Weak coupling limit for Schrödinger-type operators with degenerate kinetic energy for a large class of potentials

Abstract

We improve results by Frank, Hainzl, Naboko, and Seiringer [12] and Hainzl and Seiringer [20] on the weak coupling limit of eigenvalues for Schrödinger-type operators whose kinetic energy vanishes on a codimension one submanifold. The main technical innovation that allows us to go beyond the potentials considered in [12, 20] is the use of the Tomas-Stein theorem.

arXiv comment27 pages. Title changed, corrected misprints, included further details in some proofs, carried out some extensions of the main result (more general kinetic energies, second order in the asymptotics, improvements for potentials with further symmetries). This is a preprint of an article published in Letters in Mathematical Physics, available online at https://doi.org/10.1007/s11005-021-01385-2

References in corpus (9)

• Eigenvalue bounds for Schrödinger operators with complex potentials
• Carleman estimates and absence of embedded eigenvalues
• Critical Temperature and Energy Gap for the BCS Equation
• The BCS Functional for General Pair Interactions
• Lower Bound on the Hartree-Fock Energy of the Electron Gas
• On the number of bound states for weak perturbations of spin-orbit Hamiltonians
• Eigenvalue estimates for bilayer graphene
• Variational principle for Hamiltonians with degenerate bottom
• Quantitative bounds versus existence of weakly coupled bound states for Schrödinger type operators

Cited by in corpus (6)

• The BCS Critical Temperature at High Density
• The BCS Energy Gap at High Density
• Universality in low-dimensional BCS theory
• Asymptotics for eigenvalues of one-dimensional Dirac operators in the weak coupling limit
• On the number and sums of eigenvalues of Schrödinger-type operators with degenerate kinetic energy
• Random Schrödinger operators with complex decaying potentials