Abstract
The article concerns the study of conditions on the non-self-adjoint elliptic operator defined in the whole space ℝn, ensuring the existence and uniqueness of a constant-sign eigenfunction tending to zero at infinity. We also study the asymptotics of the corresponding eigenvalue as the coefficient in the highest-order derivative of the operator tends to zero. The result is formulated in terms connected with the variational problem for the Lagrangian on one-dimensional trajectories in the space ℝn. The explicit form of this Lagrangian is given in terms of the coefficients of the original operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
A. Piatnitski, “Asymptotic behavior of the ground state of singularly perturbed elliptic equations,” Commun. Math. Physics, 197,No. 3, 527–551 (1998).
M. A. Krasnosel'skii, E. A. Lifshits, and A. V. Sobolev, Positive Linear Systems. The Method of Positive Linear Operators, Sigma Series in Applied Math., 5, Heldermann Verlag (1989).
D. W. Stroock, An Introduction to the Theory of Large Deviations, Springer-Verlag, New York (1984).
M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, New York (1984).
Rights and permissions
About this article
Cite this article
Pyatnitskii, A.L., Shamaev, A.S. On the Asymptotic Behavior of Eigenvalues and Eigenfunctions of Non-Self-Adjoint Elliptic Operators. Journal of Mathematical Sciences 120, 1411–1423 (2004). https://doi.org/10.1023/B:JOTH.0000016058.00000.92
Issue Date:
DOI: https://doi.org/10.1023/B:JOTH.0000016058.00000.92