# American Institute of Mathematical Sciences

January  2018, 17(1): 191-208. doi: 10.3934/cpaa.2018012

## Higher order eigenvalues for non-local Schrödinger operators

 1 Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, United Kingdom 2 Center for Applied Mathematics, Tianjin University, Tianjin 300072, China 3 Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, United Kingdom

* Corresponding author

Received  April 2017 Revised  July 2017 Published  September 2017

Fund Project: The second named author is supported by Supported in part by NNSFC (11431014,11626245,11626250).

Two-sided estimates for higher order eigenvalues are presented for a class of non-local Schrödinger operators by using the jump rate and the growth of the potential. For instance, let
 $L$
be the generator of a Lévy process with Lévy measure
 $ν(\text{d} z):= \rho(z)\text{d} z$
such that
 $\rho(z)=\rho(-z)$
and
 $c_1 |z|^{-(d+\alpha_1)}≤ \rho(z)≤ c_2|z|^{-(d+\alpha_2)},\ \ |z|≤ \kappa$
for some constants
 $\kappa, c_1,c_2>0$
and
 $\alpha_1,\alpha_2∈ (0,2),$
and let
 $c_3|x|^{θ_1} ≤ V(x)≤ c_4|x|^{θ_2}$
for some constants
 $θ_1,θ_2, c_3,c_4>0$
and large
 $|x|$
. Then the eigenvalues
 $\lambda_1≤ \lambda_2≤··· \lambda_n≤ ···$
of
 $-L+V$
satisfies the following two-side estimate: there exists a constant
 $C>1$
such that
 $C n^{\frac{θ_2\alpha_2}{d(θ_2+\alpha_2)}}≥ \lambda_n ≥ C^{-1} n^{\frac{θ_1\alpha_1}{d(θ_1+\alpha_1)}},\ \ n≥ 1.$
When
 $\alpha_1$
is variable, a better lower bound estimate is derived.
Citation: Niels Jacob, Feng-Yu Wang. Higher order eigenvalues for non-local Schrödinger operators. Communications on Pure and Applied Analysis, 2018, 17 (1) : 191-208. doi: 10.3934/cpaa.2018012
##### References:
 [1] D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators Springer, 2014. doi: 10.1007/978-3-319-00227-9. [2] M. T. Barlow, R. F. Bass, Z.-Q. Chen and M. Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc., 361 (2009), 1963-1999.  doi: 10.1090/S0002-9947-08-04544-3. [3] K. Bogdan and T. Byczkowski, Potential theory for α-stable Schrödinger operators on bounded Lipschitz domains, Studia Math., 133 (1999), 53-92. [4] K. Bogdan and T. Byczkowski, Boundary potential theory for Schrödinger operators based on fractional Laplacians, In Potential Theory of Stable Processes and its Extensions. P. Graczyk, A. Stos (eds. ), Lecture Notes in Mathematics, Vol. 1980, Springer Verlag, 2009, p. 25-55. doi: 10.1007/978-3-642-02141-1. [5] B. Böttcher, R. L. Schilling and J. Wang, Lévy-type Processes: Construction, Approximation and Sample Path Properties Lecture Notes in Mathematics, Vol. 2099, Springer Verlag, 2013. doi: 10.1007/978-3-319-02684-8. [6] L. Bray and N. Jacob, Some considerations on the structure of transition densities of symmetric Lévy processes, Commun. Stoch. Anal., 10 (2016), 405-420. [7] X. Chen and J. Wang, Intrinsic ultracontractivity of Feynman-Kac semigroups for symmetric jump processes, J. Funct. Anal., 270 (2016), 4152-4195.  doi: 10.1016/j.jfa.2016.03.011. [8] Z.-Q. Chen and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains, J. Funct. Anal., 226 (2005), 90-113.  doi: 10.1016/j.jfa.2005.05.004. [9] Z.-Q. Chen and X. Zhang, Heat kernels and analyticity of non-symmetric jump diffusion semigroups, Probab. Theory Relat. Fields, 165 (2016), 267-312.  doi: 10.1007/s00440-015-0631-y. [10] H. Cycon, R. Froese, W. Kirsch and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry Springer Verlag, Berlin, 1987. [11] I. Ekeland, Convexity Methods in Hamiltonian Mechanics Ergebnisse der Mathematik und ihrer Grenzgebiete, Ser. 3, Vol. 19, Springer Verlag, 1990. [12] C. L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (New Series), 9 (1983), 129-206.  doi: 10.1090/S0273-0979-1983-15154-6. [13] P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem 2nd ed. CRC Press, Boca Raton FL, 1995. [14] P. B. Gilkey, Asymptotic Formulae in Spectral Geometry Chapman & Hall /CRC Press, Boca Baton, Fl, 2004. [15] L. Hörmander, The Analysis of Linear Partial Differential Operators Ⅳ Grundlehren der mathematischen Wissenschaften, Vol. 275, Springer Veralg, 1985. [16] V. Ya. