Local symmetries in one-dimensional quantum scattering

Abstract: We introduce the concept of parity symmetry in restricted spatial domains -- local parity -- and explore its impact on the stationary transport properties of generic, one-dimensional aperiodic potentials of compact support. It is shown that, in each domain of local parity symmetry of the potential, there exists an invariant quantity in the form of a non-local current, in addition to the globally invariant probability current. For symmetrically incoming states, both invariant currents vanish if weak commutation of the total local parity operator with the Hamiltonian is established, leading to local parity eigenstates. For asymmetrically incoming states which resonate within locally symmetric potential units, the complete local parity symmetry of the probability density is shown to be necessary and sufficient for the occurrence of perfect transmission. We connect the presence of local parity symmetries on different spatial scales to the occurrence of multiple perfectly transmitting resonances and propose a construction scheme for the design of resonant transparent aperiodic potentials. Our findings are illustrated through application to the analytically tractable case of piecewise constant potentials.

• The paper shows that perfect transmission resonances occur if and only if the probability density is completely locally parity symmetric within the interaction region.
• It employs local parity operators on non-overlapping domains to derive non-local invariants that classify scattering states and dictate resonance conditions.
• The research outlines an algorithmic blueprint for constructing aperiodic, locally symmetric potentials, paving the way for advanced quantum device engineering.

Local Parity Symmetry and One-Dimensional Quantum Scattering

Introduction and Motivation

The paper introduces the concept of local parity (LP) symmetry in the context of one-dimensional (1D) quantum scattering, focusing on systems where global spatial symmetry is absent but local symmetry exists in subdomains of configuration space. The analysis addresses stationary transport properties of arbitrary aperiodic potentials with compact support, extending the traditional treatment that primarily relies on global spatial symmetry.

LP symmetry is defined through domain-restricted parity operations: within each subdomain, a local parity transformation inverts coordinates about a central point, while outside the domain, the wavefunction is left unchanged up to a sign. These generalized parity operators reduce to the ordinary parity for the entire real line but allow for non-trivial action in composite or disordered potentials.

Formalism of Local Parity Operators

The formal structure is built upon local parity operators $\hat{\varPi}_s^{\mathcal{D}}$, parametrized by the center and extent of a domain $\mathcal{D}$. Successive, non-overlapping LP operators allow the definition of a total LP operator that acts globally but is reducible to its action on non-overlapping domains. The algebraic properties ensure two eigenvalues ($\pm 1$), degeneracy stemming from the combination of $N$ subdomains, and intricate relationships between the eigenstates' support and the associated eigenvalues of the LP operations.

Spectral equivalence is preserved with the global parity operator, but commutation between the LP operator and the Hamiltonian is only achieved weakly: it holds for states whose wavefunction fulfills specific boundary conditions dictated by the LP-structured potential. In the domain-wise symmetric potential, the stationary states are decomposable according to LP symmetry axes, leading to a set of symmetry conditions for the probability density and phase in each subdomain.

Invariant Quantities and Classification of States

The introduction of LP symmetry results in non-local, domain-specific invariants ($q_{k,n}$) in addition to the standard globally invariant probability current ($j_k$). When $j_k = 0$, one obtains LP eigenstates under symmetric asymptotic boundary conditions (SAC); these are states with zero net current. For nonzero current, relevant under asymmetric boundary conditions (AAC, i.e., scattering from one direction), the paper distinguishes states that feature perfect transmission resonances (PTRs). The key analytical result is the construct:

• For each locally parity symmetric subdomain, there exists a non-local current $q_{k,n}$, which vanishes if and only if the state is an eigenstate of the corresponding LP operator in that domain.

The conditions for LP symmetric states are established in terms of both the modulus and phase of the wavefunction, with the central criterion being local symmetry of the probability density.

