Abstract

We begin by studying the eigenvectors associated to irreducible finite birth and death processes, showing that the $i^{{\rm th}}$ nontrivial eigenvector $\varphi_i$ admits a succession of $i$ decreasing or increasing stages, each of them crossing zero. Imbedding naturally the finite state space into a continuous segment, one can unequivocally define zeros of $\varphi_i$, which are interlaced with those of $\varphi_{i+1}$. These kinds of results are deduced from the general investigation of minimax multi-set Dirichlet eigenproblems, which lead to a direct construction of eigenvectors associated to birth and death processes. This approach can be generically extended to eigenvectors of Markov processes living on trees. This enables one to interpret the eigenvalues and eigenvectors in terms of the previous Dirichlet eigenproblems and a more general conjecture is presented about related higher order Cheeger inequalities. Finally, we carefully study the geometric structure of the eigenspace associated to the spectral gap on trees.



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