Breaking of PT-symmetry in bounded and unbounded scattering systems

PT-symmetric scattering systems with balanced gain and loss can undergo a symmetry-breaking transition in which the eigenvalues of the non-unitary scattering matrix change their phase shifts from real to complex values. We relate the PT-symmetry breaking points of such an unbounded scattering system to those of underlying bounded systems. In particular, we show how the PT-thresholds in the scattering matrix of the unbounded system translate into analogous transitions in the Robin boundary conditions of the corresponding bounded systems. Based on this relation, we argue and then confirm that the PT-transitions in the scattering matrix are, under very general conditions, entirely insensitive to a variable coupling strength between the bounded region and the unbounded asymptotic region, a result that can be tested experimentally and visualized using the concept of Smith charts.

not map onto a bounded Schrödinger problem [20,[28][29][30][31][32][33], it was recently pointed out [20,34] that the relevant PT -transition occurs in the eigenvalues of the scattering (S) matrix, which relates incoming to outgoing flux channels. Specifically, it was shown that when the S matrix undergoes PT -symmetry breaking, its eigenvalues go from unimodular values, lying on the complex unit circle, to inverse-conjugate pairs of complex values. This PT -transition in the scattering problem thus raises the question of what relation it might have to a corresponding transition in bounded Hamiltonian systems of the type studied previously. A recent study of conservation laws in such PT -scattering systems suggests that a connection to the finite system with Dirichlet boundary conditions exists [34].
In this article, we derive an explicit relationship between the PT -breaking transitions of bounded and unbounded systems. Specifically, we show that the regions of the phase diagram in which an unbounded system's S matrix has either unimodular eigenvalues (PT -unbroken phase) or eigenvalues with a modulus different from one (PTbroken phase) correspond exactly to the regions in which a whole family of associated bounded systems possess specific PT -symmetric or non-PT -symmetric Robin-type boundary conditions (BC), respectively. This entry will also provide an interesting way to introduce Smith charts, familiar to microwave practitioners, into our description. The importance of Robin BC for relating the bounded and unbounded problems was noted earlier by Smilansky et al. [35,36], Robin BC in the context of PT -symmetry were studied by Krejčiřík et al. [37][38][39][40]; the correspondence which we demonstrate in the following has, however, not been shown previously. Although our proof will be presented in the context of the electromagnetic Helmholtz equation, it is also applicable to the Schrödinger equation and to other linear wave equations. We also find the surprising, but closely related result that the transition line at which the S matrix undergoes PT -breaking is unchanged by adding arbitrary P-symmetric layers ("mirrors") outside the original scatterer. In other words, the transition is determined entirely by the PT -symmetric inner portion of the optical structure.
Consider, as a starting point, the one-dimensional (1D) PT -symmetric scatterer shown in Fig. 1(a), which is similar to those used in previous studies [20,21]. This scattering system consists of a cavity, to which two semiinfinite leads of uniform real refractive index are attached on the left and right. The cavity itself consists of three layers, one with loss (L), one with gain (G), and an air gap in the middle. Its refractive index satisfies the PTsymmetry relation n(x) = n * (Px), where in this case the parity operator P performs a spatial reflection about x = L/2. We take the refractive index to be n = n 0 + ig (n = n 0 − ig) in the loss (gain) regions, and n = 1 in the air gap and in the leads. The real parameter g controls the magnitude of the balanced gain and loss. The harmonic electric field E transverse to the propagation axis satisfies L G FIG. 1. (a) One-dimensional scattering system featuring a layer of loss (blue) and gain (red), separated by an air gap (the real part of the refractive index n0 = 1 throughout). A variable coupling strength between the cavity and the asymptotic region can be introduced by two semi-transparent mirrors (green). (b) Regions of unbroken (white) and broken (grey) PT -symmetry for the scattering matrix of the system displayed in (a) (P denotes spatial reflection at x = L/2). The "exceptional line" (black) at the boundary between these two regions contains all the PT -breaking points in the scattering matrix eigenvalues occurring at specific values of kL and strength of loss and gain g. Embedding into this plot also the (real) eigenvalues below threshold of the Helmholtz equation in the bounded domain with Dirichlet (blue) or Neumann (purple) BC imposed at x = 0, L, we find that the symmetry breaking points of these eigenvalues (see colored dots) lie in close vicinity but not exactly on the exceptional line of the S-matrix eigenvalues (see inset).
