The Role of the Branch Cut of the Logarithm in the Definition of the Spectral Determinant for Non-Self-Adjoint Operators
Main Article Content
Abstract
The spectral determinant is usually defined using the spectral zeta function that is meromorphically continued to zero. In this definition, the complex logarithms of the eigenvalues appear. Hence, the notion of the spectral determinant depends on the way in which one chooses the branch cut in the definition of the logarithm. We give results for the non-self-adjoint operators that specify when the determinant can and cannot be defined and how its value differs depending on the choice of the branch cut.
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
S. Minakshisundaram, Å. Pleijel, Can. J. Math. 1, 242 (1949)
D.B. Ray, I.M. Singer, Adv. Math. 7, 145 (1971)
S. Levit, U. Smilansky, Proc. Am. Math. Soc. 65, 299 (1977)
F. Gesztesy, K. Kirsten, J. Funct. Anal. 276, 520 (2019)
C. Aldana, J.-B. Caillau, P. Freitas, J. Éc. Polytech. Math. 7, 803 (2020)
M. Bordag, B. Geyer, K. Kirsten, E. Elizalde, Commun. Math. Phys. 179, 215 (1996)
E. Aurell, P. Salomonson, Commun. Math. Phys. 165, 233 (1994)
P. Freitas, Math. Ann. 372, 1081 (2018)
A. Voros, Nuclear Physics B 165, 209 (1980)
G.V. Dunne, J. Phys. A: Math. Theor. 41, 304006 (2008)
D. Burghelea, L. Friedlander, T. Kappeler, Proc. Amer. Math. Soc. 123, 3027 (1995)
P. Freitas, J. Lipovský, Acta Phys. Pol. A 136, 817 (2019)
P. Freitas, J. Lipovský, J. Funct. Anal. 279, 108783 (2020)
J.R. Quine, S.H. Heydari, R.Y. Song, Trans. Am. Math. Soc. 338, 213 (1993)
W. Rudin, Real and complex analysis, 3rd ed., McGraw Hill, Singapore 1986, p. 483
L.V. Ahlfors, Complex analysis, 3rd ed., McGraw Hill, 1979, p. 336
A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York 1953
D. Borisov, P. Freitas, J. Diff. Eq. 247, 3028 (2009)