[Paper Review] Schrödinger operators with singular Gordon potentials

Rostyslav Hryniv, Yaroslav V. Mykytyuk|ArXiv.org|Sep 19, 2001

Spectral Theory in Mathematical Physics4 references16 citations

TL;DR

This paper introduces singular Gordon potentials in the space $W^{-1}_{2,unif}({\mathbb{R}})$, defined as distributions that are rapidly approximated by periodic functions in a weighted $L^2$ and $L^1$ sense. It proves that Schrödinger operators with such potentials have no point spectrum, extending the classical Gordon result to singular, distributional potentials including those modeling $δ$-interactions.

ABSTRACT

Singular Gordon potentials are defined to be distributions from the space W^{-1}_{2,unif}(R) that are sufficiently fast approximated by periodic ones. We prove that Schrödinger operators with singular Gordon potentials have no point spectrum and show that a rich class of quasiperiodic distributions consists of singular Gordon potentials.

Motivation & Objective

• To extend the classical Gordon theorem on absence of point spectrum to singular potentials in $W^{-1}_{2,unif}({\mathbb{R}})$, including distributional potentials like $\delta$-interactions.
• To define a new class of potentials—singular Gordon potentials—that generalize classical Gordon potentials to the setting of singular, non-locally integrable potentials.
• To establish that Schrödinger operators with such singular potentials are free of eigenvalues, ensuring purely continuous spectrum.
• To demonstrate that quasiperiodic potentials involving Liouville numbers and periodic components qualify as singular Gordon potentials.
• To provide a spectral analysis framework for singular, quasi-periodic interactions in one-dimensional quantum mechanics using a quasi-derivative regularization.

Proposed method

• Define singular Gordon potentials as $q = \sigma' + \tau$ with $\sigma \in L_{2,unif}({\mathbb{R}})$, $\tau \in L_{1,unif}({\mathbb{R}})$, satisfying rapid approximation by periodic $\sigma_m$, $\tau_m$ in $L^2$ and $L^1$ norms over intervals $[-T_m, 2T_m]$.
• Use the quasi-derivative regularization to define the Schrödinger operator $S = -d^2/dt^2 + q$ on a domain involving $W^1_2({\mathbb{R}})$ functions and a modified derivative $f^{[1]} = f' - \sigma f$.
• Apply the form-sum construction to show self-adjointness and boundedness from below of $S$ for potentials in $W^{-1}_{2,unif}({\mathbb{R}})$.
• Establish a key estimate (Theorem 4.4) on the $L^2$-norm of differences $|f(\alpha t + \theta) - f(\beta t + \theta)|$ over intervals, showing $O(T^{2s+1} (\beta - \alpha)^{2s})$ decay in terms of $W^s_{2,unif}$-norm.
• Leverage the Liouville approximation property of irrational $\alpha$ to show that $\alpha_m = R_m/T_m$ with $|\alpha - \alpha_m| \leq C m^{-T_m}$ yields exponentially decaying approximation errors.
• Combine the approximation estimate with the spectral argument to prove that the essential spectrum is continuous and no $L^2$ eigenfunctions exist.

Experimental results

Research questions

• RQ1 Can the absence of point spectrum in Schrödinger operators be extended to singular potentials that are not locally integrable?
• RQ2 What conditions on a distributional potential $q \in W^{-1}_{2,unif}({\mathbb{R}})$ ensure that the corresponding Schrödinger operator has no eigenvalues?
• RQ3 Do quasiperiodic potentials involving Liouville numbers and periodic components qualify as singular Gordon potentials?
• RQ4 Can the classical Gordon condition for eigenvalue absence be generalized to potentials with $\delta$-like singularities?
• RQ5 What role does the rate of approximation by periodic functions play in the spectral type of Schrödinger operators with singular potentials?

Key findings

• Schrödinger operators with singular Gordon potentials in $W^{-1}_{2,unif}({\mathbb{R}})$ have no point spectrum, meaning no $L^2$ eigenvalues exist.
• The class of singular Gordon potentials includes all periodic potentials in $W^{-1}_{2,unif}({\mathbb{R}})$, extending the classical result to singular settings.
• Quasiperiodic potentials of the form $q(t) = \sigma_1'(t) + \sigma_2'(\alpha t + \theta) + \tau_1(t) + \tau_2(\alpha t + \theta)$ with $\alpha$ a Liouville number are singular Gordon potentials.
• For such quasiperiodic potentials, the approximation error in $L^2$ and $L^1$ norms decays faster than any exponential, satisfying the singular Gordon condition.
• The key estimate $\|A_1 f\|_{L^2(-T,2T)} \leq C T^{2s+1} (\beta - \alpha)^{2s} \|f\|_{W^s_{2,unif}}$ underpins the proof and holds uniformly for $s \in [0,1]$.
• The result holds even if the approximation condition is required only for $C < C_q$, indicating robustness of the spectral conclusion.