[Paper Review] On the Bound States of Schr\"odinger Operators with $\delta$-interactions on Conical Surfaces

Vladimir Lotoreichik, Thomas Ourmières-Bonafos|arXiv (Cornell University)|Oct 19, 2015

Spectral Theory in Mathematical Physics35 references22 citations

TL;DR

This paper investigates Schrödinger operators with $δ$-interactions on conical surfaces in dimensions $d \geq 3$. Using spectral decomposition into angular momentum fibers, it proves that discrete spectrum exists only in $d=3$, where eigenvalues accumulate logarithmically below the essential spectrum threshold $-\alpha^2/4$. The precise asymptotic rate of accumulation is shown to be $N_{-\alpha^2/4 - \varepsilon}(H_{\alpha,C}) \sim \frac{\cot\theta}{4\pi} |\ln \varepsilon|$ as $\varepsilon \to 0^+$, and eigenvalues are non-decreasing in the cone's aperture $\theta$. For $d \geq 4$, no discrete spectrum exists.

ABSTRACT

In dimension greater than or equal to three, we investigate the spectrum of a Schr{\"o}dinger operator with a $\delta$-interaction supported on a cone whose cross section is the sphere of co-dimension two. After decomposing into fibers, we prove that there is discrete spectrum only in dimension three and that it is generated by the axisymmetric fiber. We get that these eigenvalues are non-decreasing functions of the aperture of the cone and we exhibit the precise logarithmic accumulation of the discrete spectrum below the threshold of the essential spectrum.

Motivation & Objective

• To determine the existence and structure of the discrete spectrum for Schrödinger operators with $δ$-interactions supported on conical hypersurfaces in $\mathbb{R}^d$, $d \geq 3$.
• To analyze how the geometry of the cone—specifically its aperture $θ$—influences the spectrum, particularly the eigenvalues below the essential spectrum.
• To establish sharp spectral asymptotics for the number of eigenvalues accumulating below the threshold $-\alpha^2/4$ in dimension $d=3$.
• To prove that for $d \geq 4$, no discrete spectrum exists, contrasting with the $d=3$ case and highlighting dimensional dependence.

Proposed method

• Decomposing the problem via separation of variables into angular momentum fiber components using spherical harmonics, reducing the $d$-dimensional problem to a family of 2D problems on the $(r,z)$-plane.
• Analyzing the resulting fiber operators $Q^{[l]}_{\alpha,\Gamma_\theta}$ for each angular momentum quantum number $l$, with domain $L^2(\mathbb{R}^2_+; r^{d-2} dr dz)$.
• Proving that the lowest fiber operator ($l=0$) governs the discrete spectrum in $d=3$ by showing its spectrum determines the spectrum of the full operator below $-\alpha^2/4$.
• Establishing unitary equivalence between the lowest fiber operator and a form whose domain is independent of $\theta$, enabling monotonicity analysis.
• Using variational methods and Rayleigh quotients to show that eigenvalues $E_k(H_{\alpha,C})$ are non-decreasing functions of the cone aperture $\theta$ in $d=3$.
• Applying spectral asymptotics techniques inspired by [11] to derive the precise logarithmic rate of accumulation of eigenvalues near $-\alpha^2/4$ in $d=3$.

Experimental results

Research questions

• RQ1 Does the discrete spectrum of a Schrödinger operator with a $δ$-interaction on a conical surface exist in dimensions $d \geq 4$, and if so, how many eigenvalues are there?
• RQ2 What is the precise rate at which eigenvalues accumulate below the essential spectrum threshold $-\alpha^2/4$ in dimension $d=3$?
• RQ3 How does the cone's aperture $\theta$ affect the location of the eigenvalues in $d=3$?
• RQ4 Why does the discrete spectrum vanish for $d \geq 4$ despite the presence of an attractive $δ$-interaction on a conical surface?
• RQ5 Is the lowest fiber operator ($l=0$) sufficient to capture the entire discrete spectrum in $d=3$?

Key findings

• In dimension $d=3$, the operator $H_{\alpha,C}$ has infinitely many bound states below the essential spectrum threshold $-\alpha^2/4$, i.e., $\#\sigma_{\text{dis}}(H_{\alpha,C}) = \infty$.
• For $d \geq 4$, the operator $H_{\alpha,C}$ has no discrete spectrum, i.e., $\#\sigma_{\text{dis}}(H_{\alpha,C}) = 0$, due to unitary equivalence to an orthogonal sum of fiber operators with spectra bounded below by $-\alpha^2/4$.
• The eigenvalues $E_k(H_{\alpha,C})$ in $d=3$ are non-decreasing functions of the cone's aperture $\theta$ on $(0, \pi/2)$, meaning larger cones (wider apertures) lead to higher or equal bound state energies.
• The number of eigenvalues below $-\alpha^2/4 - \varepsilon$ in $d=3$ satisfies the precise asymptotic: $N_{-\alpha^2/4 - \varepsilon}(H_{\alpha,C}) \sim \frac{\cot\theta}{4\pi} |\ln \varepsilon|$ as $\varepsilon \to 0^+$.
• The discrete spectrum in $d=3$ is entirely generated by the axisymmetric ($l=0$) fiber, and the corresponding eigenvalue problem is unitarily equivalent to a form with $\theta$-independent domain.
• The form core for the fiber operators $Q^{[l]}_{\alpha,\Gamma_\theta}$ is $C_0^\infty(\mathbb{R}^2_+)$ for $d \geq 4$ or $l \geq 0$, and $C_0^{\infty,0}(\mathbb{R}^2_+)$ for $d=3$, $l>0$, ensuring the validity of the variational approach.