Non-local approximations for the Schrödinger operator with a general point interaction via the non-relativistic limit
Abstract
We consider the one-dimensional Dirac operator DA,c with a general, not necessarily self-adjoint, point interaction at the origin, whose strength is described by an arbitrary complex 2× 2 matrix A and suitably rescaled by the speed of light c, together with its non-local approximations DA,cε obtained by replacing the Dirac distribution by a rank-one projection onto a scaled function vε. It is known that DA,cε→ DA,c as ε→ 0+ and that DA,c−c2/2 converges, as c→+∞, to the Schrödinger operator HA with the general point interaction, both in the norm resolvent sense. We prove that the first convergence is uniform with respect to c in a neighbourhood of +∞, and we provide explicit rates of convergence. Combining this with an explicit computation of the non-relativistic limit of DA,cε for fixed ε, we show that the two limits can be interchanged and that the corresponding double limit exists. As a by-product we obtain a family of non-local, rank-two approximations HAε of HA that converge to HA in the norm resolvent sense. Contrary to the relativistic setting, the coupling constants of HAε are renormalized; we give an explicit formula for this renormalization.
Related
- Diffusion Approximations to Schrödinger Bridges and the Convergence of Entropic Potentials
- A nonlocal nonlinear Schrödinger model: well-posedness, local limit, and structure-preserving asymptotically compatible Fourier approximations
- Stability in the determination of the coefficients for coupled relativistic Schrödinger system
- Operator limits of random matrices: Edge universality and sharp approximations
- Ground states for pseudo-relativistic Schrödinger equations with logarithm-law: qualitative properties and nonrelativistic limit
BibTeX
@misc{heriban2026nonlocal,
title = {{Non-local approximations for the Schrödinger operator with a general point interaction via the non-relativistic limit}},
author = {Lukáš Heriban},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05210},
eprint = {2611.05210},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05210},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Non-local approximations for the Schrödinger operator with a general point interaction via the non-relativistic limit
AU - Lukáš Heriban
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05210
UR - https://arxiv.org/abs/2611.05210
AB - We consider the one-dimensional Dirac operator D_A,c with a general, not necessarily self-adjoint, point interaction at the origin, whose strength is described by an arbitrary complex 2× 2 matrix A and suitably rescaled by the speed of light c, together with its non-local approximations D_A,c^ε obtained by replacing the Dirac distribution by a rank-one projection onto a scaled function v_ε. It is known that D_A,c^ε→ D_A,c as ε→ 0+ and that D_A,c−c^2/2 converges, as c→+∞, to the Schrödinger operator H_A with the general point interaction, both in the norm resolvent sense. We prove that the first convergence is uniform with respect to c in a neighbourhood of +∞, and we provide explicit rates of convergence. Combining this with an explicit computation of the non-relativistic limit of D_A,c^ε for fixed ε, we show that the two limits can be interchanged and that the corresponding double limit exists. As a by-product we obtain a family of non-local, rank-two approximations H_A^ε of H_A that converge to H_A in the norm resolvent sense. Contrary to the relativistic setting, the coupling constants of H_A^ε are renormalized; we give an explicit formula for this renormalization.
ER -