the Helmholtz equation with k = ω/c; solutions to this unbounded problem exist for all k ∈ R. In the leads, E can be expanded into incoming and outgoing waves: where L is the length of the cavity. The wave amplitudes { u, v} are related by the scattering matrix, u = S v. As noted, one can show [20,32] that either the eigenvalues of S are unimodular, in which case each eigenvector v j is PT -symmetric (P v * j ∝ v j ), or the eigenvalues are inverse conjugates, in which case the eigenvectors break PT -symmetry (P v * 1 ∝ v 2 ). The PT -phase diagram (i.e., the kL − g parameter space) of the scatterer is shown in Fig. 1(b). The PT -symmetric part of the diagram is shown in white, and the PT -broken part is shown in grey. These two regions are separated by an exceptional line (black), which marks the threshold of the PT -breaking transition; everywhere on this line, the S matrix is defective (its eigenvectors are linearly dependent and it cannot be diagonalized) [9,20]. Strictly speaking, due to dispersion, exact PT -symmetry cannot hold as k is varied over an interval [41], hence we imagine varying the parameters L, g to probe the PT -breaking transition [42].
Our goal is to relate the physics of the unbounded scattering problem to that of a bounded system with the same complex refractive index, n(x), defined in the finite domain x ∈ [0, L], and appropriate BC at x = 0, L. In case this BC is itself PT -symmetric, the corresponding PTsymmetric bounded problem will undergo a transition in which its discrete eigenvalues k m will go from being real, to being complex conjugate pairs. The work of Ref. [34] looked at the case of Dirichlet BCs, and found a rough correspondence, for a given value of g, between the points at which the k m are real and the intervals over which the PT -symmetry of S-matrix in the unbounded problem is unbroken. In Fig. 1(b) we plot the trajectories of the real eigenvalues k m as a function of g for both the case of Dirichlet and of Neumann BC; as is well-known, pairs of such eigenvalues eventually meet at exceptional points of the bounded problem as the PT -transition occurs (we don't plot them after they become complex). While the exceptional points of these specific bounded problems occur near the exceptional lines of the unbounded problem, close inspection shows that they do not typically occur on the exceptional lines. We now show that a more subtle relationship exists between the bounded and the unbounded problem.
