Duality of bounded and scattering wave systems with local symmetries
I. Kiorpelidis, F.K. Diakonos, G. Theocharis, V. Pagneux, O. Richoux, P. Schmelcher, P.A. Kalozoumis
DDuality of bounded and scattering wave systems with local symmetries
I. Kiorpelidis, F.K. Diakonos, G. Theocharis, V. Pagneux, O. Richoux, P. Schmelcher,
3, 4 and P.A. Kalozoumis
1, 2 Department of Physics, University of Athens, GR-15771 Athens, Greece LUNAM Universit´e, Universit´e du Maine, CNRS,LAUM UMR 6613, Avenue O.Messiaen, 72085 Le Mans, France Zentrum f¨ur Optische Quantentechnologien, Universit¨at Hamburg,Luruper Chaussee 149, 22761 Hamburg, Germany The Hamburg Centre for Ultrafast Imaging, Universit¨at Hamburg,Luruper Chaussee 149, 22761 Hamburg, Germany (Dated: September 26, 2018)We investigate the spectral properties of a class of hard-wall bounded systems, described bypotentials exhibiting domain-wise different local symmetries. Tuning the distance of the domainswith locally symmetric potential from the hard wall boundaries leads to extrema of the eigenenergies.The underlying wavefunction becomes then an eigenstate of the local symmetry transform in eachof the domains of local symmetry. These extrema accumulate towards eigenenergies which do notdepend on the position of the potentials inside the walls. They correspond to perfect transmissionresonances of the associated scattering setup, obtained by removing the hard walls. We argue thatthis property characterizes the duality between scattering and bounded systems in the presence oflocal symmetries. Our findings are illustrated at hand of a numerical example with a potentialconsisting of two domains of local symmetry, each one comprised of Dirac δ barriers. I. INTRODUCTION
The existence of symmetries provides numerous advan-tages to the study of a physical system, thereby yieldingsignificant fundamental and phenomenological insights.The usual practice for most of the studied systems is toassume a global symmetry, i.e. the symmetry holds forthe complete system under consideration. In this case,important properties, such as the band structure in pe-riodic settings [1] or the classification into even and oddeigenstates for systems with global reflection (inversion)symmetry [2] can be extracted. However, due to the finitesize of a realistic system as well as due to the inevitableexistence of defects, a globally valid symmetry constitutesan idealized scenario in nature. On the other hand, ex-act or approximate symmetries which exist in restrictedspatial subdomains of a larger system are frequently met.Such spatially localized symmetries can be intrinsic incomplex physical systems such as quasicrystals [3–5], par-tially disordered matter [6], large molecules [7, 8] or inbiological materials [9].Furthermore, contemporary technology requires struc-tures with specialized properties which are not alwayspossible to achieve in the presence of generic charac-teristics such as a global symmetry or total disorder.In such cases, local symmetries can be present by de-sign, providing tailored properties and enhanced controlin photonic multilayers [10–12], semiconductor superlat-tices [13], magnonic systems [14], as well as acoustic [15–17] and phononic [18–20] structures.With the term localsymmetries (LS) we refer to symmetries which are validin spatial subdomains of the complete (embedding) spaceand one possible way is to consider them as remnants ofa broken global symmetry.The foundations of local symmetries and their impactin a variety of scattering systems have been investigatedin a sequence of recent works. A rigorous mathemati-cal framework for the description of symmetry breakingleading to local symmetries has been developed in [21–23], where nonlocal invariant currents have been identified asremnants of broken global symmetries. In [24] it wasshown that the long-range order and complexity of lat-tice potentials generated by well-known binary aperiodicone-dimensional sequences can be encapsulated withintheir local symmetry structure, while in [25] the case ofdriven lattices was discussed. The scattering propertiesof quantum and photonic aperiodic structures were dis-cussed in [26, 27], answering the puzzling question aboutthe existence of perfect transmission resonances in ape-riodic systems and also providing a classification schemewith respect to their kind. These theoretical findings werefirstly experimentally verified in [28] in the framework ofacoustic waveguides. Apart from continuous scatteringsystems, the impact of local symmetries has been alsoinvestigated in the framework of discrete systems [29].In this context, the local symmetry partitioning revealednew possibilities for the design of flat bands and compactlocalized states [30]. Very recently, it was shown that theexistence of local