The very unusual properties of the resolvent, heat kernel, and zeta function for the operator $-d^2/dr^2 - 1/(4r^2)$
Kirsten, Klaus ; Loya, Paul ; Park, Jinsung
arXiv, 0511002 / Harvested from arXiv
Text on arXiv
In this article we analyze the resolvent, the heat kernel and the spectral zeta function of the operator $-d^2/dr^2 - 1/(4r^2)$ over the finite interval. The structural properties of these spectral functions depend strongly on the chosen self-adjoint realization of the operator, a choice being made necessary because of the singular potential present. Only for the Friedrichs realization standard properties are reproduced, for all other realizations highly nonstandard properties are observed. In particular, for $k\in \N$ we find terms like $(\log t)^{-k}$ in the small-$t$ asymptotic expansion of the heat kernel. Furthermore, the zeta function has $s=0$ as a logarithmic branch point.
Publié le : 2005-10-31
Classification:  Mathematical Physics,  High Energy Physics - Theory,  58J28,  58J52 
@article{0511002,
     author = {Kirsten, Klaus and Loya, Paul and Park, Jinsung},
     title = {The very unusual properties of the resolvent, heat kernel, and zeta
  function for the operator $-d^2/dr^2 - 1/(4r^2)$},
     journal = {arXiv},
     volume = {2005},
     number = {0},
     year = {2005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0511002}
}

Kirsten, Klaus; Loya, Paul; Park, Jinsung. The very unusual properties of the resolvent, heat kernel, and zeta
  function for the operator $-d^2/dr^2 - 1/(4r^2)$. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0511002/