[Paper Review] Schrödinger Operators with Many Bound States

David Damanik, Christian Remling|arXiv (Cornell University)|Dec 26, 2004

Spectral Theory in Mathematical Physics16 references4 citations

TL;DR

This paper develops a novel method to estimate the potential in one-dimensional Schrödinger operators from their negative eigenvalues, using a decomposition of the potential into a Weyl-type term and a remainder. The key contribution is a scaling relation between eigenvalue decay and spatial intervals, enabling inverse Lieb-Thirring inequalities and sharp spectral results, including conditions for absence of negative spectrum and existence of absolutely continuous spectrum.

ABSTRACT

Consider the Schrödinger operators $H_{\pm}=-d^2/dx^2\pm V(x)$. We present a method for estimating the potential in terms of the negative eigenvalues of these operators. Among the applications are inverse Lieb-Thirring inequalities and several sharp results concerning the spectral properties of $H_{\pm}$.

Motivation & Objective

• To establish a systematic method for estimating the potential $ V(x) $ of one-dimensional Schrödinger operators from their discrete negative eigenvalues.
• To remove the technical $ L^2 $-local boundedness assumption used in prior work, such as in Damanik-Killip's finite eigenvalue result.
• To derive inverse Lieb-Thirring inequalities and spectral control results from eigenvalue data alone.
• To characterize the spectral type (absolutely continuous, singular continuous) on $ (0,rown) $ based on eigenvalue decay rates.
• To construct potentials with prescribed discrete spectra and controlled spectral behavior on $ (0,rown) $.

Proposed method

• Introduce a decomposition $ V = W' + W^2 + R $, where $ \|W\|_{L^2(I_n)} \lesssim |I_n|^{-1/2} $, with intervals $ I_n $ satisfying $ |I_n| \sim E_n^{-1/2} $.
• Use the scaling invariance $ V(x) \to g^2 V(gx), \ E \to g^2 E $ to guide the construction of intervals and eigenvalue estimates.
• Apply a perturbative argument via cut-off eigenfunctions to approximate eigenvalues on finite intervals, ensuring convergence to the full half-line spectrum.
• Leverage the interlacing property of eigenvalues under boundary condition changes to make $ \sum E_n^p < \infty $ independent of $ \alpha $.
• Use the fact that $ -d^2/dx^2 + q $ has no negative spectrum iff $ q = w' + w^2 $, to interpret the $ W' + W^2 $ term as a 'threshold' potential.
• Construct potentials as sums of localized potentials $ V_g $ or $ W_g $, placed at increasing distances $ x_n $, to control spectral type via sparsity.

Experimental results

Research questions

• RQ1 Can the potential $ V(x) $ be estimated from the negative eigenvalues of $ H_\pm = -d^2/dx^2 \pm V(x) $ without assuming $ V \in L^2_{\text{loc}} \cap L^\infty $?
• RQ2 What spectral properties emerge when the negative eigenvalues satisfy $ \sum E_n^{1/2} < \infty $?
• RQ3 Can one construct a potential with a prescribed sequence of negative eigenvalues and a specific spectral type on $ (0,\infty) $?
• RQ4 What inverse Lieb-Thirring-type bounds can be derived from eigenvalue decay rates?
• RQ5 How does the geometry of eigenvalue decay relate to the spatial structure of the potential?

Key findings

• If $ \sum E_n^{1/2} < \infty $, then there exists $ V_0 \in L^1(0,\infty) $ such that $ H_+ + V_0 $ with Dirichlet boundary conditions has no negative spectrum.
• More generally, if $ \sum E_n^p < \infty $ for $ p \geq 1/2 $, then there exists $ V_0 \in \ell_{2p}(L^1) $ such that $ H_+ + V_0 \geq 0 $.
• Under $ \sum E_n^{1/2} < \infty $, the operator $ H_+ $ has purely absolutely continuous spectrum on $ (0,\infty) $.
• For any sequence $ e_n > 0 $ with $ e_n \to 0 $ and $ \sum e_n = \infty $, a potential $ V \leq 0 $ exists such that $ H_+ $ has eigenvalues $ E_n \leq e_n $, and $ H_- $ has no negative spectrum.
• If $ \sum e_n^{1/4} = \infty $, a potential $ V $ can be constructed such that $ H_\pm $ have eigenvalues $ E_n \leq e_n $, and the spectrum of $ H_+ $ on $ (0,\infty) $ is purely singular continuous.
• When the placement distances $ x_n $ satisfy $ x_n / x_{n+1} \to 0 $, and $ \sum g_n = \infty $, the spectral type on $ (0,\infty) $ is singular continuous, as per known results on sparse potentials.