symmetries plays a crucial role in thereal space control of edge states in aperiodic chains [31] aswell as in the wave delocalization and transport betweendisorder and quasi-periodicity [32]. Thus, the concept oflocal symmetries, even being a recent one, has alreadyled to a rich phenomenology and revealed properties offundamental importance.Even though local symmetries in continuous scatteringand discrete systems have been extensively investigatedin the aforementioned works, their consequences and ef-fects in continuous bounded systems remains unexplored.The link between bounded and unbounded systems is along standing subject of study in quantum mechanics andin the more general context of wave physics [33]. Severalmethods have been employed to describe how a boundedsystem is connected to its open counterpart [34], sincespectral properties usually can be measured when the sys-tem is coupled to an environment. Nonetheless, this linkhas never been studied under the prism of local symme-tries. a r X i v : . [ c ond - m a t . o t h e r] S e p As a first step in this direction we explore in this workthe properties of one-dimensional bounded systems withtwo locally symmetric potential barriers, focusing on thecase of local reflection symmetries. We define as setup thetwo locally symmetric potential barriers along with thedistance which separates them, while the term system isused to describe the entire potential landscape consistingof both the setup and the bounding hard walls. Tun-ing the distance of the setup from the left hard wall, weprove the existence of spectral extrema where the mirrorsymmetry of the wavefunction is restored inside each re-flection symmetric potential barrier. We also establish alink between the spectral properties of a generic boundedsystem with two domains of local symmetry and the prop-erties of the respective scattering system. In particular,we find that certain eigenenergies of the bounded systemcorrespond to the energies where perfect transmission res-onances (PTRs) manifest in the transmittance of its scat-tering counterpart and we prove that these eigenenergiesare unaffected by the position change of the setup insidethe walls. These theoretical findings are numerically ver-ified for a system with two domains of local reflectionsymmetry comprised of Dirac δ -barriers.The paper is organized as follows: In Section II wesummarize the key ingredients and present some basicresults of scattering theory in systems with local symme-tries. Also we introduce the setups which we will employ,both in the scattering and in the bounded context. InSection III we focus on the properties of a bounded sys-tem with local symmetries comprised of Dirac δ barriersand discuss the relevant properties. We also discuss theconnection between certain bounded states and PTRs.In Section IV we generalize rigorously our results for ageneric system with two domains of local symmetry ofarbitrary potential shape. Our results are summarized inSection V. II. SCATTERING IN SYSTEMS WITH LOCALSYMMETRIES - AN OVERVIEWA. Perfect transmission resonances
Scattering potentials possessing a global mirror sym-metry and the possibility of the occurrence of perfecttransmission resonances (PTRs) have been directly linkedto each other in several studies [35, 36]. On the otherhand, the lack of such a symmetry usually leads to anonvanishing reflection of an incoming scattering wave.However, the existence of PTRs in aperiodic [12] struc-tures possessing no global mirror symmetry has been re-ported. Recently, we established [27] a classification ofthe possible PTRs which occur in non-globally symmet-ric systems. In particular, for a system with local sym-metries the PTRs can be classified according to the sym-metry of the wavefunction modulus u ( x ) = | ψ ( x ) | . If u ( x ) is reflection symmetric within the domains of localreflection symmetry then it is called a symmetric PTR( s -PTR) whereas if u ( x ) does not obey this local sym-metry is called asymmetric PTR ( a -PTR). In the s -PTRcase each domain of local symmetry is individually trans- parent. For a -PTRs the system is transparent only as awhole.Figure 1 (a) shows the transmittance of the setupshown in Fig. 2 (a). The two peaks correspond to an s - and an a -PTR, as their wavefunction moduli indicatein Fig. 1 (b) and (c), respectively. Note here, that thePTRs shown in Fig. 1 (a) do not occur by chance. Thesetup is designed according to prescription based on localsymmetries and the parameters (indicated in Fig. 2) aresuitably tuned in order to emerge at the specific frequen-cies. Also different kind of tuning is required for an a -and a s -PTR, respectively. This design technique and athorough investigation of the scattering properties of thesystem which corresponds to the transmittance shown inFig. 1 (a) can be found in Ref. [28]. FIG. 1. (a) Transmittance of the setup shown in Fig. 1 (a).The two peaks correspond to a s - and an a -PTR, respectively.(b) and (c) illustrate the magnitude to the wavefunction atthe wavenumbers of those PTRs. Here D = 1 . L =0 . B. Symmetry induced invariant currents
