  Size-invariant shape transformation is a technique of changing the shape of a
domain while preserving its sizes under the Lebesgue measure. In quantum
confined systems, this transformation leads to so-called quantum shape effects
in the physical properties of confined particles associated with the Dirichlet
spectrum of the confining medium. Here we show that the geometric couplings
between levels generated by the size-invariant shape transformations cause
nonuniform scaling in the eigenspectra. In particular, the nonuniform level
scaling is characterized by two distinct spectral features: lowering of the
first eigenvalue (ground state reduction) and changing of the spectral gaps
(energy level splitting or degeneracy formation depending on the symmetries).
We explain the ground state reduction by the increase in local breadth (i.e.
parts of the domain becoming less confined) that is associated with the
sphericity of these local portions of the domain. We accurately quantify the
sphericity using two different measures: the radius of the inscribed $n$-sphere
and the Hausdorff distance. Due to Rayleigh-Faber-Krahn inequality, the greater
the sphericity, the lower the first eigenvalue. Then, level splitting or
degeneracy, depending on the symmetries of the initial configuration, becomes a
direct consequence of size-invariance dictating the eigenvalues to have the
same asymptotic behavior due to Weyl law. Furthermore, we find that the ground
state reduction causes a quantum thermal avalanche which is the underlying
reason for the peculiar effect of spontaneous transitions to lower entropy
states in systems exhibiting the quantum shape effect. Unusual spectral
characteristics of size-preserving transformations can assist in designing
confinement geometries that could lead to classically inconceivable quantum
thermal machines.
