The Fichera layer is the region of unit width obtained by removing a closed octant from a parallel open octant in three dimensions — the prototype non-smooth (polyhedral) quantum layer, with three convex and three non-convex edges and two corners. Dauge, Lafranche and Ourmières-Bonafos (2018) proved that the essential spectrum of the Dirichlet Laplacian on this layer starts at the ground state of a two-dimensional broken (L-shaped) guide, that the discrete spectrum below that threshold is finite, and found numerically exactly one bound state, without an analytic existence proof. Bakharev and Nazarov (2021) then proved existence of at least one bound state for the Fichera layer and the cross in every dimension n≥3n\ge3, and conjectured the bound state is unique; Bakharev and Matveenko (2024) proved that every polyhedral layer of their class has finite discrete spectrum with threshold set by the smallest dihedral angle, exhibiting both empty and nonempty examples. The surviving open problem is to determine the exact number of bound states of the Fichera layer, its exterior rounded-edge variant (for which even existence is unproven), and the cross — proving or disproving the uniqueness conjecture — and likewise for the higher-dimensional corner and cross analogues.

Background

An infinite layer is the region of constant thickness around an unbounded surface; the Dirichlet Laplacian on such domains models a quantum particle confined by hard walls, and its bound states are isolated eigenvalues below the threshold of the essential spectrum, corresponding to waves trapped purely by geometry. In a straight cylinder Ω=ω×R\Omega=\omega\times\mathbb{R} with bounded cross-section ω\omega, the spectrum is purely continuous and starts at the first Dirichlet eigenvalue of ω\omega; deforming the guide (bending, widening, corners) can create eigenvalues below this threshold — the curvature- and corner-induced bound states studied in quantum waveguide theory (two-dimensional strips and three-dimensional tubes).

The present problem concerns three-dimensional layers, the next step in complexity after waveguides. For a smooth conical layer — the constant-width layer around a cone-like surface that flattens out at infinity — the discrete spectrum is infinite, with a counting function growing logarithmically as the threshold is approached (Dauge, Ourmières-Bonafos and Raymond, 2015). The polyhedral case behaves very differently, and its prototype is the Fichera layer

Λ={x∈R3: −1<min⁡(x1,x2,x3)<0},\Lambda=\{x\in\mathbb{R}^3:\ -1<\min(x_1,x_2,x_3)<0\},

formed by three perpendicular quarter-plane walls of unit width; equivalently, an open octant from which a parallel closed octant is removed. It has six edges (three convex, three non-convex) and two corners, and is named after Fichera because near its non-convex corner it coincides with the classical Fichera cube, the standard model domain for the interaction of edge and corner singularities in elliptic boundary value problems. Dauge, Lafranche and Ourmières-Bonafos proved that the essential spectrum of the Dirichlet Laplacian LΛL_\Lambda is the half-line [λ1(Γ),+∞)[\lambda_1(\Gamma),+\infty), where Γ={x∈R2: −1<min⁡(x1,x2)<0}\Gamma=\{x\in\mathbb{R}^2:\ -1<\min(x_1,x_2)<0\} is the two-dimensional broken guide (an L-shaped waveguide of width 1) and λ1(Γ)≈0.929π2\lambda_1(\Gamma)\approx0.929\pi^2 is its unique eigenvalue below π2\pi^2; they also proved that the discrete spectrum of LΛL_\Lambda below λ1(Γ)\lambda_1(\Gamma) is finite, by a Born–Oppenheimer strategy (a scale-separation reduction borrowed from molecular physics) that bounds the eigenvalue count by that of an effective one-dimensional Schrödinger operator and applies a Bargmann-type estimate (a classical bound on the number of bound states of a one-dimensional Schrödinger operator). Finite-element computations adapted to the edge and corner singularities exhibited exactly one eigenvalue, about 3% below the threshold in relative terms, but no analytic existence proof was given. The same paper proves finiteness for a variant Λ♯\Lambda^\sharp with rounded exterior edges (the layer of width 1 lying outside the boundary of the positive octant), whose essential spectrum starts at λ1(Γ♯)≈0.9865π2\lambda_1(\Gamma^\sharp)\approx0.9865\pi^2, the ground state of the two-dimensional broken guide with rounded exterior corner, with numerics showing a single bound state only about 0.5% below the threshold; and for the cross Y\mathcal{Y} (three mutually perpendicular thickened planes of width 1), whose essential spectrum starts at λ1(X)≈0.6596π2\lambda_1(X)\approx0.6596\pi^2, the ground state of the planar cross XX formed by two perpendicular strips of width 1, with numerics showing a single bound state of approximate value 0.5165π20.5165\pi^2; for the associated two-dimensional domains (broken guide and planar cross) the existence of exactly one bound state is known.

