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Sluggish quantum mechanics of noninteracting fermions with spatially varying effective mass

Giuseppe Del Vecchio Del Vecchio1,*, Manas Kulkarni2,†, Satya N. Majumdar3,‡, and Sanjib Sabhapandit4,§

  • *Contact author: giuseppedelvecchiodelvecchio@gmail.com
  • †Contact author: manas.kulkarni@icts.res.in
  • ‡Contact author: satyanarayan.majumdar@cnrs.fr
  • §Contact author: sanjib@rri.res.in
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APS Open Sci. 1, 000173 – Published 15 October, 2026

DOI: https://doi.org/10.1103/mbp5-cd7c

Abstract

We analyze a class of one-dimensional quantum systems characterized by a position-dependent kinetic term arising as the continuum limit of an inhomogeneous tight-binding model with spatially varying hopping amplitudes. In this limit, the Schrödinger equation takes the so-called BenDaniel-Duke form with an effective mass, scaling as meff(x)=meff|x|α, with α>0, leading to a framework we term “sluggish quantum mechanics,” where particle motion is progressively suppressed at larger distances. For the case with no external confinement, we obtain the eigenfunctions and the quantum propagator exactly. In the presence of a confining potential Vext(x)=12meff(x)ω2x2=12meffω2|x|α+2, we show that the model remains exactly solvable and obtain the spectral properties and the propagator analytically. We then investigate the problem of N noninteracting spinless fermions in the trap, determining the many-body ground-state wavefunction and the joint probability density function of the positions of the N fermions. We show that the many-body quantum probability density in the ground state forms a determinantal point process whose correlation kernel can be computed for any N, giving access to the average density as well as higher-order correlation functions for any finite N. Moreover, we analyze the scaling form of this kernel in the large-N limit in the bulk, near the edge, and close to the origin. Our results show that the scaled average density profile for large N has a finite support symmetric with respect to the origin, but has a nonmonotonic shape with a vanishing minimum at the origin for any α>0. One of the key findings of our work is that the scaled kernel near the origin x=0 for α>0 is neither the Bessel nor the Airy kernel (that are standard for trapped fermions), but is new, and is given by a sum of two Bessel kernels with different indices. Our results thus provide a framework relevant to engineered optical lattices with position-dependent tunneling.

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APS Open Sci. 1, 000173– Published 15 October, 2026
Vol. 1
  • Received 14 April 2026
  • Accepted 29 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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