Siegert Outgoing Boundary Condition

• Siegert Outgoing Boundary Condition is defined by requiring the wave function to be regular at the origin and to exhibit only outgoing behavior, resulting in complex-energy Gamow states.
• It creates a pole condition in scattering theory by linking the zeros of the Jost function to resonances, effectively connecting non-Hermitian Hamiltonians with R-matrix and quantum defect theories.
• Its implementation via finite-radius matching and Bloch operators enables practical numerical analyses in quantum mechanics and electromagnetic resonance problems.

The Siegert outgoing boundary condition is the requirement that a solution of the stationary wave equation be regular in the interaction region and asymptotically contain only outgoing components, with no incoming wave from infinity. In quantum scattering this condition defines Siegert or Gamow states, produces complex energies of the form $E=E_R-i\Gamma/2$, and identifies resonances with poles of the $S$-matrix or Green’s function. In R-matrix theory it is implemented through matching to the logarithmic derivative of an outgoing external solution at a finite channel radius; in open-system formulations it is equivalent to an explicitly non-Hermitian effective Hamiltonian; in multichannel quantum defect theory it appears as the algebraic condition $1-\mathcal{R}_{nn}L_n=0$; and in periodic electromagnetism it becomes an outgoing Rayleigh–Bloch condition in each diffraction channel (Vaandrager et al., 2023, Hatano, 2014, Hategan et al., 2016, Ndangali et al., 2011, Rapedius, 2011).

1. Definition and spectral classification

For the radial Schrödinger equation

$H_\ell u_\ell = E u_\ell,$

with

$H_{\ell} = T_0 + T_{\ell} + V_S(r) + V_C(r),$

the regular solution $\phi_\ell(k,r)$ is defined by regularity at the origin, while its large-$r$ behavior admits the asymptotic decomposition

$$\phi_\ell(k,r) \xrightarrow[r\to\infty]{} \frac{i}{2}\Big[ f_\ell(k)\, I_\ell(kr,\eta) - f_\ell(-k)\, O_\ell(kr,\eta) \Big].$$

Here $f_\ell(k)$ is the single-channel Jost function, and $I_\ell$ and $S$0 are incoming and outgoing Coulomb waves. The Siegert outgoing boundary condition is

$S$1

which, for a short-range potential, means that the asymptotic solution is proportional to the outgoing wave $S$2 and contains no incoming component (Vaandrager et al., 2023).

Siegert states are therefore complex-energy eigenstates characterized by regularity at the origin and purely outgoing behavior at large distance. In one-dimensional scattering the same condition is written by requiring asymptotically only waves propagating away from the scattering region on both sides. This differs from the bound-state condition: bound states are square-integrable and decay at infinity, whereas Siegert states are generally non-normalizable in the usual sense and may grow exponentially in space when $S$3 (Hatano, 2014, Rapedius, 2011).

For $S$4, the classification used in the literature summarized here is:

State type	Location in $S$5-plane	Interpretation
Bound state	$S$6	Square-integrable pole
Virtual state	$S$7	Pole on negative imaginary axis
Resonance	$S$8	Decaying Gamow state
Mirror resonance	$S$9	Reflected lower-half-plane pole

For resonances, writing $1-\mathcal{R}_{nn}L_n=0$0 with $1-\mathcal{R}_{nn}L_n=0$1 implies $1-\mathcal{R}_{nn}L_n=0$2 with $1-\mathcal{R}_{nn}L_n=0$3, and the time dependence

$1-\mathcal{R}_{nn}L_n=0$4

exhibits exponential decay (Vaandrager et al., 2023).

2. Jost-function and scattering-matrix formulation

In the Jost-function formulation, the Siegert outgoing boundary condition is equivalent to the vanishing of the coefficient of the incoming solution. In the convention above, inserting the asymptotic expansion into the outgoing condition yields

$1-\mathcal{R}_{nn}L_n=0$5

Thus Siegert states are precisely the zeros of the Jost function in the complex $1-\mathcal{R}_{nn}L_n=0$6-plane (Vaandrager et al., 2023).

