Bound states without potentials: localization at singularities

Abstract

Bound state formation is a classic feature of quantum mechanics, where a particle localizes in the vicinity of an attractive potential. This is typically understood as the particle lowering its potential energy. In this article, we discuss a paradigm where bound states arise purely due to kinetic energy considerations. This phenomenon occurs in certain non-manifold spaces that consist of multiple smooth surfaces that intersect one another. The intersection region can be viewed as a singularity where dimensionality is not defined. We demonstrate this idea in a setting where a particle moves on $M$ spaces ($M=2, 3, 4, \ldots$), each of dimensionality $D$ ($D=1, 2$ and $3$). The spaces intersect at a common point, which serves as a singularity. To study quantum behaviour in this setting, we discretize space and adopt a tight-binding approach. We generically find a ground state that is localized around the singular point, bound by the kinetic energy of `shuttling' among the $M$ surfaces. We draw a quantitative analogy between singularities on the one hand and local attractive potentials on the other. To each singularity, we assign an equivalent potential that produces the same bound state wavefunction and binding energy. The degree of a singularity ($M$, the number of intersecting surfaces) determines the strength of the equivalent potential. With $D=1$ and $D=2$, we show that any singularity creates a bound state. This is analogous to the well known fact that any attractive potential creates a bound state in 1D and 2D. In contrast, with $D=3$, bound states only appear when the degree of the singularity exceeds a threshold value. This is analogous to the fact that in three dimensions, a threshold potential strength is required for bound state formation. We discuss implications for experiments and theoretical studies in various domains of quantum physics.