Isospectral Operators

Abstract

For a large class of integral operators or second-order differential operators, their isospectral (or cospectral) operators are constructed explicitly in terms of $h$-transform (duality). This provides us a simple way to extend the known knowledge on the spectrum (or the estimation of the principal eigenvalue) from a smaller class of operators to a much larger one. In particular, an open problem about the positivity of the principal eigenvalue for birth–death processes is solved in the paper.

Keywords

Isospectral / Harmonic function / Integral operator / Differential operator

References

[1] Chen, M.F.: From Markov Chains to Non-equilibrium Particle Systems. World Scientific. $2^{\text{ nd }}$ ed. ($1^{\text{ st }}$ ed., 1992). (2004)

[2] Chen MF. Speed of stability for birth-death processes. Front. Math. China. 2010, 5 3 379-515

[3] Chen, M.F.: Basic estimates of stability rate for one-dimensional diffusions. In: Barbour, A.D., Chan, H.P., Siegmund, D. (eds) Probability Approximations and Beyond, Lecture Notes in Statistics vol. 205, pp. 75–99. Springer, New York (2012a)

[4] Chen MF. Lower bounds of the principal eigenvalue in dimension one. Front. Math. China. 2012, 7 4 645-668

[5] Chen, M.F. and Zhang, Y.H.: Unified representation of formulas for single birth processes. (2014, Preprint)

[6] Jansen, S. and Kurt, N.: On the notion(s) of duality for Markov processes. arXiv:1210.7193. (2012)

[7] Murata M. Structure of positive solutions to $(-\Delta +V)u=0$ in $\mathbb{R}^n$. Duke Math. J.. 1986, 53 4 869-943

[8] Pinsky RG. Positive Harmonic Functions and Diffusion. 1995 Cambridge: Cambridge University Press

[9] Pinsky RG. Explicit and almost explicit spectral calculations for diffusion operators. J. Funct. Anal.. 2009, 256 10 3279-3312

[10] Wang J. Sharp bounds for the first eigenvalue of symmetric Markov processes and their applications. Acta Math. Sin. Eng. Ser.. 2012, 28 10 1995-2010

[11] Zettl A. Sturm-Liouville Theory. 2005 Providence: AMS

[12] Zhang, X.: On the eigenvalues of birth-death processes with killing. (2013, Preprint)