## Duke Mathematical Journal

### On positivity, criticality, and the spectral radius of the shuttle operator for elliptic operators

Yehuda Pinchover

#### Article information

Source
Duke Math. J., Volume 85, Number 2 (1996), 431-445.

Dates
First available in Project Euclid: 19 February 2004

https://projecteuclid.org/euclid.dmj/1077243254

Digital Object Identifier
doi:10.1215/S0012-7094-96-08518-X

Mathematical Reviews number (MathSciNet)
MR1417623

Zentralblatt MATH identifier
0901.35016

#### Citation

Pinchover, Yehuda. On positivity, criticality, and the spectral radius of the shuttle operator for elliptic operators. Duke Math. J. 85 (1996), no. 2, 431--445. doi:10.1215/S0012-7094-96-08518-X. https://projecteuclid.org/euclid.dmj/1077243254

#### References

• [1] M. Brelot, On topologies and boundaries in potential theory, Enlarged edition of a course of lectures delivered in 1966. Lecture Notes in Mathematics, Vol. 175, Springer-Verlag, Berlin, 1971.
• [2] K. L. Chung, On stopped Feynman-Kac functionals, Seminar on Probability, XIV (Paris, 1978/1979) (French), Lecture Notes in Math., vol. 784, Springer, Berlin, 1980, pp. 347–356.
• [3] K. L. Chung and S. R. S. Varadhan, Kac functional and Schrödinger equation, Studia Math. 68 (1980), no. 3, 249–260.
• [4] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II: Partial differential equations, (Vol. II by R. Courant.), Interscience Publishers (a division of John Wiley & Sons), New York-Lon don, 1962.
• [5] F. Gesztesy and Z. Zhao, On critical and subcritical Sturm-Liouville operators, J. Funct. Anal. 98 (1991), no. 2, 311–345.
• [6] F. Gesztesy and Z. Zhao, On positive solutions of critical Schrödinger operators in two dimensions, J. Funct. Anal. 127 (1995), no. 1, 235–256.
• [7] M. A. Krasnosel'skii, Positive solutions of operator equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd. Groningen, 1964.
• [8] W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 43–77.
• [9] M. Murata, Structure of positive solutions to $(-\Delta+V)u=0$ in $\bf R\sp n$, Duke Math. J. 53 (1986), no. 4, 869–943.
• [10] Y. Pinchover, On positive solutions of second-order elliptic equations, stability results, and classification, Duke Math. J. 57 (1988), no. 3, 955–980.
• [11] Y. Pinchover, Criticality and ground states for second-order elliptic equations, J. Differential Equations 80 (1989), no. 2, 237–250.
• [12] Y. Pinchover, On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 3, 313–341.
• [13] Y. Pinchover, On the localization of binding for Schrödinger operators and its extension to elliptic operators, J. Anal. Math. 66 (1995), 57–83.
• [14] R. G. Pinsky, Positive harmonic functions and diffusion, Cambridge Studies in Advanced Mathematics, vol. 45, Cambridge University Press, Cambridge, 1995.
• [15] R. G. Pinsky, Second order elliptic operators with periodic coefficients: criticality theory, perturbations, and positive harmonic functions, J. Funct. Anal. 129 (1995), no. 1, 80–107.
• [16] Z. Zhao, Subcriticality, positivity and gaugeability of the Schrödinger operator, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 513–517.
• [17] Z. Zhao, Subcriticality and gaugeability of the Schrödinger operator, Trans. Amer. Math. Soc. 334 (1992), no. 1, 75–96.