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Life times and lower eigenvalues of an operator of small diffusion. (Russian) Zbl 0755.34086

The eigenvalue problem (1) \({\mathcal L}_ \varepsilon v=-\varepsilon v''+\varphi'v'=\lambda v\), \(v(A)=v(B)=0\) is considered, where \(\varepsilon\) is a small positive parameter, and the potential \(\varphi\) is a Morse function. The authors investigate the behaviour of the smallest eigenvalues as \(\varepsilon\to 0\). It is assumed that \(\varphi\) has its global maximum at the endpoints: \(\varphi(A)=\varphi(B)\), and that \(\varphi\) has \(N\) points of local minima. It is shown that the \((N+1)\)-st eigenvalue \(\lambda_{N+1}(\varepsilon)\) satisfies \(\lambda_{N+1}(\varepsilon)\geq\text{const}>0\), whereas the first eigenvalue \(\lambda_ 1(\varepsilon)\) tends to zero exponentially if \(\varepsilon\to 0\). This also holds for the second eigenvalue \(\lambda_ 2(\varepsilon)\) if \(N\geq 2\). In the proof the matrix of transition probabilities of an associated Markov process is considered.

MSC:

34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
47E05 General theory of ordinary differential operators
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