  Real physical systems are only understood, experimentally or theoretically,
to a finite resolution so in their analysis there is generally an ignorance of
possible short-range phenomena. It is also well-known that the boundary
conditions of wavefunctions and fields can be used to model short-range
interactions when those interactions are expected, a priori. Here, a real-space
approach is described wherein an artificial boundary of ignorance is imposed to
explicitly exclude from analysis the region of a system wherein short-distance
effects may be obscure. The (artificial) boundary conditions encode those
short-distance effects by parameterizing the possible UV completions of the
wavefunction. Since measurable quantities, such as spectra and cross sections,
must be independent of the position of the artificial boundary, the boundary
conditions must evolve with the radius of the boundary in a particular way. As
examples of this approach, an analysis is performed of the non-relativistic
free particle, harmonic oscillator, and Coulomb potential, and some known
results for point-like (contact) interactions are recovered, however from a
novel perspective. Generically, observables differ from their canonical values
and symmetries are anomalously broken compared to those of idealized models.
Connections are made to well-studied physical systems, such as the binding of
light nuclei and cold atomic systems. This method is arguably more physically
transparent and mathematically easier to use than other techniques that require
the regularization and renormalization of delta-function potentials, and may
offer further generalizations of practical use.
