Curvature-induced bound states in quantum wires

Abstract: A classical particle under spatial constraints is strictly confined to live on a specific space manifold or path, but this assumption is incompatible with the zero-point fluctuations of a quantum particle. One way to describe quantum mechanics under constraints is the confinement potential approach (CPA). For a non-relativistic particle, the CPA maps the problem onto the solution of a Schrödinger-type equation in an isometrically embedded Riemannian submanifold of Euclidean space while the motion along orthogonal directions are decoupled and spatially confined. This approach respects quantum uncertainty, and one of its key results is the appearance of geometry- and metric-induced potentials that affect the stationary states and the dynamics of the particle. For particles constrained to different spaces, such as structures hosting sharp bents, vertices, wedges, conical apices, tips, or self-intersections, a formalism beyond the CPA is needed. Here, a step towards a CPA extension for irregular spaces is presented. After classifying the possible geometric irregularities concerning the CPA formalism, the presentation is focused on a sharply bent quantum wire modeled as an embedded curve with singular (but absolute integrable) curvature. For a subclass fulfilling the additional requirement that the geometric potential is a distribution of first order, a solution scheme for the confined Schrödinger equation is presented based on singular Sturm-Liouville theory and operator theoretic methods. The analytical considerations and numerical simulations evidence the existence of curvature-induced bound states with non-differentiable wave functions localized around the singular point, with an extension well beyond the singularity. Furthermore, a multitude of scattering states appear that may affect the transport and optical properties of the system.

• The paper demonstrates that singular curvature in quantum wires induces localized bound states with non-differentiable wavefunctions.
• It employs a regularization method based on singular Sturm-Liouville theory to establish self-adjoint Hamiltonians and discrete energy spectra.
• Numerical results show a negative ground state energy localized near sharp bends, with implications for nanoscale transport and optical properties.

Curvature-Induced Bound States in Quantum Wires

Introduction and Motivation

The study provides a rigorous extension of the confinement potential approach (CPA) for quantum particles constrained to move on one-dimensional submanifolds—specifically quantum wires with singular geometric features. Traditional CPA, built upon works such as Jensen and Koppe [Jensen1971] and da Costa [Costa1981], prescribes an effective dimensionally-reduced quantum theory. However, it mandates smoothness restrictions on the embedding manifold, typically requiring a $C^2$ Riemannian structure. Singularities—sharp bends, vertices, or points of diverging curvature—are ubiquitously observed in realistic nanostructures but fall outside the regularity domain of standard CPA. This paper classifies such geometric pathologies, investigates their physical implications, and introduces a mathematically robust scheme based on singular Sturm-Liouville theory to analyze curvature-induced phenomena in quantum wires, particularly bound states localized near singularities.

Classification of Geometric Irregularities

A foundational contribution is the codification of degenerate configuration spaces arising in physically relevant constrained quantum systems. The paper categorizes several classes of irregularities:

• Deficient differentiability: Manifolds lacking requisite smoothness for well-defined geometric quantities.
• Non-manifold points: Edges, vertices, or tip singularities destroying the manifold structure.
• Self-intersections and immersions: Non-injective parametrizations with non-unique local frames.
• Metric degeneracies: Collapse of tangent spaces, resulting in critical or singular points in the sense of singularity theory, necessitating sub-Riemannian analysis.

These classes are formally encapsulated, motivating the necessity for an analytic extension beyond the field of regular CPA.

Singular Curvature and Extension of the CPA

Focusing on planar quantum curves with sharply localized bends, the study proposes to treat singularities via limiting sequences of regularized curves. Let $\mathcal{M}$ be a plane curve, parametrized by arc length $s$ on interval $J\subset\mathbb{R}$, whose curvature $\kappa(s)$ is integrable ($\kappa \in L^1(J)$) but diverges at an isolated point, generically $s=0$. The limiting case is mathematically ill-defined in the classical sense, yet physically corresponds to the configuration of a sharply bent quantum wire.

