[Paper Review] On contact interactions as limits of short-range potentials

Gerhard Bräunlich, Christian Hainzl+1 more|arXiv (Cornell University)|May 16, 2013

Cold Atom Physics and Bose-Einstein Condensates2 references3 citations

TL;DR

This paper establishes the norm resolvent convergence of Schrödinger operators with short-range potentials to a point interaction Hamiltonian in three dimensions, even when the potential has a non-vanishing positive integral in the limit. It proves that the resolvent converges to a self-adjoint extension characterized by the scattering length, under conditions that include a repulsive core and controlled decay of the potential's negative part.

ABSTRACT

We reconsider the norm resolvent limit of $-Δ+ V_\ell$ with $V_\ell$ tending to a point interaction in three dimensions. We are mainly interested in potentials $V_\ell$ modelling short range interactions of cold atomic gases. In order to ensure stability the interaction $V_\ell$ is required to have a strong repulsive core, such that $\lim_{\ell o 0} \int V_\ell >0$. This situation is not covered in the previous literature.

Motivation & Objective

• To analyze the norm resolvent limit of Schrödinger operators with short-range potentials in 3D when the $ L^1 $-norm of the potential does not vanish in the limit.
• To address the physical relevance of contact interactions in cold atomic Fermi gases, where stability requires a strong repulsive core, implying $ \lim_{\ell\to 0}\int V_\ell > 0 $.
• To extend previous results on contact interactions by removing the restrictive assumption that $ \|V_\ell\|_{L^1} \to 0 $, which excludes physically relevant systems with repulsive cores.
• To rigorously derive the limiting Hamiltonian as a self-adjoint extension of $ -\Delta $, uniquely determined by the scattering length $ a $.

Proposed method

• Uses the Birman-Schwinger operator formalism to analyze the resolvent convergence, with $ B_\ell = V_\ell^{1/2} \frac{1}{p^2} |V_\ell|^{1/2} $ as the central object.
• Applies a recently derived formula for the scattering length: $ a(V_\ell) = \frac{1}{4\pi} \left\langle |V_\ell|^{1/2} \middle| (1 + B_\ell)^{-1} V_\ell^{1/2} \right\rangle $.
• Imposes Assumptions (A1)–(A5), including controlled decay of the negative part $ V_\ell^{-} $, zero-energy resonance conditions, and integrability with $ |x|^2 $-weight.
• Employs spectral estimates on $ X_\ell^{-} $, the Birman-Schwinger operator for the negative part, to control non-self-adjointness and obtain norm bounds.
• Uses the identity $ \| (J_\ell + X_\ell)\phi_\ell \|_{L^2} \geq \sqrt{2}|e_\ell| $ to derive decay rates for the normalized eigenfunction $ \phi_\ell $.
• Applies a refined perturbation analysis to show $ \| (1 - P_{J_\ell \phi_\ell}) P_{\phi_\ell^{-}} \| = O(|e_\ell|^{1/2}) $, crucial for resolvent convergence.

Experimental results

Research questions

• RQ1 Can the norm resolvent limit of $ -\Delta + V_\ell $ converge to a point interaction Hamiltonian when $ \|V_\ell\|_{L^1} \not\to 0 $?
• RQ2 How can one rigorously define contact interactions in 3D when the potential has a strong repulsive core, as required for stability in cold atomic gases?
• RQ3 What conditions on $ V_\ell $ ensure that the scattering length $ a(V_\ell) $ converges to a finite limit $ a $, even when $ \int V_\ell \to \text{const} > 0 $?
• RQ4 How can one control the non-self-adjoint nature of the Birman-Schwinger operator $ B_\ell $ in the presence of a large repulsive core?

Key findings

• The resolvent converges in norm: $ \frac{1}{-\Delta + V_\ell - k^2} \to \frac{1}{-\Delta - k^2} - \frac{4\pi}{a^{-1} + ik} |g_k\rangle\langle g_k| $ as $ \ell \to 0 $, for $ \Im(k) > 0 $.
• The limiting scattering length $ a = \lim_{\ell\to 0} a(V_\ell) $ is finite and uniquely determines the self-adjoint extension of $ -\Delta $.
• The repulsive core is handled via the condition $ \|V_\ell^{-}\|_{L^1} = O(e_\ell) $, with $ e_\ell \to 0 $, ensuring the negative part remains small relative to the core.
• The normalized zero-energy resonance state $ \phi_\ell $ satisfies $ \|\phi_\ell^{+}\|_{L^2}^2 = O(e_\ell) $, showing the positive part is small in norm.
• The norm of the projection error $ \| (1 - P_{J_\ell \phi_\ell}) P_{\phi_\ell^{-}} \| = O(|e_\ell|^{1/2}) $, which is essential for convergence.
• The $ |x| $-weighted integrability conditions (A4) ensure sufficient decay for the resolvent convergence to hold in the operator norm.