Rellich-Type Uniqueness Theorem

• Rellich-type uniqueness theorems are a class of results defining strict decay conditions that force eigenfunctions of elliptic PDEs to vanish in unbounded or exterior domains.
• They extend classical results from the Laplacian in Euclidean space to discrete settings, symmetric spaces, and operators like Schrödinger and Maxwell, using tools such as spherical harmonics and hypergeometric ODE analysis.
• Advanced methodologies, including Carleman estimates and Fourier analytic techniques, underpin the proofs and have significant implications for scattering theory, inverse problems, and spectral analysis.

A Rellich-type uniqueness theorem is a foundational result in analysis and mathematical physics characterizing the rigidity of solutions to elliptic partial differential equations (PDEs) and their discrete analogues in unbounded or exterior domains. The central assertion is that, under appropriate spectral and integrability or decay hypotheses, any sufficiently decaying eigenfunction of a given operator on a noncompact domain is necessarily trivial. Such theorems have been extended from the prototypical setting of the Laplacian in Euclidean space to symmetric spaces, stratified media, Schrödinger and Maxwell operators (both continuous and discrete), and generalized oscillators. The methodologies span techniques from spherical and spectral theory, commutator analysis, Carleman estimates, and advanced complex variable methods.

1. Classical Formulation and Extensions

The classical Rellich uniqueness theorem applies to the Helmholtz equation $(-\Delta-\lambda)u=0$ in an exterior domain in $\mathbb{R}^n$ with $L^{2}$-type decay conditions. Any solution decaying sufficiently rapidly at infinity must vanish identically outside a compact set. Banerjee–Garofalo extended the theorem to $L^{p}$ integrability for $1 \leq p \leq 2n/(n-1)$. In higher generality, Rellich-type theorems have been established for discrete Laplacians, operators with growing or stratified coefficients, domains in rank-one symmetric spaces, and more (Vesalainen, 2014, Isozaki et al., 2012, Tagawa, 2022, Ganguly, 16 Nov 2025, Isozaki et al., 16 Dec 2024, Humaikani et al., 24 Apr 2025).

2. Rank-One Symmetric Spaces and Non-Euclidean Uniqueness

On rank-one symmetric spaces of noncompact type $X=G/K$, the natural Laplace–Beltrami operator $\Delta_X$ exhibits fundamentally different spectral and asymptotic behavior due to exponential volume growth (encoded by the geometric parameter $\rho$). The sharp Rellich-type theorem in this geometry states: for $1\leq p\leq2$, $\lambda\in\mathbb{C}\setminus i\mathbb{Z}$ with $|{\rm Im}\,\lambda|\leq \gamma_p\rho$ ($\gamma_p=2/p-1$), any $f\in C^2(\Omega)$, $\Omega$ the exterior of a ball, solving $\Delta_X f+(\lambda^2+\rho^2)f=0$ and $\|f\|_{L^p(\Omega)}<\infty$ is identically zero. The proof employs spherical harmonic expansions, reduction to a hypergeometric ODE for radial components, and the analysis of their asymptotics. The uniqueness range $1\leq p\leq2$ stands in contrast to the $p$-dependent range in Euclidean settings, reflecting the impact of the exponential metric volume (Ganguly, 16 Nov 2025).

3. Discrete Operators and Lattice Models

Rellich-type uniqueness theorems have been rigorously established for discrete Laplacians and Maxwell operators on cubic and hypercubic lattices. For the discrete Schrödinger operator, if $u:\mathbb{Z}^d\rightarrow \mathbb{C}$ solves

$(\Delta_d + V - \lambda) u(n) = 0$

outside a finite region, with a minimal averaged $L^2$ decay condition $\lim_{R\to\infty}\frac1R\sum_{R_0<|n|<R}|u(n)|^2=0$, then $u$ vanishes outside a large ball. For discrete Maxwell operators with anisotropic medium, if $u$ satisfies the same type of decay in the Besov-space $B^*_0(\mathbb{Z}^3;\mathbb{C}^6)$ and the equation outside a compact, then $u$ vanishes identically at infinity for spectral parameters in the appropriate regime dictated by the coefficients (Isozaki et al., 16 Dec 2024, Isozaki et al., 2012).

4. Generalized Oscillator and Growth Characterizations

For the generalized oscillator Hamiltonian $H=-\Delta+2|x|^{2b}+V(x)$, Rellich-type theorems characterize the exponential rate of permitted growth or decay of eigenfunctions. If an $L^2$ solution $\phi$ of $(H-\lambda)\phi=0$ decays faster than $e^{-r^{b+1}/(b+1)}$ or grows faster than $e^{+r^{b+1}/(b+1)}$, then $\phi\equiv0$. Methods avoid microlocal or pseudodifferential analysis, relying instead on commutator techniques and explicit radial flow conjugate operators (Tagawa, 2022).

5. Stratified Media, Inhomogeneous Domains, and Junctions

In two-dimensional stratified media comprising unions of half-planes (possibly forming junctions), the Rellich-type theorem asserts that any $L^2$ solution of $-\Delta u - k(x,y)^2 u=0$ must vanish identically, with no need for explicit boundary conditions or Sommerfeld radiation constraints. The proof employs a generalized Fourier transform adapted to stratified operators and exploits analytic continuation and spectral results for one-dimensional Hamiltonians parametrizing the transversal variables (Humaikani et al., 24 Apr 2025).

6. Decay and Support Conditions; Carleman and Fourier–Complex Methods

Extensions to unbounded domains with nonstandard geometries (half-spaces, cones, etc.) and more general operators have been established using Carleman estimates and Fourier analytic division. Criteria for theorems include super-exponential decay of inhomogeneity (e.g., $e^{\gamma|x|}f\in L^2$ for all $\gamma>0$), “exponentially thin” support, and minimal $L^2$-averaged vanishing. In discrete analogues, domains may be taken as cones or exterior to balls, with decay criteria formulated in terms of sequences and support in $\ell^2$ or Besov-type spaces. Analytic continuation arguments, Paley–Wiener theorems, and Hartogs’ extension play key technical roles (Vesalainen, 2014, Isozaki et al., 2012).

7. Applications and Spectral Implications

Rellich-type uniqueness theorems have direct consequences for the absence of embedded eigenvalues, uniqueness of continuation, discreteness of non-scattering energies, and absence of trapped or guided modes in waveguides or photonic lattice structures under appropriate geometric or spectral assumptions. These results are central to the mathematical analysis of scattering theory, inverse problems in periodic and random media, and the qualitative spectral geometry of elliptic operators.

Selected Cited Sources:

• Rank-one symmetric spaces: (Ganguly, 16 Nov 2025)
• Generalized oscillator: (Tagawa, 2022)
• Stratified media: (Humaikani et al., 24 Apr 2025)
• Discrete Schrödinger: (Isozaki et al., 2012)
• Extensions to unbounded domains: (Vesalainen, 2014)
• Discrete Maxwell: (Isozaki et al., 16 Dec 2024)