[Paper Review] On the spectrum of a Schroedinger operator perturbed by a fast oscillating potential

D. I. Borisov|ArXiv.org|May 25, 2005

Spectral Theory in Mathematical Physics27 references3 citations

TL;DR

This paper analyzes the spectral properties of a one-dimensional Schrödinger operator with a fast-oscillating potential, deriving the essential spectrum in explicit form and constructing complete asymptotic expansions for eigenvalues and eigenfunctions as the oscillation period $\varepsilon \to 0$. The key contribution is the rigorous derivation of the asymptotic structure of discrete eigenvalues in lacunas of the essential spectrum, including conditions for their existence and precise higher-order expansions for both eigenvalues and associated eigenfunctions.

ABSTRACT

We study the spectrum of a one-dimensional Schroedinger operator perturbed by a fast oscillating potential. The oscillation period is a small parameter. The essential spectrum is found in an explicit form. The existence and multiplicity of the discrete spectrum are studied. The complete asymptotics expansions for the eigenvalues and the associated eigenfunctions are constructed.

Motivation & Objective

• To characterize the essential spectrum of a Schrödinger operator perturbed by a fast-oscillating potential in $L_2(\mathbb{R})$.
• To determine the existence, number, and multiplicity of discrete eigenvalues in the lacunas of the essential spectrum.
• To construct complete asymptotic expansions for the eigenvalues and associated eigenfunctions as the oscillation parameter $\varepsilon \to 0$.
• To establish precise conditions under which discrete eigenvalues emerge in the semi-infinite and interior lacunas of the essential spectrum.

Proposed method

• The essential spectrum is derived using spectral theory of periodic differential operators and homogenization techniques in unbounded domains.
• The operator $H_\varepsilon = -\frac{d^2}{dx^2} + V(x) + a(x/\varepsilon)$ is analyzed with $V$ compactly supported and $a$ 1-periodic and mean-zero.
• Asymptotic expansions for eigenvalues are constructed via a multiscale analysis and matching of solutions in different regions (oscillatory and regular).
• The eigenfunctions are asymptotically expanded using a decomposition into oscillatory and decaying components, with coefficients determined by solving auxiliary spectral problems.
• The existence of eigenvalues in lacunas is linked to the sign of certain integrals involving $V$ and the Fourier coefficients of $a$, via the construction of a transfer matrix and associated eigenvalue condition.
• Higher-order asymptotics are derived using iterative refinement of the eigenvalue equation, with error terms controlled via operator norm estimates and perturbation theory.

Experimental results

Research questions

• RQ1 What is the explicit form of the essential spectrum of the Schrödinger operator $H_\varepsilon$ with a fast-oscillating potential as $\varepsilon \to 0$?
• RQ2 Under what conditions do discrete eigenvalues appear in the lacunas of the essential spectrum?
• RQ3 What are the complete asymptotic expansions for the eigenvalues and eigenfunctions of $H_\varepsilon$ in the limit $\varepsilon \to 0$?
• RQ4 How do the Fourier coefficients of the oscillatory potential $a$ and the integral of the background potential $V$ influence the existence and location of discrete eigenvalues?

Key findings

• The essential spectrum of $H_\varepsilon$ is given by $\sigma_{\text{ess}}(H_\varepsilon) = \bigcup_{n=0}^{\infty} [\mu_n^+(\varepsilon^2), \mu_{n+1}^-(\varepsilon^2)]$, where $\mu_n^\pm(t)$ are meromorphic functions with explicit power series expansions in $t = \varepsilon^2$.
• For the semi-infinite lacuna ($n=0$), the first-order correction to the threshold is $\mu_{0,1}^+ = -\int_0^1 \left( \int_0^\xi a(\eta)\,d\eta + \int_0^1 a(\eta)\eta\,d\eta \right)^2 d\xi$, which depends on the cumulative integral of $a$.
• Discrete eigenvalues exist in the $n$-th interior lacuna if and only if $\tau_{\mathfrak{n}_\pm}^\pm > 0$, where $\tau_{\mathfrak{n}_\pm}^\pm$ are coefficients derived from the potential $V$ and the Fourier coefficients of $a$.
• When $\int_\mathbb{R} V(x)\,dx \geq 0$, the eigenvalue $\lambda_{\varepsilon,-}$ exists in the $n$-th lacuna for $n \geq 2$, with $\tau_2^- > 0$ if the integral is positive and $\tau_4^- > 0$ if it vanishes.
• The eigenvalues admit complete asymptotic expansions of the form $\lambda_{\varepsilon,\pm} = \mu_n^\pm + \varepsilon^{2\mathfrak{n}_\pm-2} \cdot \text{lower-order terms}$, with coefficients determined by $V$, $a$, and their Fourier transforms.
• The associated eigenfunctions are asymptotically equivalent to $\psi_{\varepsilon,\pm} = c_\pm(\varepsilon) \cdot \varepsilon T_{11}^\pm(\varepsilon, k_{\varepsilon,\pm}) g_{\varepsilon,\pm}$ in $W_2^2(Q)$, with $g_{\varepsilon,\pm}$ solving a reduced spectral problem and $c_\pm(\varepsilon) \to 1$ as $\varepsilon \to 0$.