Another important finding of our theoretical frame-work on local symmetries is the existence of symmetry-induced currents which are spatially invariant in domainswhere a certain symmetry i.e. reflection, translation or PT symmetry is present. Employing a generic wave me-chanical framework we consider a generalized Helmholtzequation ψ (cid:48)(cid:48) ( x ) + U ( x ) ψ ( x ) = 0, where U ( x ) is the gen-eralized potential. In this framework, it is possible totreat in a unified way different wave mechanical systemsof e.g. photonic, acoustic and quantum mechanical ori-gin. Assuming that the potential U ( x ) obeys a symmetrytransformation U ( x ) = U [ F ( x )] within a domain D ⊆ R ( D = R corresponds to a global symmetry), it can beshown that a spatially invariant, nonlocal current exists, Q = 12 i [ σψ ( x ) ψ (cid:48) (˜ x ) − ψ (˜ x ) ψ (cid:48) ( x )] = const ∀ x, ˜ x ∈ D . (1)This quantity plays a fundamental twofold role. It pro-vides the tool to systematically describe the breaking ofdiscrete symmetries, while it also generalizes the Blochand parity theorems for systems with broken translationand reflection symmetry, respectively [21]. The quantity Q is of central importance for this study. Note, that inbounded systems -since the wavefunction is real- only theinvariant current Q exists. On the other hand, in scatter-ing systems where the wavefunction ψ ( x ) is complex anadditional invariant quantity˜ Q = 12 i [ σψ ∗ ( x ) ψ (cid:48) (˜ x ) − ψ (˜ x ) ψ (cid:48)∗ ( x )] = const ∀ x, ˜ x ∈ D (2)emerges. C. Description of the setup
Let us now describe the setup which we will usethroughout this work. Figure 2 (a) illustrates a scatter-ing system comprised of seven Dirac δ barriers, formingtwo reflection symmetric potential subparts denoted as V and V (coloured areas). The lengths of V and V are d and d , respectively, while their separating distance is L .Moreover, a and a stand for the positions of the reflec-tion centers of each subpart. The parameters c i , i = 1 , δ barriers and r, t are thereflection and transmission coefficients. A detailed studyof the scattering properties of this system and their ex-perimental verification can be found in Ref. [28]. On theother hand, Fig. 2 (b) shows the corresponding boundedsystem, where the aforementioned setup is delimited byhard wall boundaries. The general characteristics remainthe same. However, the distance (cid:96) which determines thedistance from the left wall plays a crucial role and in thefollowing will serve as our tuning parameter. In orderto preserve the local symmetries of the system for anyvalue of (cid:96) -namely the domains D and D being alwaysreflection symmetric- it should hold L = (cid:96) + ˜ (cid:96) [see Fig. 2(b)]. III. LOCAL SYMMETRIES IN BOUNDEDSYSTEMS
Several connections between bounded and scatteringsystems can be investigated. In this work we focus onbounded systems and how they can be linked to theirscattering counterparts from the perspective of
PTRs and local symmetries . To this end, we consider the boundedversion of our system as shown in Fig. 2 (b). We remindthe reader that for the lengths (cid:96), ˜ (cid:96) it holds L = (cid:96) + ˜ (cid:96) .With this choice -and employing (cid:96) as our tuning param-eter with (cid:96) ∈ [0 , L ]- we ensure that for any value of (cid:96) thesystem is always decomposable into two locally symmet-ric domains D and D . Keeping L fixed and varying (cid:96) we expect that the spectrum of the allowed wavenumberswill change continuously. During this variation we willexamine the spectral properties which emerge due to thelocal symmetries. FIG. 2. (a) Scattering and (b) bounded systems with localsymmetries. Each potential subparts V , V is reflection sym-metric. The distances between the delta functions are equalto L = 0 . L = 0 . L = 0 . δ barriers are equal to c = 7 . c = 12 . Figure 3 (a) shows the first six states of the boundedsystem [see Fig 2 (b)] for L = 0 . (cid:96)/L varies within the range [0 , κ remains unaffected by the position ofthe setup inside the box. This is an important observa-tion because eigenenergies with this property emerge inthe respective scattering counterpart as PTRs, offeringthe ground for establishing a duality between open andclosed systems. Figure 3 (b) shows the transmittanceof the corresponding scattering system. The dashed lineindicates an a -PTR peak at k = 15 . L values in the vicinity of L = 0 . L < . κ ( (cid:96)/L ) curve exhibits a minimum.Exactly at L = 0 . κ = 15 .
