First eigenvalue of birth-death processes with killing

Abstract

In this paper, we present an explicit and computable lower bound for the first eigenvalue of birth-death processes with killing. This estimate is qualitatively sharp for birth-death processes without killing. We also establish an approximation procedure for the first eigenvalue of the birth-death process with killing by an increasing principal eigenvalue sequence of some birth-death processes without killing. Some applications of our results are illustrated by many examples.

Keywords

First eigenvalue / birth-death processes (with killing) / Schrödinger operator with difference form

References

[1] Chen M. F. The principal eigenvalue for jump processes. Acta Math Sin Engl Ser, 2000, 16: 361-368

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

[3] Chung K. L., Zhao Z. From Brownian Motion to Schrödinger’s Equation, 1995, Berlin: Springer-Verlag

[4] Davies E. B. Spectral Theory and Differential Operators, 1995, Cambridge: Cambridge University Press

[5] Reed M., Simon B. Methods of Modern Mathematical Physics, 1972, London: Academic Press

[6] Simon B. Functional Integration and Quantum Physics, 1979, London: Academic Press

[7] Wang J. First Dirichlet eigenvalue of transient birth-death processes. Preprint, 2008