ИСТИНА

The class of three-diagonal Jacobi matrix with exponentially increasing elements is considered. Under some assumptions the matrix corresponds to unbounded self-adjoint operator $L$ in the weighted space $l_2(\omega)$ with the scalar product $(x,y)=\sum_{k=1}^\infty \omega_k x_k\overline{y_k}$. The weight can arise indefinite metric in some cases. We proved that the eigenvalue problem for this operator is equivalent to the eigenvalue problem of Sturm--Liouville operator with discrete self-similar weight. The asymptotic formulas for eigenvalues are obtained. These formulas differ for cases of definite and indefinite metrics. \textbf{Theorem.} 1) Spectrum of operator $L$ is discrete. If $L$ is self-adjoint in Hilbert space then all eigenvalues are positive and simple. There exists a positive number $c$ such that eigenvalues $\lambda_n$ enumerated in increasing order satisfy the following asymptotic formula $$ \lambda_n=c q^n(1+o(1)) \quad (n\to +\infty). $$ 2) If $L$ is self-adjoint in Krein space then all eigenvalues are simple. There exists a positive number $c$ such that positive eigenvalues $\lambda_n$ enumerated in increasing order satisfy the asymptotic formula $$ \lambda_{n+1}=c q^{2n}(1+o(1)) \quad (n\to +\infty) $$ and negative eigenvalues $\lambda_{-n}$ enumerated in increasing order by absolute values satisfy the following asymptotic formula $$ \lambda_{-(n+2)}=-c q^{2n+1}(1+o(1)) \quad (n\to +\infty). $$