Abstract
In this work we introduce a Poincaré determinant type for operators on the torus \({\mathbb {T}}^n\). As an application we establish the existence of nontrivial solutions for elliptic equations of the form \((-\Delta )^{\frac{\nu }{2}}u+Qu=0\) on \({\mathbb {T}}^n\) by using the Hill’s method.
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References
Bornemann, F.: On the numerical evaluation of Fredholm determinants. Math. Comp. 79(270), 871–915 (2010)
Borodin, A., Corwin, I., Remenik, D.: Log-gamma polymer free energy fluctuations via a Fredholm determinant identity. Comm. Math. Phys. 324(1), 215–232 (2013)
Bothner, T., Its, A.: Asymptotics of a Fredholm determinant corresponding to the first bulk critical universality class in random matrix models. Comm. Math. Phys. 328(1), 155–202 (2014)
Delgado, J., Ruzhansky, M.: Fourier multipliers, symbols and nuclearity on compact manifolds. J. Anal. Math. 135(2), 757–800 (2018)
Delgado, J., Ruzhansky, M.: \({L}^p\)-nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups. J. Math. Pures Appl. 102, 153–172 (2014)
Delgado, J., Ruzhansky, M.: Schatten classes on compact manifolds: Kernel conditions. J. Funct. Anal. 267, 772–798 (2014)
Delgado, J., Ruzhansky, M.: Schatten classes and traces on compact groups. Math. Res. Letters. 24, 979–1003 (2017)
Gesztesy, F., Latushkin, Y., Zumbrun, K.: Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves. J. Math. Pures Appl. (9) 90(2), 160–200 (2008)
Gohberg, I., Goldberg, S., Krupnik, N.: Traces and determinants of linear operators. In: Operator Theory: Advances and Applications, vol. 116. Birkhäuser Verlag, Basel (2000)
Hill, G.: On the part of the motion of lunar perigee which is a function of the mean motions of the sun and the moon. Acta Math. 8, 1–3 (1886)
McKean, H.P.: Fredholm determinants and the Camassa-Holm hierarchy. Comm. Pure Appl. Math. 56(5), 638–680 (2003)
Poincaré, H.: Sur les déterminants d’ordre infini. Bull. Soc. Math. Fr. 14, 77–90 (1886)
Ruzhansky, M., Turunen, V.: Pseudo-differential operators and symmetries. Background analysis and advanced topics, volume 2 of Pseudo-Differential Operators. Theory and Applications. Birkhäuser Verlag, Basel, (2010)
Zhao, L., Barnett, A.: Robust and efficient solution of the drum problem via Nyström approximation of the Fredholm determinant. SIAM J. Numer. Anal. 53(4), 1984–2007 (2015)
Acknowledgements
The author was supported by Grant CI-71234 Vic. Inv. Universidad del Valle. I would also like to thank an anonymous referee for the careful review.
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Delgado, J. A Poincaré determinant on the torus. J. Pseudo-Differ. Oper. Appl. 13, 29 (2022). https://doi.org/10.1007/s11868-022-00461-y
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DOI: https://doi.org/10.1007/s11868-022-00461-y