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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.