Consider an incident pair of waves v corresponding to an eigenvector of the S matrix, denoted v i , with complex eigenvalue σ i : Let ψ i (x) denote the corresponding field, obtained by inserting the coefficients of v i and u i into Eq. (2) and by solving Eq. (1). At the boundaries x = 0, L, this scattering eigenfunction must obey We now ask what choice of BC a bounded system must have in order to possess an eigenfrequency k m which is equal to the value of k chosen for the unbounded system. Note that for this BC also the eigenfunction E m (x) coincides with the scattering field ψ i (x) within the scatterer, x ∈ [0, L]. It is easy to see that Eqs. (4)-(5) define Robin BCs, of the form where the Robin parameter λ is related to σ i by We have thus found, by construction, an exact mapping between one of the scattering eigenstates of the unbounded system, at any arbitrary wavevector k and gain-loss parameter g, and a particular eigenstate of a specific bounded system with Robin BC whose eigenfrequency equals k at the same value of g. Since the S matrix depends parametrically on k and g, the relevant bounded system has a different BC at each point in the phase diagram and a real eigenfrequency k m = k. Furthermore, the construction also works in the opposite direction: given any bounded problem obeying the BC (6), and a choice of any one of its real eigenfrequencies k m , the corresponding scattering matrix S(k m , g) must have an eigenvalue given by the right hand equation in (7). This mapping is completely general for any 1D Helmholtz equation and n(x); we did not use Hermiticity or PT -symmetry in deriving Eq. (7). A transformation very similar to Eq. (7) is, in fact, used extensively in microwave engineering and known there under the name of "Smith charts" [43,44]. This concept maps the normalized impedance z of a one-port system (one input, one output port) to its reflection coefficient ρ through a Möbius transformation, ρ = (z − 1)/(z + 1). Rewriting (7) as −σ i = (iλ/k − 1)/(iλ/k + 1), we immediately see the equivalence of the two transformations if we interpret −σ i as ρ (the minus sign is due to our convention for the S matrix) and iλ/k as z. Note, however, that differently from the conventional concept of Smith charts, our approach from above applies to systems with an arbitrary number of ports. What we thus find is that a subdivision of a multi-port scattering problem into independent scattering matrix eigenchannels allows us to assign to each of these channels its own single-port Möbius transformation and with it a corresponding Smith chart. We speculate that this approach might also find applications in multi-port microwave scattering problems.
For the PT -symmetric case considered here, the mapping (7) has the following immediate implications: In  7). Only the real eigenvalues below the PTthreshold are shown as a function of the gain/loss-parameter g. (Altogether ten different values of the boundary parameter λ ∈ [−500, +500] were used.) The envelope of the eigenvalues in this bounded system corresponds exactly to the exceptional line (black), which separates the unbroken (white) from the broken (grey) PT -phase in the corresponding unbounded scattering problem. Results are shown for (a) a two-layer setup as displayed in Fig.1(a) as well as for (b) a more complicated system featuring altogether six layers of loss and gain (see insets).
the PT -symmetric phase of the S matrix, both σ i 's are unimodular, and the λ i 's are real. In this case the Möbius transformation in Eq. (7) maps values of σ on the complex unit circle onto the entire real λ axis and vice versa. With real λ i , the Robin BC are Hermitian and satisfy P-and T -symmetry separately, even though the heterostructure itself does not. It follows that the union of the trajectories of all real eigenvalues k m in the bounded PT -problem coincide with the unimodular phase of S in the kL−g plane, as the real λ varies from −∞ to ∞ and g varies from zero to ∞. Thus there is no simple correspondence between the PT -transition in scattering and any specific bounded problem; each bounded problem, however, contains information about the PT -phase diagram. This statement is illustrated in Fig. 2(a),(b) in which we vary g up to the transition point of the bounded system for many different real values of λ, and for two different PT -structures of increasing complexity.
Note that the above statement does not imply that the phase boundary of the PT scattering problem (every point of which is an exceptional point of S) coincides with the union of all exceptional points of the bounded problem [see the inset in Fig. 1(b)]. Since the latter points cannot occur in the PT -broken phase of S, they are, however, enclosed by the phase boundary of S. In the PT -broken phase of S the σ i have left the unit circle and the two corresponding λ i form a complex conjugate pair at each point in the kL−g plane. Whereas the BC are then non-Hermitian and non-PT -symmetric, the equivalence between the bounded and the unbounded quantities, k m = k, E m (x) = ψ i (x), still holds and may be visualized using the concept of 3D Smith charts recently introduced in [44]. Note how, in this way, the bounded-unbounded mapping from above provides important information on the boundary between phases where the scattering matrix features eigenvalues on or away from the unit circle, respectively. The more trivial cases are realized for hermitian systems, where the scattering matrix eigenvalues are always on the unit circle, or for systems with only gain or only loss, where these eigenvalues never fall on the unit circle.