The partial-wave $1-\mathcal{R}_{nn}L_n=0$7-matrix is

$1-\mathcal{R}_{nn}L_n=0$8

Accordingly, zeros of $1-\mathcal{R}_{nn}L_n=0$9 are poles of the $H_\ell u_\ell = E u_\ell,$0-matrix. For real potentials one also has the symmetry

$H_\ell u_\ell = E u_\ell,$1

This links the outgoing-wave formulation, the analytic structure of the Jost function, and the pole structure of scattering observables into a single statement: the Siegert boundary condition is the pole condition for scattering (Vaandrager et al., 2023).

A recurring clarification concerns the distinction between physically meaningful poles and other singularities. The zeros of $H_\ell u_\ell = E u_\ell,$2 define Siegert states and correspond to $H_\ell u_\ell = E u_\ell,$3-matrix poles. By contrast, poles of $H_\ell u_\ell = E u_\ell,$4 are zeros of the $H_\ell u_\ell = E u_\ell,$5-matrix and were described as “false poles,” with no direct physical meaning in the same sense (Vaandrager et al., 2023). In multichannel R-matrix language this same structure is expressed through the denominator factor $H_\ell u_\ell = E u_\ell,$6: the absence of an incoming wave is identified with the vanishing of the corresponding Jost factor, so that the multichannel Siegert condition is written as $H_\ell u_\ell = E u_\ell,$7 (Hategan et al., 2016).

This analytic characterization also dispels a common misconception that resonances require a separate definition unrelated to the ordinary stationary equation. The same stationary equation is used; what changes is the boundary condition. The resulting spectrum is not the self-adjoint spectrum associated with square-integrable states, but the pole spectrum selected by outgoing asymptotics (Hatano, 2014).

3. Finite-radius implementation, Bloch operators, and Siegert pseudostates

In computational R-matrix theory, the exact boundary condition at infinity is replaced by an outgoing condition at a finite channel radius $H_\ell u_\ell = E u_\ell,$8,

$H_\ell u_\ell = E u_\ell,$9

under the assumption that the potential is negligible for $H_{\ell} = T_0 + T_{\ell} + V_S(r) + V_C(r),$0. Space is split into an internal region $H_{\ell} = T_0 + T_{\ell} + V_S(r) + V_C(r),$1, where the wave function is expanded on a finite square-integrable basis, and an external region $H_{\ell} = T_0 + T_{\ell} + V_S(r) + V_C(r),$2, where the solution is matched to its asymptotic form (Vaandrager et al., 2023).

The Bloch operator

$H_{\ell} = T_0 + T_{\ell} + V_S(r) + V_C(r),$3

allows one to impose a prescribed logarithmic derivative at the boundary. The R-matrix is then

$H_{\ell} = T_0 + T_{\ell} + V_S(r) + V_C(r),$4

and matching at $H_{\ell} = T_0 + T_{\ell} + V_S(r) + V_C(r),$5 gives

$H_{\ell} = T_0 + T_{\ell} + V_S(r) + V_C(r),$6

with

$H_{\ell} = T_0 + T_{\ell} + V_S(r) + V_C(r),$7

Choosing $H_{\ell} = T_0 + T_{\ell} + V_S(r) + V_C(r),$8 imposes the outgoing external logarithmic derivative at the channel radius and is the R-matrix realization of the Siegert outgoing boundary condition at $H_{\ell} = T_0 + T_{\ell} + V_S(r) + V_C(r),$9 (Vaandrager et al., 2023).

This finite-radius construction changes the analytic structure of the problem. For the exact, non-truncated potential, the Jost function may possess both zeros and poles. After truncation, the numerically calculated Jost function no longer contains the exact Jost poles; instead, a discrete set of zeros appears in their vicinity. These additional zeros are the Siegert pseudostates. They satisfy the same outgoing boundary condition at the truncation radius, but they do not have a direct physical counterpart in the full non-truncated problem (Vaandrager et al., 2023).

The computed behavior described for supersymmetry-generated test potentials is specific. For an $\phi_\ell(k,r)$0-wave potential with exact Jost function

$\phi_\ell(k,r)$1

the zero at $\phi_\ell(k,r)$2 is reproduced accurately, while the pole at $\phi_\ell(k,r)$3 is replaced by a horizontal line of equally spaced Jost zeros. As the radius $\phi_\ell(k,r)$4 increases, the number of pseudostates in a given $\phi_\ell(k,r)$5 interval increases and their spacing decreases. The same phenomenon appears for the resonance-bearing $\phi_\ell(k,r)$6- and $\phi_\ell(k,r)$7-wave test potentials, with good agreement for the resonance zero and pseudostate structures near the nearest missing pole (Vaandrager et al., 2023).