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics Springer Verlag, Berlin 1998. doi: 10.1007/978-3-662-12496-3. [17] V. Ya. Ivrii, 100 years of Weyl's law, Bull. Math. Sci., 6 (2016), 379-452.  doi: 10.1007/s13373-016-0089-y. [18] N. Jacob, Pseudo-Differential Operators and Markov Processes, Vol Ⅰ-Ⅲ, Imperial College Press, 2001-2005. doi: 10.1142/9781860947155. [19] N. Jacob, V. Knopova, S. Landwehr and R. L. Schilling, A geometric interpretation of the transition density of a symmetric Lévy process, Sci. China Ser. A Math, 55 (2012), 1099-1126.  doi: 10.1007/s11425-012-4368-0. [20] N. Jacob and E. Rhind, Aspects of micro-local analysis in the study of Lévy-type generators, (in preparation). [21] K. Kaleta, Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval, Studia Math., 209 (2012), 267-287.  doi: 10.4064/sm209-3-5. [22] K. Kaleta and T. Kulczycki, Intrinsic ultracontractivity for Schrödinger operators based on fractional Laplacian, Potential Anal., 33 (2010), 313-339.  doi: 10.1007/s11118-010-9170-4. [23] K. Kaleta and J. Lörinczi, Pointwise eigenfuction estimates and intrinsic ultra-contractivity-type properties of Feynman-Kac semigroups for a class of Lévy processes, Ann. Prob., 43 (2015), 1350-1398.  doi: 10.1214/13-AOP897. [24] K. Kaleta and J. Lörinczi, Fall-off of eigenfunctions for non-local Schrödinger operators with decaying potentials, Potential Anal., 46 (2017), 647-688.  doi: 10.1007/s11118-016-9597-3. [25] T. Kulczycki, Eigenvalues and eigenfunctions for stable processes, In Potential Theory of Stable Processes and its Extensions. P. Graczyk, A. Stos (eds. ), Lecture Notes in Mathematics, Vol. 1980, Springer Verlag, 2009, p. 73-86. [26] M. Kwaśnicki, Spectral gap estimate for stable processes on arbitrary bounded open sets, Probab. Math. Stat., 28 (2008), 163-167. [27] J. Lörinczi, K. Kaleta and S. O. Durugo, Spectral and analytic properties of non-local Schrödinger operators and related jump processes, Commun. Appl. Ind. Math. 6 (2015), e-534, 22pp. [28] J. Lörinczi and J. Malecki, Spectral properties of the massless relativistic harmonic oscillator, J. Diff. Equations, 253 (2012), 2846-2871.  doi: 10.1016/j.jde.2012.07.010. [29] F. Nicola and L. Rodino, Global Pseudo-Differential Calculus on Euclidean Spaces Birkhäuser Verlag, Basel 2010. doi: 10.1007/978-3-7643-8512-5. [30] M. Reed and B. Simon, Methods in Modern Mathematical Physics Vol. Ⅳ. Analysis of Operators, Academic Press, New York 1978. [31] E. Rhind, Forthcoming PhD-thesis Swansea University. [32] M. A. Shubin, Pseudodifferential Operators and Spectral Theory Transl. from the Russian, Springer Verlag, Berlin 1987. doi: 10.1007/978-3-642-96854-9. [33] B. Simon, Schrödinger operators with purely discrete spectrum, Meth. Funct. Anal. Top., 15 (2009), 61-66. [34] R. Song and Z. Vondracek, Potential theory of subordinate Brownian motion, In Potential Theory of Stable Processes and its Extensions. P. Graczyk, A. Stos (eds. ), Lecture Notes in Mathematics, Vol. 1980, Springer Verlag, 2009, p. 87-179. [35] F.-Y. Wang, Functional inequalities for empty essential spectrum, J. Funct. Anal., 170 (2000), 219-245.  doi: 10.1006/jfan.1999.3516. [36] F. -Y. Wang, Functional inequalities and spectrum estimates: the infinite measure case J. Funct. Anal. 194 (2002), 288-310 doi: 10.1006/jfan. 2002. 3968. [37] F. -Y. Wang, Functional Inequalities, Markov Semigroups and Spectral Theory Science Press, Beijing, 2005. [38] F.-Y. Wang and J. Wu, Compactness of Schrödinger semigroups with unbounded below potentials, Bull. Sci. Math., 132 (2008), 679-689.  doi: 10.1016/j.bulsci.2008.06.004. [39] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenschwingungeneines beliebig gestalteten elastischen Körpers, Rend. Circ. Mat. Palermo, 39 (1915), 1-49.