The mapping between bounded and unbounded problems suggests a further, previously unknown property of the PT -transition in scattering. This property can be uncovered by observing that adding thin regions of real index of refraction (like a delta-function) to the scattering region symmetrically at each end (which preserves PT -symmetry) does not change the scattering matrix eigenstates inside the mirrors (apart from a global amplitude). Instead, the mirrors just shift the boundary parameter λ of the corresponding bounded problem without mirrors by a real value, λ →λ = λ+µ, µ ∈ R (where µ just depends on the reflectivity of the mirror). Changing the BC in this sense, however, does not change the location of the exceptional line since the PT -symmetric phase of S is the union of all real values of λ as mentioned, from −∞ to ∞. This result can be conveniently visualized with a Smith chart (see Fig. 3), where the shift of λ can be seen to just rotate both scattering matrix eigenvalues on the unit circle, which leaves the gain/loss-strength g at which they coalesce invariant [compare Fig. 3

(a) and (b)].
We thus arrive at the conjecture that many different PTscattering problems have the same PT -phase diagram, if they differ only by the addition of dielectric "mirrors" at the two ends. This conjecture can be proved rigorously in 1D for arbitrary lossless dielectric structures added to the original PT -cavities (see appendix A for this proof which also holds for thick dielectric structures featuring several dielectric layers). We have tested this "mirror theorem" also numerically by adding lossless mirrors [see Fig. 1(a)] to the ends of the six-layer scattering system in Fig. 2(b), and find that its complicated phase boundary is reproduced to within the numerical accuracy of the computation.
The above "mirror theorem" also features an interesting manifestation in the ratio of the incoming (outgoing) amplitudes ξ i ≡ v i,1 /v i,2 (= u i,1 /u i,2 ) of the scattering eigenstates in 1D, whose modulus and phase display a bifurcation at the phase boundary of S, respectively [34]. As we show in appendix A, ξ i is invariant not just at the exceptional line with the addition of lossless symmetric mirrors; it is so everywhere in the phase space, both in the PT -symmetric phase and broken phase. More surprisingly, this property holds even when the symmetric mirrors added are dissipative or amplifying, which clearly violates the global PT -symmetry of the heterostructure. This finding implies that the phase boundary of S is also invariant in this general situation and we find, indeed, that the scattering eigenvalues σ i still meet at exactly the same exceptional line as in the case without such mirrors. For thin non-hermitian mirrors with loss or gain, this situation can again be understood by the shift which these mirrors induce on the Robin BC parameter λ →λ = λ + µ with µ now being complex rather than real as before. As visualized conveniently on a Smith chart, the real part of µ leads again to a rotation of the scattering matrix eigenvalues σ i , but its imaginary part shrinks (expands) the circle below (beyond) the unit circle on which they rotate in the PT -unbroken phase [see Fig. 3(c)]. Both of these operations do, however, leave the critical gain/loss-strength g at which the two scattering matrix eigenvalues σ i coalesce invariant. Since both the scattering eigenvectors and eigenvalues still coalesce at the original exceptional line, we arrive at the very general result that this line is entirely unaffected by the symmetric mirrors, even if they are absorbing or amplifying. In appendix A we provide a rigorous proof of this result even for thick stacks of absorbing or amplifying mirrors and check this result also explicitly numerically (see Fig. 5). The generalization of the mirror theorem to non-hermitian mirrors is particularly important for two reasons: First, it shows that the exceptional points in a PT -symmetric system are not necessarily a result of the global PTsymmetry; they persist even when the symmetry is broken when the absorptive or amplifying mirrors are added. Second, it paves the way for the experimental verification of the "mirror theorem," since in practice the mirrors can never be absolutely loss-free. Our mapping approach to connecting bounded and unbounded scattering problems suggests that some form of the mirror theorem could also hold in higher dimensions; we will now