2005/12/28 by Mikhail Shubin, Shubin, Mikhail, Toshikazu Sunada +1
Engineering · Mathematics · Physics and Astronomy · #47L90 #81Q10 #82B20 #Advanced Numerical Analysis Techniques #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.MP #math.SP #msc:47L90 #msc:81Q10 #msc:82B20
paper · pdf · doi:10.48550/arxiv.math-ph/0512088
31 pages, minor corrections made
openalex publication_date 2005/12/28 · arxiv created 2006/04/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We discuss, from a geometric standpoint, the specific heat of a solid. This is a classical subject in solid state physics which dates back to a pioneering work by Einstein (1907) and its refinement by Debye (1912). Using a special quantization of crystal lattices and calculating the asymptotic of the integrated density of states at the bottom of the spectrum, we obtain a rigorous derivation of the classical Debye T3 law on the specific heat at low temperatures. The idea and method are taken from discrete geometric analysis which has been recently developed for the spectral geometry of crystal lattices.