# American Institute of Mathematical Sciences

October  2016, 36(10): 5709-5720. doi: 10.3934/dcds.2016050

## Matsaev's type theorems for solutions of the stationary Schrödinger equation and its applications

 1 School of of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450046, China

Received  October 2015 Revised  March 2016 Published  July 2016

Our aim in this paper is to give lower estimates for solutions of the stationary Schrödinger equation in a cone, which generalize and supplement the result obtained by Matsaev's type theorems for harmonic functions in a half space. Meanwhile, some applications of this conclusion are also given.
Citation: Lei Qiao. Matsaev's type theorems for solutions of the stationary Schrödinger equation and its applications. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5709-5720. doi: 10.3934/dcds.2016050
##### References:
 [1] V. S. Azarin, Generalization of a theorem of Hayman's on a subharmonic function in an $n$-dimensional cone (Russian), Mat. Sb. (N.S.), 108 (1965), 248-264. [2] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [3] N. V. Govorov and M. I. Zhuravleva, On an upper bound of the module of a function analytic in a half-plane and in a plane with a cut (Russian), Izv. Severo-Kavkaz. Nauchn. Tsentra Vyssh. Shkoly Estestv. Nauk., 4 (1973), 102-103. [4] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. [5] A. I. Kheyfits, Growth of Schrödingerian subharmonic functions admitting certain lower bounds, Advances in Harmonic Analysis and Operator Theory, Oper. Theory, Adv. Appl., 229 (2013), 215-231. doi: 10.1007/978-3-0348-0516-2_12. [6] I. F. Krasičkov-Ternovskiĭ, Estimates for the subharmonic difference of subharmonic functions. II, Math. USSR-Sb., 32 (1977), 32-59. [7] B. Ya. Levin, Lectures on Entire Functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996. [8] B. Ya. Levin and A. I. Kheyfits, Asymptotic behavior of subfunctions of time-independent Schrödinger operator, in Some Topics on Value Distribution and Differentiability in Complex and P-adic Analysis (eds. A. Escassut, W. Tutschke and C. C. Yang), Science Press, 11 (2008), 323-397. [9] N. K. Nikol'skiĭ, Selected Problems of the Weighted Approximation and Spectral Analysis, American Mathematical Society, Providence, R.I., 1976. [10] L. Qiao, Integral representations for harmonic functions of infinite order in a cone, Results Math., 61 (2012), 63-74. doi: 10.1007/s00025-010-0076-7. [11] L. Qiao and G. Deng, A theorem of Phragmén-Lindelöf type for subfunctions in a cone, Glasg. Math. J., 53 (2011), 599-610. doi: 10.1017/S0017089511000164. [12] L. Qiao and G. Pan, Integral representations of generalized harmonic functions, Taiwanese J. Math., 17 (2013), 1503-1521. [13] L. Qiao and G. Pan, Lower-bound estimates for a class of harmonic functions and applications to Masaev's Type theorem, Bull. Sci. Math., 140 (2016), 70-85. doi: 10.1016/j.bulsci.2015.02.005. [14] L. Qiao and Y. Ren, Integral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone, Monats. Math., 173 (2014), 593-603. doi: 10.1007/s00605-013-0506-1. [15] A. Yu. Rashkovskiĭ and L. I. Ronkin, Subharmonic functions of finite order in a cone. I. General theory, (Russian) Teor. Funktsiĭ Funktsional. Anal. i Prilozhen., 54 (1990), 74-89. doi: 10.1007/BF01097287. [16] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447-526. doi: 10.1090/S0273-0979-1982-15041-8. [17] G. M. Verzhbinskiĭ and V. G. Maz'ya, Asymptotic behavior of the solutions of second order elliptic equations near the boundary. I. (Russian), Sibirsk. Mat. Ž., 12 (1971), 1217-1249. [18] Y. Zhang, G. Deng and K. Kou, On the lower bound for a class of harmonic functions in the half space, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 1487-1494. doi: 10.1016/S0252-9602(12)60117-9.

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##### References:
 [1] V. S. Azarin, Generalization of a theorem of Hayman's on a subharmonic function in an $n$-dimensional cone (Russian), Mat. Sb. (N.S.), 108 (1965), 248-264. [2] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [3] N. V. Govorov and M. I. Zhuravleva, On an upper bound of the module of a function analytic in a half-plane and in a plane with a cut (Russian), Izv. Severo-Kavkaz. Nauchn. Tsentra Vyssh. Shkoly Estestv. Nauk., 4 (1973), 102-103. [4] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. [5] A. I. Kheyfits, Growth of Schrödingerian subharmonic functions admitting certain lower bounds, Advances in Harmonic Analysis and Operator Theory, Oper. Theory, Adv. Appl., 229 (2013), 215-231. doi: 10.1007/978-3-0348-0516-2_12. [6] I. F. Krasičkov-Ternovskiĭ, Estimates for the subharmonic difference of subharmonic functions. II, Math. USSR-Sb., 32 (1977), 32-59. [7] B. Ya. Levin, Lectures on Entire Functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996. [8] B. Ya. Levin and A. I. Kheyfits, Asymptotic behavior of subfunctions of time-independent Schrödinger operator, in Some Topics on Value Distribution and Differentiability in Complex and P-adic Analysis (eds. A. Escassut, W. Tutschke and C. C. Yang), Science Press, 11 (2008), 323-397. [9] N. K. Nikol'skiĭ, Selected Problems of the Weighted Approximation and Spectral Analysis, American Mathematical Society, Providence, R.I., 1976. [10] L. Qiao, Integral representations for harmonic functions of infinite order in a cone, Results Math., 61 (2012), 63-74. doi: 10.1007/s00025-010-0076-7. [11] L. Qiao and G. Deng, A theorem of Phragmén-Lindelöf type for subfunctions in a cone, Glasg. Math. J., 53 (2011), 599-610. doi: 10.1017/S0017089511000164. [12] L. Qiao and G. Pan, Integral representations of generalized harmonic functions, Taiwanese J. Math., 17 (2013), 1503-1521. [13] L. Qiao and G. Pan, Lower-bound estimates for a class of harmonic functions and applications to Masaev's Type theorem, Bull. Sci. Math., 140 (2016), 70-85. doi: 10.1016/j.bulsci.2015.02.005. [14] L. Qiao and Y. Ren, Integral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone, Monats. Math., 173 (2014), 593-603. doi: 10.1007/s00605-013-0506-1. [15] A. Yu. Rashkovskiĭ and L. I. Ronkin, Subharmonic functions of finite order in a cone. I. General theory, (Russian) Teor. Funktsiĭ Funktsional. Anal. i Prilozhen., 54 (1990), 74-89. doi: 10.1007/BF01097287. [16] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447-526. doi: 10.1090/S0273-0979-1982-15041-8. [17] G. M. Verzhbinskiĭ and V. G. Maz'ya, Asymptotic behavior of the solutions of second order elliptic equations near the boundary. I. (Russian), Sibirsk. Mat. Ž., 12 (1971), 1217-1249. [18] Y. Zhang, G. Deng and K. Kou, On the lower bound for a class of harmonic functions in the half space, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 1487-1494. doi: 10.1016/S0252-9602(12)60117-9.
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