008 be-comes invariant with respect to the position of the setupwithin the walls and this manifests through the flat solidline. Remarkably, this κ value is identical to the k valueof the a -PTR in the transmittance of (b). For L > . κ ( (cid:96)/L ) curves exhibit maxima. All extrema approachthe flat line of L = 0 . L value, both the spectrum ofthe bounded system and the transmittance of the scatter-ing system will change. Nevertheless, the aforementionedcorrespondences can be identified. In Fig. 4 the respec-tive properties of the system for L = 0 .
239 are discussed.Figure 4 (a) shows five states of the spectrum and howthese change as (cid:96)/L varies within the range [0 , (cid:96)/L shifts correspondsto the s -PTR shown in the transmittance of Fig. 4 (b). In FIG. 3. (a) Spectrum showing the first 6 states of the bounded system shown in Fig 2 (b) for L = 0 . (cid:96) varies in the range [0 , L ] the wavenumber κ varies continuously. The insets zoom into the curves to provide a betterresolution. (b) Transmittance for the corresponding open system. The dashed line at k = 15 .
008 indicates an a -PTR. (c) Thefourth state of spectrum of the bounded system for different L values. For L = 0 . k = 15 .
008 and the invariant bounded state κ = 15 . Fig. 4 (c) the dependence of the second state as L changesis shown. The pattern is the same as in the previous ex-ample. For L = 0 .
239 the wavenumber κ = 8 .
237 remainsconstant as the (cid:96)/L changes. For this L value the trans-mittance exhibits a s -PTR peak at the same wavenumber k = 8 . L (cid:54) = 0 . κ ( (cid:96)/L ) possess extrema, which‘saturate’ to the flat line.Another very interesting property which is observedhere relates the extrema of the κ ( (cid:96)/L ) curves with thelocal symmetries of the setup and the form of the wave-function. In particular, at every extremum of the κ ( (cid:96)/L )curves, the wavefunction becomes an eigenstate of the lo-cal reflection symmetry transform and follows the localsymmetries of the setup. Figures 3 (c) and (d) illustratethis case. In particular, Fig. 3 (d) shows the wavefunc-tion at the maximum of the curve for L = 0 . (cid:52) ). It is clear thatthe wavefunction is (locally) parity definite within thelocal symmetry domains D and D of the setup. Thesame holds for the system in Fig. 4. The wavefunctionin Fig. 3 (d) corresponds to the minimum of the κ ( (cid:96)/L )curve for L = 0 .
21 (see the (cid:79) ) and is locally parity def-inite following the symmetries of the D and D . Thiscorrespondence between the κ ( (cid:96)/L ) extrema and the lo-cal symmetry properties of the wavefunction provides the-possibly- first systematic attempt to investigate the man-ifestation of local symmetries in bounded systems.In fact, there are κ ( (cid:96)/L ) curves which may possess morethan one extrema, as those shown in Fig. 5 (b). Thesecurves correspond to the seventh state of a bounded sys-tem for several L values around L = 0 . L at κ = 20 . a -PTR in the corresponding scat-tering setup, as indicated in Fig. 5 (a), a finding whichsupports the duality between open and closed systemsat the PTR wavenumbers. Figures 5 (c), (d) illustratethe wavefunction at the two extrema of the first curve (marked with the up and down triangles). In both cases,it becomes parity definite within the two domains of localsymmetry D and D . Note that the fifth state in Fig. 4(a) also possesses two extrema. However, we showed thecase of a different setup in order to stress further the cor-respondence between PTRs and translationally invariantbound states.Note that this bound-scattering duality and the mani-festation