demonstrate a two-dimensional example of this (some general relations that hold for arbitrary 2D PT -symmetric scattering problems are provided in appendix C). Consider the case of a twodimensional PT -symmetric disk of radius R as shown in Fig. 4(a). To evaluate a scattering matrix for such a system, we envision a circular boundary with radius B > R [see Fig. 4(a)] outside of which we define an appropriate scattering basis as the product of normalized incoming (−) and outgoing (+) Hankel functions, H ± n (kr) ≡ H ± n (kr)/H ± n (kB), and normalized trigonometric functions χ n,1 (ϕ) = A 1 sin(nϕ) , χ n,2 (ϕ) = A 2 cos(nϕ). For the infinite dimensional scattering matrix defined in this basis there are typically many eigenvalue pairs which go through a PT -transition, leading to a very complicated phase diagram [a small detail of which is shown in Fig. 4(b)]. Still, we can represent the scattering eigenstates in the bounded domain r < B as the eigenstates of a boundary value problem (with the boundary at r = B). For this purpose we first expand the scattering matrix eigenstates S v i = σ i v i for r > B in the above basis ψ i = n,η v i n,η χ n,η (H − n +σ i H + n ). Making a Robin-ansatz for the boundary conditions of these states ψ i (as in 1D), we find that the corresponding factors λ i appearing here do, in general, not just depend on the eigenvalue σ i but also on the angular position ϕ on the boundary. The resulting angle-dependent Robin boundary condition takes the following form, These coefficients do, however, lose their n-dependence if we choose the boundary of our finite domain in the far-field, i.e., B R. In this case the Hankel-functions can be approximated by n-independent cylindrical waves, H ± n (kr) ≈ e ±ik(r−B) / r/B for r > B and the expressions in Eq. (9) drastically simplify. As a result, the terms λ i in the Robin boundary condition, Eq. (8), are then given by the coefficients Λ i n which are independent of n and thus of ϕ. Neglecting contributions of lower order than r −1/2 we find, which is exactly the same expression, Eq. (7), which we have previously obtained in 1D. In the same way as we have argued in 1D that a mirror placed symmetrically around the PT -system does not change the exceptional line, we can now make the same conjecture for each individual eigenvalue of a 2D scattering matrix evaluated in the far-field. Hence the complicated PT -phase diagrams as in Fig. 4(b) should not change with the addition of concentric mirrors in the far-field. Again we confirm this conjecture by numerical tests involving the structure shown in Fig. 4(a). The mirror theorem indicates that the PT -transition in scattering is quite a subtle phenomenon. If we think of the PT -symmetric scattering region as a resonator, adding (non-absorbing) external mirrors greatly enhances the Q value (i.e., the cavity lifetime) of such a resonator, but apparently has no effect on its phase boundary. The reasoning that having the waves stay in the resonator much longer would allow them to feel the presence of gain and loss more strongly and would thus lead to a PT -transition at smaller g therefore proves incorrect. Our results thus dramatically illustrate that the PT -breaking transition in scattering is not a resonance phenomenon and that it does not depend on quantities like the round-trip gain/loss that are used to estimate the lasing transition. Instead, the PT -transition is sensitive to the coupling of the gain and loss regions with each other, with strong coupling making the transition harder to achieve and weak coupling making it (trivially) easier. Quite on the contrary, the coupling to the external world has no effect at all on the transition, but strongly affects other features of the resonator which are sensitive to higher Q-values. Consider, e.g., those singular points in the broken symmetry phase [20,28,31,32,45,46] at which one eigenvalue of the 1D S-matrix goes to infinity, corresponding to the laser threshold, and the other one goes to zero, corresponding to coherent perfect absorption (CPA) [20]. If one adds highly reflecting dielectric mirrors to a simple low-Q PTresonator as in Fig. 1(a), these singular CPA-Laser points are pulled down almost to the PT -phase boundary which itself, however, stays unchanged. This behavior, details of which are shown in Fig. 6 of appendix B, nicely illustrates that PT -symmetry breaking and the lasing transition are very different phenomena.
