Spectral properties of size-invariant shape transformation.

Area of Science:

• Quantum mechanics
• Mathematical physics

Background:

• Size-invariant shape transformation preserves domain size under Lebesgue measure.
• This transformation induces quantum shape effects in confined particles.
• These effects are linked to the Dirichlet spectrum of the confining medium.

Purpose of the Study:

• To investigate how geometric couplings from size-invariant shape transformations affect eigenspectra.
• To characterize the resulting nonuniform level scaling and its spectral features.
• To explore the implications for quantum thermal machines.

Main Methods:

• Analyzing eigenspectra under size-invariant shape transformations.
• Quantifying domain sphericity using inscribed n-sphere radius and Hausdorff distance.
• Applying Rayleigh-Faber-Krahn inequality and Weyl's law.

Main Results:

• Geometric couplings cause nonuniform eigenspectra scaling.
• Observed ground-state reduction and spectral gap alterations (splitting/degeneracy).
• Ground-state reduction linked to increased local domain breadth and sphericity.
• Level splitting interpreted as geometric analogs of Stark and Zeeman effects.
• Ground-state reduction drives quantum thermal avalanches and spontaneous transitions to lower entropy states.

Conclusions:

• Size-invariant transformations create unique spectral characteristics.
• Sphericity quantifies ground-state reduction via Rayleigh-Faber-Krahn inequality.
• Weyl's law explains eigenvalue asymptotic behavior and level splitting.
• Findings suggest potential for designing novel quantum thermal machines with unprecedented capabilities.