of local symmetries on the extrema are not sys-tem specific. Our conclusions will be rigorously provenand generalized for systems with two locally symmetricpotentials domains of arbitrary shape in the following sec-tion. To conclude this Section we summarize our keyfindings in the following statement which holds for thegeneral case: “Consider a bounded system which consistsof two domains of local symmetry D , D , each one witha reflection symmetric potential V , V of arbitrary shapeand finite support. Between this system and its scatter-ing counterpart the following duality holds: (i) Startingfrom a bounded system: If a bound state with wavenum-ber κ is invariant with respect to translations of the setupinside the cavity, then it corresponds to a PTR ( a or s ) in the corresponding scattering system with incomingwavenumber k = κ . (ii) Starting from a scattering sys-tem: The existence of an a -PTR in a scattering system at k is equivalent to a bound state with wavenumber κ = k which is invariant under translations of the (same) setupinside the cavity. The reason that we discriminate s -from a -PTRs is because the wavenumber k of an a -PTRwill always emerge as an eigenstate with wavenumber κ in the corresponding bounded system. This one-to-onecorrespondence between scattering and bounded systemsexists because the a -PTR occurs for a specific distance L between the two locally symmetric scatterers. For the s -PTR, on the other hand, this one-to-one correspondencebetween scattering and bounded systems would not bepossible because it appears in the transmittance for any FIG. 4. (a) Spectrum showing 5 states of the bounded system shown in Fig 2 (b), this time for L = 0 . (cid:96) varies in [0 , L ]the wavenumber κ varies continuously. (b) Transmittance for the corresponding open system. The dashed line at k = 8 . s -PTR. (c) The second state of spectrum of the bounded system for different L values. For L = 0 . k = 8 .
237 and the invariant bounded state κ = 8 . distance L between the locally symmetric scatterers [27].In this case, all bounded systems’ spectra for all different L values would have an eigenstate at the same κ , whichis not possible. Nevertheless, if the bounded system hasan eigenstate at a κ value which coincides with the k wavenumber of an s -PTR of its scattering counterpart,then the equivalence between the s -PTR and the boundstate translation invariance is preserved (case shown inFig. 4). (iii) All extrema which emerge in the κ ( (cid:96)/L ) curves correspond to states which are eigenstates of thethe local reflection symmetry transform and the wavefunc-tion is parity definite inside D and D .” IV. GENERALIZATION FOR AN ARBITRARYBOUNDED SYSTEM WITH TWO DOMAINS OFLOCAL SYMMETRY
In this Section we will generalize the results presentedabove for arbitrary bounded systems with two domainsof local symmetry. In order to prove the above statementwe employ the transfer matrix (TM) approach to connectthe wave fields in the regions I , II and III of the system,as shown in Fig. 6. Since I , II and III are potential freeregions, the wavefunction will be of the form, ψ m ( x ) = A m e iκx + B m e − iκx ; m = I, II, III. (3)The connection between ψ I and ψ II is provided by theTM, which reads for a Hermitian system, M D j = (cid:20) w j z j z ∗ j w ∗ j (cid:21) ; j = 1 , . (4)Here j = 1 , D and D , respectively. For the TMelements it holds that w j = 1 /t j and z j = r ∗ j /t ∗ j , where t j and r j are the transmission and reflection amplitudesof the j -th potential unit, respectively. FIG. 5. (a) Transmittance of the setup shown in Fig. 1 (a)for L = 0 . L = 0 . k = 20 .