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##### References:
 [1] D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators Springer, 2014. doi: 10.1007/978-3-319-00227-9. [2] M. T. Barlow, R. F. Bass, Z.-Q. Chen and M. Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc., 361 (2009), 1963-1999.  doi: 10.1090/S0002-9947-08-04544-3. [3] K. Bogdan and T. Byczkowski, Potential theory for α-stable Schrödinger operators on bounded Lipschitz domains, Studia Math., 133 (1999), 53-92. [4] K. Bogdan and T. Byczkowski, Boundary potential theory for Schrödinger operators based on fractional Laplacians, In Potential Theory of Stable Processes and its Extensions. P. Graczyk, A. Stos (eds. ), Lecture Notes in Mathematics, Vol. 1980, Springer Verlag, 2009, p. 25-55. doi: 10.1007/978-3-642-02141-1. [5] B. Böttcher, R. L. Schilling and J. Wang, Lévy-type Processes: Construction, Approximation and Sample Path Properties Lecture Notes in Mathematics, Vol. 2099, Springer Verlag, 2013. doi: 10.1007/978-3-319-02684-8. [6] L. Bray and N. Jacob, Some considerations on the structure of transition densities of symmetric Lévy processes, Commun. Stoch. Anal., 10 (2016), 405-420. [7] X. Chen and J. Wang, Intrinsic ultracontractivity of Feynman-Kac semigroups for symmetric jump processes, J. Funct. Anal., 270 (2016), 4152-4195.  doi: 10.1016/j.jfa.2016.03.011. [8] Z.-Q. Chen and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains, J. Funct. Anal., 226 (2005), 90-113.  doi: 10.1016/j.jfa.2005.05.004. [9] Z.-Q. Chen and X. Zhang, Heat kernels and analyticity of non-symmetric jump diffusion semigroups, Probab. Theory Relat. Fields, 165 (2016), 267-312.  doi: 10.1007/s00440-015-0631-y. [10] H. Cycon, R. Froese, W. Kirsch and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry Springer Verlag, Berlin, 1987. [11] I. Ekeland, Convexity Methods in Hamiltonian Mechanics Ergebnisse der Mathematik und ihrer Grenzgebiete, Ser. 3, Vol. 19, Springer Verlag, 1990. [12] C. L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (New Series), 9 (1983), 129-206.  doi: 10.1090/S0273-0979-1983-15154-6. [13] P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem 2nd ed. CRC Press, Boca Raton FL, 1995. [14] P. B. Gilkey, Asymptotic Formulae in Spectral Geometry Chapman & Hall /CRC Press, Boca Baton, Fl, 2004. [15] L. Hörmander, The Analysis of Linear Partial Differential Operators Ⅳ Grundlehren der mathematischen Wissenschaften, Vol. 275, Springer Veralg, 1985. [16] V. Ya. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics Springer Verlag, Berlin 1998. doi: 10.1007/978-3-662-12496-3. [17] V. Ya. Ivrii, 100 years of Weyl's law, Bull. Math. Sci., 6 (2016), 379-452.  doi: 10.1007/s13373-016-0089-y. [18] N. Jacob, Pseudo-Differential Operators and Markov Processes, Vol Ⅰ-Ⅲ, Imperial College Press, 2001-2005. doi: 10.1142/9781860947155. [19] N. Jacob, V. Knopova, S. Landwehr and R. L. Schilling, A geometric interpretation of the transition density of a symmetric Lévy process, Sci. China Ser. A Math, 55 (2012), 1099-1126.  