Scattering Theory and Consequences for Transport

Application of LP symmetry to 1D stationary scattering reveals several important distinctions:

• Symmetric Asymptotic Conditions (SAC): Only under SAC (waves incident from both sides) can the system host LP eigenstates with zero current, even if the potential lacks global parity.
• Asymmetric Asymptotic Conditions (AAC): For the physically relevant setting of a single incoming wave, the existence of PTRs is strictly governed by the LP symmetry of the probability density. The central theorem shown is that a PTR exists resonating within locally symmetric potential units if and only if its probability density is completely LP symmetric within the interaction region.

This generalizes the known fact for globally symmetric scatterers to systems with multiple, nested, or hierarchical local parity symmetries. The probability density at PTRs aligns with an LP decomposition of the potential, and the notion of reducibility emerges: some PTRs can be associated with finer or coarser scales of LP domains, corresponding to resonance within different symmetric groupings of scatterers.

Construction Principles for Resonant Transparency

The paper formulates an algorithmic construction paradigm for globally aperiodic but locally parity-symmetric potentials exhibiting PTRs at prescribed energies. The approach involves:

1. Decomposition of the system into all possible LP symmetric groupings (resonators) at various scales.
2. Imposing the PTR condition ($r = 0$) for each resonator at the target resonance energy, leading to a system of complex algebraic equations in the scatterer parameters.
3. In the case of piecewise constant (PWC) potentials (barrier arrays), the transfer matrix method enables analytic expressions. The LP symmetry constraints reduce the solution space, yielding device designs—such as asymmetric double-barriers or aperiodic multi-barrier structures—where PTRs can be engineered by appropriate selection of potential heights, widths, and positions.

The construction protocol is robust and highlights the role of local, rather than global, order in controlling transmission properties.

Numerical Examples and Reduction to Practice

The analytic framework is illustrated with several examples:

• Asymmetric double-barrier systems displaying both zero-current states (under SAC) and PTRs (under AAC), with probability densities tracking LP symmetry within the defined domains.
• Potentials composed of locally symmetric barrier clusters (resonators), showing multiple PTRs with densities reducible at different symmetry scales.
• Periodic arrays with engineered defects, where local symmetry decompositions identify the support for PTRs beyond what can be predicted by global order.

These scenarios underscore that identification of local symmetries, and their mapping via LP decomposition, is fundamental for understanding and engineering transport in quantum and wave systems with complex structure.

Theoretical and Practical Implications

The findings have significant theoretical implications:

• The classification of scattering states by their LP properties, combined with the existence or non-existence of non-local invariants, provides a deeper taxonomy of quantum states in complex media.
• The necessity and sufficiency result for PTRs introduces a general principle for transparency in aperiodic systems, directly linking observable transport properties to local structural symmetries.
• The formalism offers a generalized symmetry-based methodology, applicable not only in quantum scattering, but also in electromagnetic and acoustic analogues (e.g., photonic multilayers, electronic transport in nanostructures).

Practically, the analytic construction of PTR-supporting structures extends the design space for quantum devices (e.g., electron waveguides, resonant tunneling diodes) and classical analogs (e.g., aperiodic photonic filters, phononic crystals). The LP principle enables systematic engineering of transparent energies irrespective of the absence of global symmetry.

Long-term, extensions to higher dimensions and alternative local symmetry types could influence the analysis and synthesis of complex materials, quasi-crystals, and disordered systems, potentially revising the approach to disorder-induced localization and band structure predictions.

Conclusion

The local parity framework developed here rigorously unpacks the influence of domain-wise inversion symmetry on stationary quantum transport. The analytical apparatus—comprising LP operators, the associated non-local invariants, and the LP decomposition—uncovers a direct connection between local symmetries and resonant transparency. The theoretical results, algorithmic construction scheme, and demonstrative numerical studies collectively shift the focus from global to local symmetry analysis, laying the foundation for a nuanced understanding and purposeful design of quantum and wave transport phenomena in structured and aperiodic media.

Reference: "Local symmetries in one-dimensional quantum scattering" (1207.1043)