4. Open quantum systems and effective non-Hermitian Hamiltonians

In open quantum systems, the Siegert boundary condition can be imposed directly on the full Schrödinger equation or encoded through an effective Hamiltonian obtained by eliminating continuum degrees of freedom. For the T-type quantum dot model, consisting of a dot state side-coupled to an infinite one-dimensional lead, imposing outgoing asymptotics on the lead states,

$\phi_\ell(k,r)$8

together with the lead dispersion relation

$\phi_\ell(k,r)$9

yields the discrete-lattice outgoing relation

$r$0

The infinite-dimensional problem then collapses to the finite-dimensional eigenvalue problem

$r$1

with

$r$2

The secular equation $r$3 gives the resonant spectrum (Hatano, 2014).

The Feshbach projection method produces exactly the same matrix,

$r$4

for the same model. The Siegert-boundary approach and the effective non-Hermitian Hamiltonian approach are therefore algebraically equivalent in this setting. The continuum wave function reconstructed from the lead Green’s function,

$r$5

has the same outgoing form as the one obtained by imposing the Siegert condition directly (Hatano, 2014).

This equivalence is tied to the non-Hermitian character of resonant problems. The tight-binding Hamiltonian is Hermitian in the Hilbert space spanned by bound and scattering states, but resonant Siegert states lie outside that Hilbert space because their amplitudes diverge exponentially as $r$6 for lower-half-plane $r$7. In that generalized setting the full Hamiltonian is implicitly non-Hermitian, while $r$8 makes the same non-Hermiticity explicit in a contracted space (Hatano, 2014). A common misconception is therefore that non-Hermiticity is introduced artificially by projection; the model calculations show that it is already implicit in the outgoing-state extension of the full problem.

5. Multichannel R-matrix theory and quantum defect theory

In the Bloch–Lane–Robson formalism, the Siegert condition is expressed as a matching relation between the internal R-matrix and the external outgoing logarithmic derivative. The Bloch operator acts at the channel radius and enforces the channel boundary condition. Projecting onto channels leads to the determinant condition

$r$9

which gives the complex pole energies of the Siegert states (Hategan et al., 2016).

For a selected channel $$\phi_\ell(k,r) \xrightarrow[r\to\infty]{} \frac{i}{2}\Big[ f_\ell(k)\, I_\ell(kr,\eta) - f_\ell(-k)\, O_\ell(kr,\eta) \Big].$$0 embedded in a multichannel system, couplings to the remaining channels are absorbed into a reduced R-matrix element $$\phi_\ell(k,r) \xrightarrow[r\to\infty]{} \frac{i}{2}\Big[ f_\ell(k)\, I_\ell(kr,\eta) - f_\ell(-k)\, O_\ell(kr,\eta) \Big].$$1, and the single-channel Siegert equation becomes

$$\phi_\ell(k,r) \xrightarrow[r\to\infty]{} \frac{i}{2}\Big[ f_\ell(k)\, I_\ell(kr,\eta) - f_\ell(-k)\, O_\ell(kr,\eta) \Big].$$2

In the one-channel limit this reduces to

$$\phi_\ell(k,r) \xrightarrow[r\to\infty]{} \frac{i}{2}\Big[ f_\ell(k)\, I_\ell(kr,\eta) - f_\ell(-k)\, O_\ell(kr,\eta) \Big].$$3

This point is emphasized explicitly: a Siegert state is not described by a pole of the Wigner R-matrix itself, but by the equation $$\phi_\ell(k,r) \xrightarrow[r\to\infty]{} \frac{i}{2}\Big[ f_\ell(k)\, I_\ell(kr,\eta) - f_\ell(-k)\, O_\ell(kr,\eta) \Big].$$4, or by its reduced multichannel form (Hategan et al., 2016).

