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arXiv:2607.08320

Approximate eigenfunctions for some aperiodic crystals

Long Meng

math-phcond-mat.mes-hallmath.FAmath.SPquant-phSubmitted Jul 9, 2026• Announced Jul 10, 2026• Revised Sep 14, 2026 (v2)
v1 · Jul 9, 2026v2 · Sep 14, 2026
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Abstract

In this paper, we consider a broad class of continuum two-scale Hamiltonians \begin{align*} H_\varepsilon:=T(-i\nabla_x+\mathbf A(x,\varepsilon x))+V(x,\varepsilon x),\qquad x\in \mathbb{R}^d \end{align*} where $T$ represents either a Dirac operator or a Schrödinger operator, and $x\mapsto \mathbf A(x,X)$ and $x\mapsto V(x,X)$ are $\mathbb L$-periodic with respect to some lattice $\mathbb L\subset\mathbb{R}^d$. No periodicity assumption is imposed on the second variable $X$. Let \begin{align*} \mathbb{R}^d\times \mathbb{R}^d\ni (k,X) \mapsto h(k,X):=T(-i\nabla_x+k+\mathbf A(x,X))+V(x,X) \end{align*} be a family of operators acting on $L^2(\mathbb{R}^d/\mathbb{L})$ with periodic boundary conditions. We assume only local spectral information near one point $(k_0,X_0)$: an isolated $J$-dimensional Bloch bands at energy $e_0$, possibly with internal crossings, and a Hermitian matrix-valued homogeneous function of degree $m\in\{1,2\}$. From these data we derive an effective Hamiltonian $\mathfrak{h}$. Then every sufficiently localized eigenpair $(\vec v,μ)$ of $\mathfrak{h}$ gives an approximate eigenfunction $Θ_\varepsilon$ such that for $\varepsilon$ small enough, \[ \|Θ_\varepsilon\|_{L^2(\mathbb{R}^d)}=|Ω|^{-1/2}+O(\varepsilon^{1/2}),\qquad \|(H_\varepsilon-e_0-\varepsilon^{m/2}μ)Θ_\varepsilon\|_{L^2(\mathbb{R}^d)} =O(\varepsilon^{m/2+1/4}). \]