In summary, we have uncovered a close link between the phase transitions in the scattering matrix S of an unbounded PT -symmetric system and the corresponding transitions in the underlying bounded systems. The most interesting result which follows from this relation is the fact that under very general conditions the PT -thresholds in the scattering matrix are unchanged by external mirrors which increase the Q values of the scattering system. This prediction should be directly testable in the PTsymmetric scattering experiments that have recently been realized.
We would like to thank S. Burkhardt, R. El-Ganainy, U. Günther, and M. Liertzer for helpful discussions.  In the main text we argue that the exceptional line of the scattering matrix and the incoming (outgoing) amplitude ratio ξ of scattering eigenstates are strictly independent of the coupling strength to the asymptotic regions if symmetric mirrors are added at both boundaries of the 1D scattering system. Here we prove this independence explicitly, whether or not the mirrors are lossless.
The 1D scattering matrix S connecting incoming ( v) with outgoing ( u) coefficients is defined by where r L(R) denote the reflection coefficients for injection from the left (right) and t the transmission amplitude. It is easy to see, that its eigenvalues and the incoming (outgoing) amplitude ratios ξ ≡ v 1 /v 2 (= u 1 /u 2 ) of scattering matrix eigenstates are given by respectively. Along the exceptional line, the two eigenvalues and eigenvectors coalesce, i.e., the square root on the right-hand sides of (A2),(A3) vanish. Thus, the scattering coefficients satisfy [20] Y ≡ at the PT -phase transition points, and we note that r L ,r R are always π out-of-phase with t in a PT -symmetric heterostructure. Note that Y is the sum of the two offdiagonal elements in the corresponding transfer matrix M that connects the coefficients corresponding to the asymptotic regions to the right and to the left of the heterostructure. Accordingly, M is defined by We now vary the coupling of the PT -symmetric heterostructure to the unbounded asymptotic regions by introducing two symmetric mirrors of arbitrary complexity and extension (one on each side). The refractive indices n L , n R of the left and right mirrors satisfy n L (x) = n R (Px). We refer to the transfer matrices of the left mirror, of the original heterostructure, and of the right mirror, as M L , M , and M R , respectively. The total transfer matrix of the new structure is then M = M L M M R .
Using the P-symmetry of n L and n R , we find that the transfer matrices of the mirrors are connected by M L = P M −1 R P where P is the matrix representation of the parity operator P, given by This relation leads to One particular implication of this relation is that the two scattering eigenvalues σ 1,2 still coalesce at the original exceptional line, where Y = Y = ±2i. In addition, we note that the incoming/outgoing amplitude ratios ξ can be rewritten as Y /2 ± Y 2 /4 + 1. Thus they are also invariant in this case everywhere in the kL − g plane and coalesce at the original exceptional line. Therefore, we come to the conclusion that the exceptional line is invariant with the addition of the symmetric mirrors. Note that we do not impose any condition on the realness of the refractive indices n L , n R , thus the above conclusion holds for lossless mirrors, as well as for absorptive/amplifying mirrors. In the former case where n L , n R are real, the system is still PT -symmetric and the exceptional line is still the phase boundary of the PT -symmetric and broken phases of S. In the latter case where the mirrors destroy the overall PT -symmetry of the resulting system, i.e. P S † P S = 1, the eigenvalues of the S-matrix generally do not lie on the unit circle and no symmetry breaking occurs. Nevertheless, the same exceptional line persists and so do the incoming/outgoing amplitude ratios ξ everywhere in the kL − g plane. To FIG. 5. Difference between the two scattering matrix eigenvalues D(kL, g) ≡ |σ1 − σ2| for the PT -symmetric two-layer system shown in Fig. 1(a) with two slabs of width L/4 attached on either side (see also smaller inset). The slabs feature randomly chosen complex refractive indices that satisfy nR(x) = nL(Px) where R/L stands for for the slab on the right/left side of the PT -symmetric structure. The lines where D=0, i.e., where σ1 = σ2, coincide exactly with the exceptional line of the inner PT -symmetric part of the system (orange dashed line).
