₂

The Schrödinger equation for the scattering problem of potential V ₂ is

for positive energies E=k2, where f±(k,r) are the two linearly independent unnormalized Jost solutions of Hamiltonian H₂, which behave asymptotically as outgoing and incoming spherical waves ²⁴. They are obtained from the Darboux transformation in (5) with n = 2 and ϕE are the free particle wave solutions e±ikr with eigenvalue E=k2:

f±(k,r)=W(ϕ,∂ϕ,e±ikr)W(ϕ,∂ϕ).

As the last column of the Wronskian in the numerator is proportional to e±ikr, the above expression can be written as

with w±(k,r) a complex function of real arguments k and r defined as

Using the expressions (9) and (19) in (18), we can write the unnormalized Jost solutions in explicit form as

and from the linear behaviour of γ in r for large values of r we find the asymptotic behaviour:

Hence, to obtain the two Jost solutions of H ₂ normalized to unit flux at infinity, we must divide (20) by the factor -(k2-q2),

and the Jost solutions exhibit a simple pole at k = q.

In Appendix A, it is shown that the previous results can be obtained in an alternative way using the confluent case of the intertwining method of SUSY QM.

Hamiltonian H ₂ has its spectrum defined for positive energies and, for each spectral point, there exist two linearly independent eigenfunctions. However, at the point k = q the two unnormalized Jost solutions coalesce. The Wronskian of the unnormalized Jost solutions can be obtained with the asymptotic behaviour given in (21):

which vanishes at the point k = q and therefore the unnormalized Jost solutions are linearly dependent at that spectral point. An eigenfunction is lost and the basis of linearly independent eigenfunctions of the Hamiltonian H ₂ appears to be incomplete. In its place a Jordan chain of two generalized eigenfunctions is formed. The subspace spanned by the generalized eigenfunctions is in the domain of H ₂ for E=q2.

4. Poles of the Jost solutions and Jordan chain

To obtain the generalized eigenfunctions, we rewrite the normalized Jost solution as a decomposition of its pole in a sum of singular and regular parts. From (22) and using (20) we can write

which can also be written as

with the following defined functions:

and

where ψB(q,r) is a square integrable function of r, while ψR±(k,r) are analytic functions of k = q, and behave asymptotically in 𝑟 as outgoing and incoming spherical waves. The explicit form of ψR±(k,r) obtained form (27) is

and its expression in k = q is given by

After some algebraic manipulation the previous expression can be written as

and ψA(q,r), the associated eigenfunction to ψB(q,r), is defined as

which is a bounded, non-normalizable function.

As eigenfunctions of the Hamiltonian H ₂ , the normalized Jost solutions satisfy the time independent Schrödinger equation

for all k, except k = q where F±(k,r) is not defined. To explore the limit k→q we substitute (25) in (32):

Multiplying (33) by (k2-q2) and taking the limit k→q we obtain:

the square integrable solution ψB(q,r) representing the bound state embedded in the continuum is an eigenfunction of H₂ with energy E=q2. Figure 1 shows |ψB(q,r)|2 as a function of 𝑟.

Using the result (34) in (33) and taking the limit k→q we get

However, as the additive term proportional to ψB(q,r) in (30) satisfies (34) it can be omitted and we can write

with ψA(q,r) the associated eigenfunction. Thus, ψB(q,r) and ψA(q,r) are generalized eigenfunctions of the Hamiltonian H ₂ , and they form a Jordan chain of rank two for E=q2. The Jordan chain is the result of the coalescence of two energy levels (25), for a direct proof see Appendix B.

5. Pseudo-unitary time evolution of the generalized eigenfunctions

The two generalized eigenfunctions ψB(q,r) and ψA(q,r) belong to the same spectral point, E=q2; in consequence, they evolve in time together. Hence, it should be convenient to introduce a matrix notation to deal with the two together. From (34) and (36) we can write them as

where

is the two component vector of the doublet, and

is the 2 x 2 energy matrix.

The time dependent generalized eigenfunctions are

where Uq,t is the 2 x 2 matrix of time dependent coefficients and gives the time evolution of the wave function Ψ(q,r).