99. (b)Variation of the seventh state with (cid:96)/L of the spectrum ofthe respective bounded system for several L choices around L = 0 . κ vs (cid:96)/L curve exhibits one minimum andone maximum. For L = 0 . κ = 20 . (cid:96) with a value coinciding with the wavenum-ber k of the a -PTR. (c), (d) Wavefunctions at the maximumand minimun of the first curve (indicated by up and down tri-angles). Here the wavefunction becomes an eigenstate of thelocal symmetry transform. Positioning the first wall at x = 0, the wavefunctionshould be zero there i.e. ψ (0) = 0, leading to the condi-tion A I = − B I and subsequently to, A II B II = − z c e − iκa + w ∗ c z ∗ c e iκa + w c , (5)connecting regions I and II . Note here that the index c inthe TM elements (referring to “centered”) corresponds tothe TM for D centered at x = 0. The shift to the actualposition of D in our setup is realized by the phases inEq. (5), with a being the position of the reflection axis FIG. 6. Schematic of a bounded system with hard wall bound-ary conditions containing two domains of local symmetry D , D . In the regions I, II, III the potential vanishes. of D .In the same manner, taking the wavefunction zero atthe second wall ψ ( D ) = 0, we find the condition B III = − A III e ikD which, in turn, leads to the relation, A II B II = w c − z c e − iκa e iκD z ∗ c e iκa − w ∗ c e iκD , (6)connecting the plane wave coefficients in regions II and III . a and a correspond to the mirror symmetry centerof the two scatterers (see Fig. 6). In turn, Eq. (5) andEq. (6) yield, G = z c e − iκa + w ∗ c z ∗ c e iκa + w c + w c − z c e − iκa e iκD z ∗ c e iκa − w ∗ c e iκD = 0 , (7)which involves only the wave number κ and the charac-teristic parameters of the system which are included inthe TM elements w jc , z jc ( j = 1 , F . Since we areinterested in the behaviour of κ with respect to (cid:96) it is suf-ficient to calculate the total derivative of F with respectto (cid:96) and then calculate the derivative dκ/d(cid:96)d F d(cid:96) = ∂ F ∂(cid:96) + ∂ F ∂κ dκd(cid:96) = 0 , (8)which leads to dκd(cid:96) = − ∂ F ∂(cid:96)∂ F ∂κ . (9)Therefore the behaviour of the wavenumber κ with re-spect to (cid:96) can be investigated via the term ∂ F /∂(cid:96) . Notethat in order to find d F /d(cid:96) we have expressed a , withrespect to (cid:96), d , d and L , namely a = (cid:96) + d / a = (cid:96) + d + L + d /
2. Then, the latter is written as ∂ F /∂(cid:96) = 2 iκ (cid:2) e iκ P (cid:0) w ∗ c z ∗ c + z ∗ c w c e − iκD (cid:1) + e − iκ P (cid:0) w c z c + z c w ∗ c e iκD (cid:1)(cid:3) , (10)where P = 2 (cid:96) + d . In the following we will show thatthis equation has a very instructive form regarding theemergence of local symmetries. To this end, we employthe existence of the symmetry induced invariant current Q for reflection symmetry, as defined in Eq. (1). Given the plane wave solution in the potential free regions I , II and III [see Eq. (3)] we find that the form for Q and Q in the LS domains D and D is, Q = κ (cid:0) A I A II e iκa + B I B II e − iκa (cid:1) , (11a) Q = κ (cid:0) A II A III e iκa + B II B III e − iκa (cid:1) , (11b)where A j , B j are the plane wave coefficients in regions I, II and
III and κ is the wavenumber of the spe-cific eigenstate. We stress here that Q can be calcu-lated taking into account only the potential free regions I, II, III , since -due to the symmetry- it is independentof the exact potential form (see Ref. [21]). Note also thatEqs. (11a), (11b) has no dependence on x , signaling itsspatial invariance. Focusing on the setup shown in Fig. 1we find that Q for the domain D reads as follows Q = kA I B II (cid:18) A II B II e iκa + e − iκa (cid:19) , (12)where we have used the condition B I = − A I . The be-haviour of Q determines the LS properties of the wave-function in the domain D . In particular, if Q = 0 thewavefunction inside D will be parity definite. Substitut-ing Eq. (5) into Eq. (12) and setting Q = 0 we find z c = 12 (cid:0) w c e − iκ P − w ∗ c e iκ P (cid:1) , (13)where we have used the property z c = − z ∗ c which holdsfor the TM of reflection symmetric potentials. Followingthe same procedure for the domain D and for Q = 0 wefind, z c = 12 (cid:0) w c e iκ P e − iκD − w ∗ c e − iκ P e κkD (cid:1) . (14)We stress here that when Eqs. (13) and (14) hold, thenthe wavefunction is parity definite inside D and D , re-spectively.The next step is to substitute Eqs. (13), (14) intoEq. (10). This allows to focus on the behaviour of thequantity ∂ F /∂(cid:96) when the wavefunction is parity definitein both domains D , D . After some algebraic manipu-lations, we find that ∂ F /∂(cid:96) = 0 and, in turn, that dκ/d(cid:96) = 0 . (15)Therefore, the restoration of LS in the wavefunction (i.e.the field is parity definite