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 & 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.  Google Scholar [2] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar [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. Google Scholar [4] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964.  Google Scholar [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.  Google Scholar [6] I. F. Krasičkov-Ternovskiĭ, Estimates for the subharmonic difference of subharmonic functions. II, Math. USSR-Sb., 32 (1977), 32-59. Google Scholar [7] B. Ya. Levin, Lectures on Entire Functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996.  Google Scholar [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.  Google Scholar [9] N. K. Nikol'skiĭ, Selected Problems of the Weighted Approximation and Spectral Analysis, American Mathematical Society, Providence, R.I., 1976.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] L. Qiao and G. Pan, Integral representations of generalized harmonic functions, Taiwanese J. Math., 17 (2013), 1503-1521.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447-526. doi: 10.1090/S0273-0979-1982-15041-8.  Google Scholar [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. Google Scholar [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.  Google Scholar

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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.  Google Scholar [2] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar [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. Google Scholar [4] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964.  Google Scholar [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.  Google Scholar [6] I. F. Krasičkov-Ternovskiĭ, Estimates for the subharmonic difference of subharmonic functions. II, Math. USSR-Sb., 32 (1977), 32-59. Google Scholar [7] B. Ya. Levin, Lectures on Entire Functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996.  Google Scholar [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.  Google Scholar [9] N. K. Nikol'skiĭ, Selected Problems of the Weighted Approximation and Spectral Analysis, American Mathematical Society, Providence, R.I., 1976.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] L. Qiao and G. Pan, Integral representations of generalized harmonic functions, Taiwanese J. Math., 17 (2013), 1503-1521.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447-526. doi: 10.1090/S0273-0979-1982-15041-8.  Google Scholar [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. Google Scholar [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.  Google Scholar
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