To regularize, a family $\{\mathcal{M}_\varepsilon\}$ with smooth, bounded curvatures $\{\kappa_\varepsilon\}$ is constructed such that, away from the singularity, all geometric and analytic data converge pointwise, and crucially, the distributional limit of the geometric potential and its primitive (entering the Schrödinger equation) converge in $L^2(J)$. This notion of admissible regularization is necessary to obtain well-posed limits of the quantum Hamiltonian.

Figure 1: Four representatives of the regularized curve family $\mathcal{M}$0 converging pointwise to the degenerate singular-curve, parameterized by opening angle $\mathcal{M}$1.

The technical core leverages singular Sturm-Liouville theory, particularly results by Savchuk and Shkalikov [A.M.Savchuk1999, Savchuk2003]. By adapting the notion of quasi-derivatives to account for distributional potentials, self-adjoint realizations of the Hamiltonian with singular geometric potentials are defined, ensuring the existence of real, discrete spectra and $\mathcal{M}$2 orthonormal bases of eigenfunctions.

Spectral and Physical Consequences: Emergence of Bound States

Numerical and analytic study of the regularized problems yields several salient physical results. For the prototypical singular curvature $\mathcal{M}$3, with $\mathcal{M}$4, the geometric potential diverges as $\mathcal{M}$5. In the limiting case $\mathcal{M}$6, one has a one-dimensional Coulomb-like attraction centered at the bend.

Figure 2: Curvature functions $\mathcal{M}$7 interpolating between bounded regular and divergent singular limits.

The spectrum of the regularized Schrödinger operators is computed as the regularization parameter $\mathcal{M}$8:

• Ground state energy: Becomes negative and converges monotonically, indicating the realization of a bound state localized near the bend.
• Excited states: Remain above threshold (positive energy) in the large-interval limit, corresponding to delocalized scattering states.

Figure 3: The lowest four eigenenergies of the regularized Hamiltonian versus the regularization parameter, evidencing convergence and the development of a negative ground state.

Figure 4: Probability densities $\mathcal{M}$9 for the ground state localized near the singularity, narrowing and approaching non-differentiability as the curvature regularization sharpens.

The dependence of the ground state energy on the total geometric bend (quantified by the "turn" or integrated curvature) is encapsulated in the final figure.

Figure 5: Ground state energy in the degenerate limit as a function of the total turn of the curve, demonstrating monotonic behavior and eventual loss of binding at large opening angles.

Implications and Outlook

This work establishes that singularities in the geometry of quantum wires—realizable experimentally via controlled mechanical deformation—are not merely mathematical pathologies but induce robust and observable physical effects: localization and the existence of curvature-induced bound states with non-differentiable wavefunctions. The generality of the strategy, based on operator-theoretic regularization and singular Sturm-Liouville analysis, allows for rigorous treatment of quantum mechanics on manifolds with mild singularities, extending CPA beyond its classical regime.

Practical implications are direct for the design and interpretation of transport, optical, and spectroscopic properties of nanostructures, where sharp bends or kinks unavoidably arise. From the theoretical viewpoint, this scheme can be combined with advanced functional analytic tools to handle more complex geometric degeneracies, possibly in higher codimensions or under additional physical (e.g., electromagnetic, spin-orbit) interactions.

Future developments include the incorporation of electron-electron interactions (notoriously relevant in 1D systems [giamarchi_book]), generalization to quantum graphs and networks with manifold singularities, and treatment of curvature effects in the presence of topological and gauge structures, with anticipated consequences on transport anomalies and localization phenomena.

Conclusion

This paper rigorously extends the quantum mechanics of constrained particles to include sharp geometric singularities by coupling singular Sturm-Liouville theory with operator-theoretic regularization. It demonstrates the emergence of curvature-induced bound states in quantum wires with singular bends, characterized by non-differentiable, localized wave functions even in the presence of integrable but divergent curvatures. The work provides foundational tools to analyze and engineer quantum phenomena in realistic nanostructures with geometric irregularities and necessitates future studies on correlated systems and multidimensional generalizations.

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Reference:

"Curvature-induced bound states in quantum wires" (2604.01856)