Persistence and disappearance of negative eigenvalues in dimension two

TL;DR

This work analyzes how negative eigenvalues at the bottom of the spectrum behave for Schrödinger operators in two dimensions as a coupling parameter tends to zero, revealing a sharp dichotomy between persistent and disappearing eigenvalues tied to zero-energy resonances. Using a detailed radial analysis with Bessel/Hankel functions and resolvent techniques, it provides explicit low-energy scattering phase asymptotics that depend on whether the zero-energy state is s-, p-, or nonexistent, and demonstrates how persistent eigenvalues shape prominent Breit–Wigner features in the scattering phase. The circular well serves as the principal model, yielding precise resonance locations and asymptotics; these are then extended to more general radial perturbations via Birman–Schwinger reductions and resonances on the Riemann surface of the logarithm. The results sharpen the understanding of zero-energy spectral phenomena in 2D and offer practical tools for predicting low-energy scattering behavior and resonance phenomena in planar quantum systems and subwavelength resonator contexts.

Abstract

We compute asymptotics of eigenvalues approaching the bottom of the continuous spectrum, and associated resonances, for Schrödinger operators in dimension two. We distinguish persistent eigenvalues, which have associated resonances, from disappearing ones, which do not. We illustrate the significance of this distinction by computing corresponding scattering phase asymptotics and numerical Breit--Wigner peaks. We prove all of our results for circular wells, and extend some of them to more general problems using recent resolvent techniques.

Figures (6)

• Figure 1: Plot of $\lambda_\varepsilon$ in $\mathbb C \setminus -i(0,\infty)$ for $a^2 = j_{\ell-1,1}^2 - \varepsilon$, with $\ell=1$ (left) $\ell=2$ (middle) and $\ell=3$ (right). The values of $\varepsilon$ range from $\varepsilon = -0.81$ to $\varepsilon=0.81$. When $\varepsilon<0$, $\lambda_\varepsilon$ is on the positive imaginary axis and corresponds to an eigenvalue, when $\varepsilon>0$, $\lambda_\varepsilon$ is in the fourth quadrant and is a resonance.
• Figure 2: Plot of $\lambda_\varepsilon$ for $\ell=0$ mode, with $\varepsilon \in \{-k\Delta\varepsilon : k=2,\ldots,25\}$, with $\Delta\varepsilon = 0.3$ in the first plot and $\Delta\varepsilon=0.1$ in the second plot. Approx. is $\lambda_\varepsilon = i \frac{2}{\rho} \exp(\frac{2}{\rho^2\varepsilon} - \gamma)$.
• Figure 3: Graphs of $\sigma'(\lambda)$ for $-\Delta - a^2 \mathbbm{1}_1$, with $a^2 = j_{1,1}^2-\varepsilon$ on the left (perturbing a zero eigenvalue) and $a^2 = j_{0,1}^2-\varepsilon$ (perturbing a $p$-resonance at zero) on the right. The cases $\varepsilon \ne 0$ all correspond to case (1) of Theorem \ref{['t:spintro']}. Case (2) of Theorem \ref{['t:spintro']} is $\varepsilon=0$ on the right and Case (3) of Theorem \ref{['t:spintro']} is $\varepsilon=0$ on the left. The behavior for larger $\lambda$ is in Figure \ref{['fig:sigP0']}.
• Figure 4: On the left, the first three resonances of $-\frac{d^2}{dx^2} + 10\delta_1$ on the half line. On the right, a graph of $\sigma'(\lambda)$. The vertical lines are at $\mathop{\mathrm{Re}}\nolimits \lambda_1$, $\mathop{\mathrm{Re}}\nolimits \lambda_2$, $\mathop{\mathrm{Re}}\nolimits \lambda_3$, and the dashed curve is the approximation $\frac{-1}{\pi} - \frac{\mathop{\mathrm{Im}}\nolimits \lambda_1}{\pi|\lambda - \lambda_1|^2}$.
• Figure 5: Graphs of $\sigma'(\lambda)$ for $-\Delta - a^2 \mathbbm{1}_1$, with $a^2 = j_{1,1}^2-\varepsilon$ on the left (perturbing a zero eigenvalue) and $a^2 = j_{0,1}^2-\varepsilon$ (perturbing a $p$-resonance at zero) on the right. The vertical lines are at $\mathop{\mathrm{Re}}\nolimits \lambda_\varepsilon$, with $\lambda_\varepsilon$ as in Figure \ref{['f:3foldres']}. Each $\varepsilon=0.09$ curve has a maximum when $\lambda \approx \mathop{\mathrm{Re}}\nolimits \lambda_\varepsilon$, with $\sigma'(\mathop{\mathrm{Re}}\nolimits \lambda_\varepsilon)\approx 367.37$ on the left, and $\sigma'(\mathop{\mathrm{Re}}\nolimits \lambda_\varepsilon)\approx 17.048$ on the right. The dashed curves are Breit--Wigner approximations $\frac{-\mathop{\mathrm{Im}}\nolimits \lambda_\varepsilon}{\pi|\lambda - \lambda_\varepsilon|^2} - 0.8\sqrt \lambda$ and $\frac{-\mathop{\mathrm{Im}}\nolimits \lambda_\varepsilon}{\pi|\lambda - \lambda_\varepsilon|^2} + \log \lambda$, with the non-resonance terms $\sqrt \lambda$ and $\log \lambda$ (meant to be analogous to $-L/\pi$ in \ref{['e:bw1d']}) chosen ad hoc by eye to improve the fit.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4