illustrate the generality of this result explicitly, we show in Fig. 5 the difference between the two scattering matrix eigenvalues, D(kL, g) ≡ |σ 1 − σ 2 | for a system composed of two mirrors of width L/4 featuring randomly chosen complex refractive index distributions n L (x) and n R (x) = n L (Px) attached to the two-layer system displayed in Fig. 1(a). As can be seen by comparison to Fig. 1(b), the union of all the points where D = 0 is the exceptional line of the original system without mirrors. At discrete points in its phase space, a PT -symmetric scatterer can act simultaneously as a laser at threshold and a coherent perfect absorber of incident light [20,31]. Since these "CPA-Laser" points correspond to one eigenvalue of the S-matrix going to zero and the other going to infinity, they can occur only within the PT -broken phase of S. Although the "mirror theorem" (see main text and appendix A) shows that adding P-symmetric mirrors at the scattering boundaries does not alter the exceptional line, they do affect the position of the CPA-Laser points.
To demonstrate this invariance, we consider again a 1D PT -symmetric scatterer (see inset of Fig. 6) and modify its dielectric function by with a real parameter µ that controls the mirror reflectivity. This change in the dielectric function corresponds to placing dielectric mirrors of vanishing width at the scattering boundaries x = 0, L. (The reason for using zero-width mirrors is to avoid the complication of additional internal resonances.) Fig. 6 shows the trajectories of the CPA-Laser points as µ is varied, for a simple heterojunction of two equallength gain-loss segments with refractive index n 0 ± ig (and with n = 1 outside). For µ = 0, the CPA-Laser points are located at some distance above the exceptional line, as discussed in Ref. [20]. Increasing or decreasing µ pulls the CPA-Laser points toward the exceptional line, which can be interpreted as an effect of the increase in the Q-factor of the cavity.
Appendix C: Boundary conditions for two-dimensional scattering setups In this section we provide further details on the connection between the PT -thresholds in bounded and unbounded two-dimensional (2D) systems. Following the arguments for 1D setups, we will find that also in 2D appropriate Robin BCs can be found which give rise to eigenstates of the S matrix.
We consider an arbitrarily shaped 2D PT -symmetric cavity through the boundary of which waves can go in and out from and to infinity. The m-th eigenstate of the corresponding scattering matrix, written as a coefficient vector v m and associated to the eigenvalue σ m , is decomposed outside the cavity into a scattering basis φ n as follows, As was shown in [20], the eigenvectors of S satisfy P T v ∝ v below threshold and P T v m ∝ v m above threshold for an associated pair of eigenvectors ( v m , v m ), respectively. The connection between the normal derivative and the wave function of a scattering matrix eigenstate at the system boundary can formally be written as follows .

(C2)
Below the PT -threshold, where σ * = σ −1 , we can rewrite the boundary-value-function λ(σ) in vector notation and suppressing ( x) as follows (C3) where we further used P * = P = P T for the matrix representation P of the parity-operator P and v T ∝ v † P . Note, that the action of P and taking the normal derivative commute due to the P-symmetry of the boundary. From (C3) we see, that λ(σ) is a PT -symmetric function λ(σ, x) = λ * (σ, P x), leading to PT -symmetric Robin boundary conditions for the wave function. Above the PT -threshold of an associated pair (σ m , σ m ) of S matrix eigenvalues, we find for the expression of λ(σ m ) using v T m ∝ v † m P and σ m = 1/σ * m , From Eqs.(C4) and (C3) we may thus conclude, that for 2D PT -systems the situation is very similar to the 1D case: Below the PT -threshold of a pair of Seigenvalues, the corresponding eigenstates feature different, PT -symmetric BC, above threshold the BC are non-PT -symmetric, but pairwise connected.