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51 pages, 1 figures

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In this paper, we consider a broad class of continuum two-scale Hamiltonians \begin{align*}
H_\varepsilon:=T(-i\nabla_x+\mathbf A(x,\varepsilon x))+V(x,\varepsilon x),\qquad x\in \mathbb{R}^d \end{align*} where TTT represents either a Dirac operator or a Schrödinger operator, and x↦A(x,X)x\mapsto \mathbf A(x,X)x↦A(x,X) and x↦V(x,X)x\mapsto V(x,X)x↦V(x,X) are L\mathbb LL-periodic with respect to some lattice L⊂Rd\mathbb L\subset\mathbb{R}^dL⊂Rd. No periodicity assumption is imposed on the second variable XXX.
Let \begin{align*}
\mathbb{R}^d\times \mathbb{R}^d\ni (k,X) \mapsto h(k,X):=T(-i\nabla_x+k+\mathbf A(x,X))+V(x,X) \end{align*} be a family of operators acting on L2(Rd/L)L^2(\mathbb{R}^d/\mathbb{L})L2(Rd/L) with periodic boundary conditions. We assume only local spectral information near one point (k0,X0)(k_0,X_0)(k0​,X0​): an isolated JJJ-dimensional Bloch bands at energy e0e_0e0​, possibly with internal crossings, and a Hermitian matrix-valued homogeneous function of degree m∈{1,2}m\in\{1,2\}m∈{1,2}. From these data we derive an effective Hamiltonian h\mathfrak{h}h. Then every sufficiently localized eigenpair (v⃗,μ)(\vec v,μ)(v,μ) of h\mathfrak{h}h gives an approximate eigenfunction ΘεΘ_\varepsilonΘε​ such that for ε\varepsilonε small enough, [ |Θ_\varepsilon|_{L^2(\mathbb{R}^d)}=|Ω|^{-1/2}+O(\varepsilon^{1/2}),\qquad
|(H_\varepsilon-e_0-\varepsilon^{m/2}μ)Θ_\varepsilon|_{L^2(\mathbb{R}^d)}
=O(\varepsilon^{m/2+1/4}). ]
An exactly solvable lattice with periodic inverse-square singularities exhibits two physically distinct mechanisms of spectral-gap closure. The model arises from the generalized trigonometric Pöschl-Teller potential in the coupling range where both local solutions at each singularity are square-integrable. A transmitting, current-preserving self-adjoint matching condition connects neighboring cells and generates a nontrivial Bloch spectrum. For a generic lattice with two inequivalent singular sublattices, the spectrum contains two alternating families of gaps, and the band edges reproduce the scale-invariant spectra of the associated finite singular system. One gap family closes when the two sublattices become equivalent. In this symmetric limit, however, the primitive period halves, so the apparent band crossing results from Brillouin-zone folding. A different family closes at a critical but genuinely asymmetric configuration. Because the primitive period is preserved, adjacent bands then meet at the center of the original Brillouin zone in a genuine massless Dirac-like cone. Near either critical limit, a small finite gap acts as an effective Dirac mass: the dispersion remains quadratic near a band edge but crosses over to a massive Dirac-like form beyond that narrow region. The band curvatures define effective electron and hole masses. Exact Bloch eigenfunctions further reveal that isospectral lattices can display different single-particle probability densities. This model therefore provides an analytically controlled one-dimensional setting in which conventional quadratic carriers, massive Dirac-like band dispersions, and a tunable massless contact coexist within the same singular lattice.
We show that the thermal properties of periodic lattice systems can be approximated to that of a finite cluster with appropriate boundary conditions which include a modified Hamiltonian for the boundary variables. Imposing lattice invariance on the correlation of the local site variables is sufficient to obtain the free parameters of the boundary Hamiltonian. The degree of accuracy of the calculation depends on the interaction range allowed in the boundary Hamiltonian and the range up to which the correlation of the site variables are made lattice invariant. The Bethe approximation can be considered a trivial case of this general method for clusters of one lattice site. The reliability of the method is demonstrated with the results obtained for the two-dimensional Ising model, where a cluster of four spins and invariance conditions up to second neighbors is sufficient to reproduce some nonuniversal thermal properties of the model with an accuracy comparable or better than other more complex numerical methods.
We study wave propagation in 2D honeycomb structures with a non-commensurate or ``irrational'' line defect or edge. Our model is a Schrödinger operator which interpolates, across the edge, between two distinct bulk (asymptotic) Hamiltonians with a common spectral gap about the ``Dirac point'' of an unperturbed honeycomb operator. We seek edge states, eigenstates that are bounded and oscillatory parallel to the edge, and decaying in the transverse direction. For non-commensurate edges, the rigorous definition of these states is nontrivial due to the lack of translation invariance along the edge. To address this, we exploit quasiperiodicity along the edge by expressing the Hamiltonian as the restriction of a 3D (degenerate elliptic) Hamiltonian describing a 3D medium with a 2D interface within which there is periodicity. Via multiscale analysis, we construct approximate edge states in this 3D setting and obtain by restriction 2D edge states which are quasiperiodic along the irrational edge. These edge states are seeded by eigenfunctions of an effective Dirac operator, which has an infinite block-diagonal structure due to the non-commensurate geometry. A consequence is that infinitely many edge state eigenpairs arise, whose energies are dense in the perturbed bulk spectral gap. In a forthcoming paper, we rigorously construct these gap-filling edge states under a Diophantine condition. The main result here is a key tool in this construction: a resolvent expansion for the 3D Hamiltonian, whose leading term is the resolvent of the block-diagonal Dirac operator. The validity of this expansion requires an omnidirectional non-resonance (no-fold) condition on the dispersion functions of the unperturbed honeycomb Hamiltonian. This condition is satisfied in the strong binding regime. In contrast with earlier works on commensurate edges, the omnidirectional condition is independent of the edge.
We present a semiclassical analysis of the high-energy eigenstates of an electron inside a closed resonator. An asymptotic method of the construction of the energy spectrum and eigenfunctions, localized in the small neighborhood of a periodic orbit, is developed in the presence of a homogeneous magnetic field and arbitrary scalar potential. The isolated periodic orbit is confined between two interfaces which could be planar, concave or even convex. Such a system represents a quantum electronic resonator, an analog of the well-known high-frequency optical or acoustic resonator with eigenmodes called 'bouncing ball vibrations'. The first step in the asymptotic analysis involves constructing a solitary localized asymptotic solution to the Schrodinger equation (electronic Gaussian beam—wavepackage). Then, the stability of a closed continuous family of periodic trajectories confined between two reflecting surfaces of the resonator boundary was studied. The asymptotics of the eigenfunctions were constructed as a superposition of two electronic Gaussian beams propagating in opposite directions between two reflecting points of the periodic orbits. The asymptotics of the energy spectrum are obtained by the generalized Bohr–Sommerfeld quantization condition derived as a requirement for the eigenfunction asymptotics to be periodic. For one class of periodic orbits, localized eigenstates were computed numerically by the finite element method using FEMLAB and proved to be in a very good agreement with those computed semiclassically.