Bakharev and Nazarov subsequently proved that the corner domain Θ1n={x∈Rn: ∣min⁡jxj∣<1}\Theta^n_1=\{x\in\mathbb{R}^n:\ |\min_j x_j|<1\} (a translate-dilate of the Fichera layer when n=3n=3) and the cross Θ2n={x∈Rn: min⁡j∣xj∣<1}\Theta^n_2=\{x\in\mathbb{R}^n:\ \min_j|x_j|<1\} each have at least one eigenvalue below the essential spectrum in every dimension n≥3n\ge3, the essential spectrum being [λ∙(Θn−1),+∞)[\lambda_\bullet(\Theta^{n-1}),+\infty); they state explicitly that the number of eigenvalues is unknown and conjecture it equals one in every dimension. Bakharev and Matveenko then studied general three-dimensional polyhedral layers of unit width built inside polyhedral angles (outer and inner polyhedral surfaces parallel, which includes the Fichera layer as a regular case): the discrete spectrum is always finite, and the essential-spectrum threshold is the first eigenvalue of an L-shaped guide whose opening equals the smallest dihedral angle of the layer; they construct a trihedral layer (two right vertex angles and a small third angle) with empty discrete spectrum and prove nonemptiness for regular angular layers. A later preprint of the same authors establishes monotonicity of the below-threshold eigenvalues with respect to angular parameters, together with non-monotone phenomena under asymmetric perturbations, and Chesnel, Nazarov and Taskinen showed for a related mixed Dirichlet–Neumann problem on a chamfered quarter of layer (a quarter of an infinite layer whose corner is cut off flat) that existence of the discrete spectrum depends sharply on the chamfer parameters. Thus, within the polyhedral layer family, the finite-versus-infinite dichotomy raised in the source paper is settled in favor of finite, and what remains open is the exact number of bound states carried by each of these layers.

Problem Statement

Determine the discrete spectrum of the Dirichlet Laplacian below the essential-spectrum threshold across the family of generalized Fichera layers. After the existence theorem of Bakharev–Nazarov and the finiteness theorem of Bakharev–Matveenko, the surviving question is the exact bound-state count. Determine the exact number of eigenvalues below the essential-spectrum threshold for the Dirichlet Laplacian in the Fichera layer Λ={x∈R3: −1<min⁡(x1,x2,x3)<0}\Lambda=\{x\in\mathbb{R}^3:\ -1<\min(x_1,x_2,x_3)<0\} (threshold λ1(Γ)≈0.929π2\lambda_1(\Gamma)\approx0.929\pi^2, the ground state of the two-dimensional broken, i.e. L-shaped, guide Γ={x∈R2: −1<min⁡(x1,x2)<0}\Gamma=\{x\in\mathbb{R}^2:\ -1<\min(x_1,x_2)<0\} of width 1); in the exterior rounded-edge variant Λ♯\Lambda^\sharp, the layer of width 1 lying outside the boundary of the positive octant (threshold λ1(Γ♯)≈0.9865π2\lambda_1(\Gamma^\sharp)\approx0.9865\pi^2, the ground state of the broken guide with rounded exterior corner); and in the cross Y\mathcal{Y} of three mutually perpendicular thickened planes of width 1 (threshold λ1(X)≈0.6596π2\lambda_1(X)\approx0.6596\pi^2, the ground state of the planar cross XX formed by two perpendicular strips of width 1); and likewise for the higher-dimensional analogues Θ1n={x∈Rn: ∣min⁡jxj∣<1}\Theta^n_1=\{x\in\mathbb{R}^n:\ |\min_j x_j|<1\} and Θ2n={x∈Rn: min⁡j∣xj∣<1}\Theta^n_2=\{x\in\mathbb{R}^n:\ \min_j|x_j|<1\} for every n≥3n\ge3. Determine the exact below-threshold eigenvalue count for the Dirichlet Laplacian in every domain listed above. The three-dimensional Λ\Lambda and Y\mathcal{Y} are the n=3n=3 members of the Θ1n\Theta^n_1 and Θ2n\Theta^n_2 families up to translate-dilate normalisation, and so require nothing beyond those families. The conjecture of Bakharev–Nazarov is that each of these has exactly one bound state, as all numerical evidence indicates; for Λ♯\Lambda^\sharp even the existence of a single bound state is still unproven.

References

  1. Monique Dauge, Yvon Lafranche, Thomas Ourmières-Bonafos, "Dirichlet spectrum of the Fichera layer", Integral Equations and Operator Theory 90 (2018), article 60. DOI: 10.1007/s00020-018-2486-y; arXiv:1711.08439. https://doi.org/10.1007/s00020-018-2486-y (preprint: https://arxiv.org/abs/1711.08439)
  2. F. L. Bakharev, A. I. Nazarov, "Existence of the discrete spectrum in the Fichera layers and crosses of arbitrary dimension", Journal of Functional Analysis 281 (2021), 109071. DOI: 10.1016/j.jfa.2021.109071; arXiv:2006.12045. https://doi.org/10.1016/j.jfa.2021.109071 (preprint: https://arxiv.org/abs/2006.12045)
  3. F. L. Bakharev, S. G. Matveenko, "Spectra of the Dirichlet Laplacian in 3-dimensional polyhedral layers", St. Petersburg Mathematical Journal 35 (2024), 597–610. DOI: 10.1090/spmj/1818; arXiv:2304.10575. https://doi.org/10.1090/spmj/1818 (preprint: https://arxiv.org/abs/2304.10575)
  4. F. L. Bakharev, S. G. Matveenko, "Monotonicity of discrete spectra of Dirichlet Laplacian in 3-dimensional layers", preprint (2025/2026). arXiv:2508.12696. https://arxiv.org/abs/2508.12696
  5. Lucas Chesnel, Sergei A. Nazarov, Jari Taskinen, "Spectrum of the Laplacian with mixed boundary conditions in a chamfered quarter of layer", Journal of Spectral Theory 14 (2024), 37–57. DOI: 10.4171/jst/493; arXiv:2303.15345. https://doi.org/10.4171/jst/493 (preprint: https://arxiv.org/abs/2303.15345)
  6. Monique Dauge, Thomas Ourmières-Bonafos, Nicolas Raymond, "Spectral asymptotics of the Dirichlet Laplacian in a conical layer", Communications on Pure and Applied Analysis 14 (2015), 1239–1258. DOI: 10.3934/cpaa.2015.14.1239. https://doi.org/10.3934/cpaa.2015.14.1239