Substitution of Ψ(r,t) in the time dependent Schrödinger equation gives the following set of coupled equations written in matrix form

Making abstraction of Ψ(q,r), we obtain

Integrating equation (42) we get

writing Hq in explicit form in (43), and computing the exponential, we obtain

Substitution of the expression (44) in (40) gives the evolution in time of the two generalized components of the doublet Ψ(r,t):

and

The component ψB(r,t), describing the time evolution of the bound state eigenfunction embedded in the continuum, exhibits a unitary evolution in time, while the component ψA(r,t) has a linear growth with time. Therefore, the wave function Ψ(r,t) grows linearly with time t. This type of behaviour has been found by Longhi et al. in a non-Hermitian Hamiltonian.

The above result is a direct consequence of the pseudo-Hermiticiy of Hamiltonian H₂ at E=q2, represented as the matrix Hq in (39). An operator is pseudo-Hermitian if there exists a linear, invertible, Hermitian operator η such that ²⁷

The general form of η satisfying (47) for Hq in (39) is

with a and b any real parameters. In turn, U(q,t) is pseudo-unitary because Hamiltonian H₂ at E=q2 is a pseudo-Hermitian operator. An operator is pseudo-unitary if its inverse and its adjoint satisfy the transformation ²⁷

as it may be verified by substitution of (44) and (48).

Therefore, the generalized eigenfunctions ψB(q,r) and ψA(q,r) in the Jordan cycle have a pseudo-unitary time evolution.

6. Cross section and scattering matrix

In this section, we will show that the scattering matrix has no singularities at the corresponding spectral point k = q, contrary to the case of conventional bound states with negative energies.

The scattering solution is defined as ²⁴:

where F±(k,r) are the Jost solutions in (22) and S(k) is the scattering matrix defined as

and F+(k,0) is the Jost function.

From (20) and (22) evaluated at r = 0 we obtain F±(k,0) and a direct substitution in (51) gives:

which can be written as

where Δ(k) is the phase shift and is given by

Taking the limit k→q we get Δ(q)=δ. Hence, the scattering matrix evaluated at k = q is finite and equal to

However, there can be singularities of S(k) for different values of k. The singularities of S(k) are the zeros of the Jost function F+(k0,0)=0:

which is a quadratic equation for k₀. In terms of α and β we have

The zeros in the fourth quadrant of the complex k-plane near the real axis may be resonances, while zeros on the imaginary positive axis correspond to bound states with negative energy.

The cross section is defined as

and with the explicit form of Δ(k) in (54) we are able to observe its behaviour and dependence on k. Figure 2 shows σ(k) as a function of k and for the chosen values of parameters α and β a resonance shape is found, belonging to the value k0=2/3-i/3, far from the value k = q = 1 of the BIC. The BIC has no effect in the cross section, only when the system is perturbed the BIC may manifest itself as a resonance ²⁰.

FIGURE 2 Cross section σ(k) for parameters values α=2,β=3 and q = 1. The Breit-Wigner peak in σ(k) corresponds to a resonance, far from the value k = q = 1 of the BIC, located at k0=23-i/3. 

7. Summary and conclusions

In this work, we presented a von Neumann-Wigner type potential V ₂ constructed by means of a two times iterated and completely degenerated Darboux transformation. The Hamiltonian H ₂ and the free particle Hamiltonian H₀ are isospectral, and to each point in their continuous spectrum corresponds two linearly independent Jost solutions, which behave at infinity as incoming and outgoing waves. However, we have shown that in the continuous spectrum of H ₂ there is a singular point, with energy E=q2, such that the two unnormalized Jost solutions are linearly dependent and coalesce to give rise to a Jordan chain of rank two of generalized eigenfunctions and a Jordan block representation of the Hamiltonian H ₂ . The normalized Jost solutions have a simple pole at wave number k = q and after a pole decomposition the Jordan chain and respective generalized eigenfunctions are obtained. One of the generalized eigenfunctions is normalizable and corresponds to the BIC, the other is a bounded, non-normalizable function associated with the BIC. Finally, we obtained the time evolution of the generalized eigenfunctions: the BIC has a unitary time evolution, while the associated eigenfunction has a linear growth in time. Together, they exhibit a pseudo-unitary behaviour characteristic of a pseudo-Hermitian system. Finally, we have shown that the BIC is not associated with a singularity of the scattering matrix S(k) and, as a result, the BIC is not observed in the cross section σ(k).