inside D , D and Q = Q =0) in any structure comprised of two barriers of arbitraryshape obeying the corresponding local symmetry, mani-fests as an extremum in the κ vs (cid:96) curve.The implications of Eq. (15) on bounded systems pro-vides also certain interesting links to their correspondingscattering counterparts. The transition from the boundedsystem to the scattering one is achieved by removingthe hard wall boundaries, leaving otherwise the systemunaffected. Then, on either side of the setup, the po-tential vanishes and the wavefunction can be describedby plane waves. Here, the asymptotic conditions aredescribed by incoming and outgoing waves of the form ψ I ( x ) = e ikx + re − ikx and ψ III ( x ) = te ikx , respectively.Note that k is the continuous wavenumber of the scatter-ing system, while r and t the transmission amplitudes.Then, we can distinguish two cases of particular inter-est which render Eq. (10) [and consequently Eq. (15)] alsoequal to zero.These cases provide the connection betweenspecial spectral points of the bounded system and thewavenumbers where PTRs occur in the transmittance ofthe respective scattering system. For reasons of clarity,we denote with k s and k a the wavenumbers where the s -and a -PTRs occur, respectively. Therefore, we have,1. κ = k s : At this κ value the corresponding scatter-ing system exhibits a s -PTR. In the case of a s -PTRboth potential parts are independently transparent(for a detailed analysis see Ref. [27]). The inde-pendency refers to the fact that their distance L isirrelevant to their transparency. In terms of the TMformalism this occurs when the anti-diagonal termsare z c ( k s ) = z c ( k s ) = 0. In this case ( κ = k s )Eqs. (10), (15) become zero independently of the (cid:96) value. Therefore, the eigenstate with κ = k s will beinvariant under translations of the setup inside thecavity and the κ vs (cid:96)/L curve will appear as a hor-izontal line. We remind here that for the scatteringsystem the s -PTR at k s will appear for any distance L between the two scatterers. On the other hand,for the bounded system not all L values will yieldan eigenstate at κ = k s . If however, an eigenstatewith κ = k s exists in the spectrum, then it will havethe aforementioned translation invariance property.2. κ = k a : At this κ value the corresponding scat-tering system exhibits an a -PTR. In the case ofan a -PTR the complete setup i.e. the combina-tion of the two subparts is transparent. Then, theoff-diagonal terms of the total TM -which is theproduct M D · M D of the two individual TMs- isgiven by z tot = w c z c + z c w ∗ c e − iκD , which (alongwith its complex conjugate) are the quantities inthe parentheses in Eq. (10). Apparently, for the re-flectionless state we have z tot = 0 and consequentlyEq. (10) (and Eq. (15) ) becomes zero.Therefore,also this κ = k a value will be unaffected by the (cid:96) variation and will appear as a horizontal line in the κ vs (cid:96) diagram. Here, contrary to the s -PTR, thedistance L plays a major role on the transparency,since an a -PTR corresponds to a specific distance L . This leads to fact that there is always a corre-spondence between an a -PTR of a scattering systemat k s and a translation invariant bound state of itsbounded counterpart at κ = k s .Inversely, it holds that a translation invariant boundstate at κ value will manifest in the transmittance of thecorresponding scattering system as a PTR. V. CONCLUSIONS
We explored a generic bounded system with hard wallboundary conditions, consisting of two locally reflectionsymmetric potential barriers. Focusing on the varia-tion of the energy eigenvalues by tuning the position ofthe potential units inside the box, we proved the exis-tence of spectral extrema where the mirror symmetryof the wavefunction is restored inside each locally sym-metric potential barrier. These extrema accumulate toeigenenergy values which coincide with the energies whereperfect transmission resonances emerge in the transmit-tance of the associated scattering system. This behaviouris a benchmark of the duality between scattering andbounded systems in the presence of locally symmetricpotential landscapes. It is exemplified in this work fora system with two domains of local symmetry comprisedof Dirac δ barriers. Our work could facilitate the design ofcavities with prescribed spectral and wavefunction prop-erties. The established duality opens the perspective oflinking and controlling bounded versus scattering setups.A bounded system can be designed to possess not onlyLS symmetric wavefunctions but its opening up to a scat-tering device leads also to an “infinite range” extensionvia PTRs to the outside region. VI. ACKNOWLEDGEMENTS
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