The same equation covers both bound and quasistationary states. Below threshold, the logarithmic derivative is real and reduces to the shift function,

$$\phi_\ell(k,r) \xrightarrow[r\to\infty]{} \frac{i}{2}\Big[ f_\ell(k)\, I_\ell(kr,\eta) - f_\ell(-k)\, O_\ell(kr,\eta) \Big].$$5

so the condition gives bound states. Above threshold, for an outgoing external solution, $$\phi_\ell(k,r) \xrightarrow[r\to\infty]{} \frac{i}{2}\Big[ f_\ell(k)\, I_\ell(kr,\eta) - f_\ell(-k)\, O_\ell(kr,\eta) \Big].$$6 is complex and the same equation produces complex roots $$\phi_\ell(k,r) \xrightarrow[r\to\infty]{} \frac{i}{2}\Big[ f_\ell(k)\, I_\ell(kr,\eta) - f_\ell(-k)\, O_\ell(kr,\eta) \Big].$$7, ანუ quasistationary Siegert poles (Hategan et al., 2016).

Within quantum defect theory, this structure becomes especially explicit for an attractive Coulomb channel below threshold:

$$\phi_\ell(k,r) \xrightarrow[r\to\infty]{} \frac{i}{2}\Big[ f_\ell(k)\, I_\ell(kr,\eta) - f_\ell(-k)\, O_\ell(kr,\eta) \Big].$$8

With a one-channel R-matrix element $$\phi_\ell(k,r) \xrightarrow[r\to\infty]{} \frac{i}{2}\Big[ f_\ell(k)\, I_\ell(kr,\eta) - f_\ell(-k)\, O_\ell(kr,\eta) \Big].$$9, the level condition yields

$f_\ell(k)$0

Above threshold, the scattering phase shift satisfies

$f_\ell(k)$1

which tends to $f_\ell(k)$2 at zero energy. Comparison gives Seaton’s theorem,

$f_\ell(k)$3

In multichannel problems the reduced R-matrix becomes complex, and the same Siegert equation yields a complex quantum defect and channel resonances. The pole factor $f_\ell(k)$4 then appears directly in the multichannel collision matrix (Hategan et al., 2016).

6. Electromagnetic analogues and resonance calculations

The outgoing-boundary construction extends beyond quantum mechanics. For TE-polarized electromagnetic scattering by a periodic dielectric structure, the time-harmonic field satisfies

$f_\ell(k)$5

together with Bloch periodicity

$f_\ell(k)$6

Using the outgoing Helmholtz Green’s function $f_\ell(k)$7 and its periodic Rayleigh–Bloch expansion, electromagnetic Siegert states are defined as poles $f_\ell(k)$8 of the meromorphically continued resolvent, satisfying

$f_\ell(k)$9

and the outgoing asymptotics

$I_\ell$0

This is the electromagnetic form of the Siegert outgoing boundary condition: in each open diffraction channel only the outgoing Rayleigh–Bloch wave is present, while closed channels are exponentially decaying (Ndangali et al., 2011).

For these periodic structures, the outgoing condition defines a non-selfadjoint resonance problem with poles in the lower half-plane, supports the analytic continuation of resonant states as a function of a coupling parameter $I_\ell$1, and describes bound states in the radiation continuum when $I_\ell$2 while the state remains square-integrable in the strip. Near such a bound state in the continuum, the field expansion at resonance contains the factor $I_\ell$3, so regions within the structure can exhibit arbitrarily large near-field amplification as $I_\ell$4 by tuning $I_\ell$5 (Ndangali et al., 2011).

A more elementary computational realization appears in the Siegert approximation method for one-dimensional tunneling decay. For a finite-range potential, the resonance state satisfies

$I_\ell$6

with complex energy $I_\ell$7. For narrow resonances, the real part $I_\ell$8 is approximated by the energy $I_\ell$9 of a transmission resonance, and the width is obtained from flux loss through the continuity equation. For symmetric barriers this gives

$S$00

while for asymmetric barriers the outgoing fluxes on both sides enter explicitly. This formulation is intended for analytical and numerical calculations of complex resonances in both the linear and nonlinear Schrödinger equation (Rapedius, 2011).

Taken together, these developments show that the Siegert outgoing boundary condition is not a specialized auxiliary device but a general resonance principle. It selects homogeneous solutions with no incoming component, converts scattering problems into non-selfadjoint eigenvalue problems, and provides a common language for Jost-function zeros, effective Hamiltonians, multichannel collision-matrix poles, quantum-defect relations, finite-radius R-matrix computations, pseudostate structures produced by truncation, and outgoing radiative modes in periodic electromagnetism (Vaandrager et al., 2023, Hatano, 2014, Hategan et al., 2016, Ndangali et al., 2011, Rapedius, 2011).