Lattice Schrödinger Operators

Updated 19 November 2025

Lattice Schrödinger operators are discrete analogues of continuum Schrödinger operators defined on regular lattices with hopping terms and potential functions.
They enable rigorous spectral decomposition through Bloch–Floquet analysis, characterizing essential and discrete spectra crucial for solid-state physics.
These operators underpin scattering theory, dispersive estimates, and continuum limits, linking discrete models to effective partial differential equations.

A lattice Schrödinger operator is a self-adjoint (or, in some contexts, non-self-adjoint) difference operator acting on functions defined on the vertices of a regular lattice (such as $\mathbb{Z}^d$ ), designed as a discrete analogue of the continuum Schrödinger operator $-\Delta + V(x)$ . The paper of these operators encompasses spectral theory, localization, scattering, and integrable system connections, and plays a central role in mathematical physics, particularly in the physics of solid-state systems and models of quantum dynamics on lattices.

1. Algebraic and Functional Framework

Lattice Schrödinger operators typically act on $\ell^2(\mathbb{Z}^d)$ or sections of vector bundles over $\mathbb{Z}^d$ . The canonical form is

$H = H_0 + V$

where $H_0$ is a convolution (hopping) operator, often a Laurent-Toeplitz operator,

$(H_0 f)(x) = \sum_{y \in \mathbb{Z}^d} e(x-y) f(y),$

with $e\in \ell^1(\mathbb{Z}^d)$ a hopping matrix, and $V$ is a multiplication operator (potential). For periodic or finite-range hopping, $H_0$ is bounded and self-adjoint if $e(-a) = \overline{e(a)}$ . Via the (unitary) Fourier transform $\mathcal{F}$ ,

$(\mathcal{F} f)(p) = \sum_{x \in \mathbb{Z}^d} f(x) e^{-i p \cdot x}, \qquad p \in [-\pi, \pi]^d,$

$H_0$ is diagonalized: $\mathcal{F} H_0 \mathcal{F}^* = \text{multiplication by } e(p) = \sum_{x \in \mathbb{Z}^d} e(x) e^{i p \cdot x}.$

$V$ may be local, finite-range, decaying, periodic, quasi-periodic, or even complex or PT-symmetric.

2. Spectral Theory and Decomposition

Essential and Discrete Spectrum

The essential spectrum $\sigma_{\text{ess}}(H)$ is determined by the unperturbed operator and is stable under compact perturbations (Weyl's theorem). For periodic hopping,

$\sigma_{\text{ess}}(H) = \sigma(H_0) = [e_{\min}, e_{\max}],$

where $e_{\min} = \inf_{p \in \mathbb{T}^d} e(p)$ , $e_{\max} = \sup_{p \in \mathbb{T}^d} e(p)$ .

Discrete spectrum consists of eigenvalues outside (and sometimes embedded within) the essential spectrum, arising from appropriate decay or localization of $V$ or from higher-rank perturbations. These may be counted via Birman–Schwinger principles, variational formulas, or trace formulae (Kholmatov et al., 2022, Bach et al., 2017, Bach et al., 2017, Korotyaev et al., 2016, Korotyaev, 2017).

Bloch–Floquet Theory

For periodic lattices and periodic potentials, the Bloch–Floquet transform decomposes the operator as a direct integral over the Brillouin zone: $U H U^* = \int_{\mathbb{T}^d}^\oplus H(p) \, dp.$ The fiber operators $H(p)$ are finite-dimensional or difference operators with periodic coefficients.

For non-self-adjoint cases with potentials supported on a half-space in reciprocal space, the Bloch eigenvalues and Fermi surfaces of the perturbed and unperturbed operators coincide, leading to purely absolutely continuous spectrum without gap opening (Veliev, 2016).

3. Discrete and Threshold Spectral Analysis

Discrete Spectrum: Weyl Bounds and Counting Functions

For potentials $V \geq 0$ decaying at infinity, the number of eigenvalues $N(\lambda)$ below the essential spectrum for $H = H_0 + \lambda V$ tracks the semiclassical phase space volume: $N_{\text{sc}}(\lambda) = \int_{[-\pi, \pi]^d} \sum_{x \in \mathbb{Z}^d} \mathbf{1}\{e(p) - \lambda V(x) < 0\} \frac{dp}{(2\pi)^d}.$ In $d \geq 3$ and for $V \in \ell^{d/2}$ , two-sided Weyl bounds hold provided slow enough growth of level sets for $V$ (Bach et al., 2017). In $d=1,2$ , additional spatial weights are necessary due to slower Green’s function decay. For certain very sparse potentials, the semi-classical correspondence can fail, with $N(\lambda)$ oscillating between $0$ and $\infty$ compared to $N_{\text{sc}}(\lambda)$ .

Pure Point Spectrum: Variational and Absence/Existence Criteria

A variational principle bounds the number of eigenvalues by the minimal support size of $V$ up to small tail error. Sufficient decay conditions guarantee absence of embedded and threshold eigenvalues. For discrete Schrödinger operators, explicit criteria for zero-energy (threshold) bound states are available: if $V(x) \lesssim -2d + c/|x|^2$ asymptotically with a precise coefficient, there is no threshold state; if $V(x) \gtrsim -2d + c/|x|^2$ and $H$ is critical at zero, a threshold bound state exists. The threshold between absence and existence is dictated by the discrete Hardy inequality, with explicit logarithmic corrections in the lattice case (Jex et al., 2023). Operator-theoretic approaches guarantee absence of threshold or embedded eigenvalues under sufficient decay (Bach et al., 2017).

Threshold Resonances and Eigenvalues

Analysis of simple and higher-rank localized perturbations reveals a sharp dichotomy: for sufficiently localized (e.g., delta) potentials, the discrete spectrum can display threshold eigenvalues (embedded at band edge), threshold resonances (non- $\ell^2$ solutions decaying at infinity), or super-threshold resonances in dimension one, determined by exploiting the Birman–Schwinger principle and explicit momentum integrals (Hiroshima et al., 2018, Kholmatov et al., 2022, Jex et al., 2023).

4. Scattering Theory and Limiting Absorption

Short-Range and Long-Range Potentials

For short-range $V$ , standard stationary and time-dependent scattering theory holds on square, hexagonal, and other periodic lattices (Ando et al., 2014, Tadano, 2018). The existence and completeness of (modified or standard) wave operators is established, and the absolutely continuous spectrum is preserved under compact perturbations. The construction employs performance of pseudodifferential calculus on the lattice, lattice analogues of the Mourre and Virial commutator estimates, and radiation conditions formulated either in position space or in microlocal scales (discrete Besov spaces), ensuring the limiting absorption principle holds away from exceptional (threshold) energies.

For long-range and slowly decaying potentials, Isozaki–Kitada modifiers are constructed in the form of lattice pseudodifferential operators, whose phase solves a lattice eikonal equation, ensuring completeness of modified wave operators outside of threshold and band-edge singularities (Tadano, 2016, Tadano, 2018).

Inverse Scattering

On $\mathbb{Z}^2$ or on the hexagonal/triangular lattice, for compactly supported $V$ , the on-shell scattering matrix at all energies determines $V$ uniquely. Explicit reconstruction algorithms exploit asymptotic expansions of the scattering amplitude in complex energy, isolating moments of $V$ via limits at infinite spectral parameter (Isozaki et al., 2011, Ando, 2011). The S-matrix is constructed from the generalized eigenfunctions and the spectral representation given by the Bloch–Floquet or trace transform.

5. Periodic, Quasi-Periodic, and Integrable Examples

Periodic and Quasi-Periodic Potentials

The spectral analysis of periodic and quasi-periodic lattice Schrödinger operators is accomplished via Floquet–Bloch theory. The Bethe–Sommerfeld property holds: in prime cases, only a finite number of spectral gaps can open for small periodic $V$ ; for certain arithmetic lattices, gaps can only occur at specified band edges, with explicit no-gap criteria depending on lattice symmetry (Fillman et al., 2018). For small analytic quasi-periodic potentials, multiscale (KAM-type) induction establishes strong Green’s function control, leading to Anderson localization and quantitative H\"older continuity of the integrated density of states (IDS), thereby resolving long-standing conjectures (Cao et al., 2022).

Lattice Integrable Systems and Orthogonal Polynomials

Special classes of 2D operators, such as cross-shaped difference operators, can be constructed to possess as formal eigenvectors multiple orthogonal polynomial solutions, generalizing the link between Jacobi matrices and orthogonal polynomials to the 2D lattice case. The spectrum and eigenfunction expansions in these integrable systems are intimately related to the properties (recurrence relations, Riemann–Hilbert analysis) of the underlying multiple orthogonal polynomials (Aptekarev et al., 2014).

6. Continuum/Scaling Limits and Dispersive Properties

Continuum Limit and Effective PDEs

Scaling limits, as lattice spacing $h\to 0$ , connect the discrete Helmholtz or Schrödinger equation to effective continuum PDEs on $\mathbb{R}^d$ , with the band structure yielding limiting operators of Schrödinger or Dirac type depending on the local structure of the lattice's dispersion relation at the relevant quasimomentum (minima, Dirac points) (Isozaki et al., 2020). Spectral projections, gauge transformations, and pseudodifferential convergence theory ensure that solutions to the lattice equations converge in weighted Sobolev spaces to solutions of the limiting equations.

Time Decay and Dispersive Estimates

On $\mathbb{Z}$ , Schrödinger and bi-Schrödinger evolution display distinct dispersive time-decay: the discrete Laplacian yields $|t|^{-1/3}$ decay ( $\ell^1\to\ell^\infty$ ), which is slower than the $|t|^{-1/2}$ in the continuum. Strikingly, the discrete bi-Laplacian (and appropriate perturbations) restore $|t|^{-1/4}$ decay, equal to the continuum rate for the same order operator, provided spectral regularity at thresholds (Huang et al., 4 Apr 2025). For wave/beam equations associated to lattice Schrödinger operators, decay rates differ according to the order of dispersive relations and thresholds.

7. Trace Formulae and Spectral Identities

Canonical trace formulae relate the sums of eigenvalues, their total real and imaginary parts, and higher spectral moments to traces of powers of the potential, with complete analogues in the lattice setting for both self-adjoint and non-selfadjoint (complex) potentials. The form and analytic continuation of the Fredholm determinant play a central role; factorization in Hardy spaces yields Blaschke product representations and precise sum rules. Eigenvalue sums are bounded in terms of $\ell^p$ norms of $V$ , and singular spectral measures on the continuous spectrum are similarly controlled (Korotyaev et al., 2016, Korotyaev, 2017, Isozaki et al., 2011).