doi: 10.1007/s11425-012-4368-0. [20] N. Jacob and E. Rhind, Aspects of micro-local analysis in the study of Lévy-type generators, (in preparation). [21] K. Kaleta, Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval, Studia Math., 209 (2012), 267-287.  doi: 10.4064/sm209-3-5. [22] K. Kaleta and T. Kulczycki, Intrinsic ultracontractivity for Schrödinger operators based on fractional Laplacian, Potential Anal., 33 (2010), 313-339.  doi: 10.1007/s11118-010-9170-4. [23] K. Kaleta and J. Lörinczi, Pointwise eigenfuction estimates and intrinsic ultra-contractivity-type properties of Feynman-Kac semigroups for a class of Lévy processes, Ann. Prob., 43 (2015), 1350-1398.  doi: 10.1214/13-AOP897. [24] K. Kaleta and J. Lörinczi, Fall-off of eigenfunctions for non-local Schrödinger operators with decaying potentials, Potential Anal., 46 (2017), 647-688.  doi: 10.1007/s11118-016-9597-3. [25] T. Kulczycki, Eigenvalues and eigenfunctions for stable processes, In Potential Theory of Stable Processes and its Extensions. P. Graczyk, A. Stos (eds. ), Lecture Notes in Mathematics, Vol. 1980, Springer Verlag, 2009, p. 73-86. [26] M. Kwaśnicki, Spectral gap estimate for stable processes on arbitrary bounded open sets, Probab. Math. Stat., 28 (2008), 163-167. [27] J. Lörinczi, K. Kaleta and S. O. Durugo, Spectral and analytic properties of non-local Schrödinger operators and related jump processes, Commun. Appl. Ind. Math. 6 (2015), e-534, 22pp. [28] J. Lörinczi and J. Malecki, Spectral properties of the massless relativistic harmonic oscillator, J. Diff. Equations, 253 (2012), 2846-2871.  doi: 10.1016/j.jde.2012.07.010. [29] F. Nicola and L. Rodino, Global Pseudo-Differential Calculus on Euclidean Spaces Birkhäuser Verlag, Basel 2010. doi: 10.1007/978-3-7643-8512-5. [30] M. Reed and B. Simon, Methods in Modern Mathematical Physics Vol. Ⅳ. Analysis of Operators, Academic Press, New York 1978. [31] E. Rhind, Forthcoming PhD-thesis Swansea University. [32] M. A. Shubin, Pseudodifferential Operators and Spectral Theory Transl. from the Russian, Springer Verlag, Berlin 1987. doi: 10.1007/978-3-642-96854-9. [33] B. Simon, Schrödinger operators with purely discrete spectrum, Meth. Funct. Anal. Top., 15 (2009), 61-66. [34] R. Song and Z. Vondracek, Potential theory of subordinate Brownian motion, In Potential Theory of Stable Processes and its Extensions. P. Graczyk, A. Stos (eds. ), Lecture Notes in Mathematics, Vol. 1980, Springer Verlag, 2009, p. 87-179. [35] F.-Y. Wang, Functional inequalities for empty essential spectrum, J. Funct. Anal., 170 (2000), 219-245.  doi: 10.1006/jfan.1999.3516. [36] F. -Y. Wang, Functional inequalities and spectrum estimates: the infinite measure case J. Funct. Anal. 194 (2002), 288-310 doi: 10.1006/jfan. 2002. 3968. [37] F. -Y. Wang, Functional Inequalities, Markov Semigroups and Spectral Theory Science Press, Beijing, 2005. [38] F.-Y. Wang and J. Wu, Compactness of Schrödinger semigroups with unbounded below potentials, Bull. Sci. Math., 132 (2008), 679-689.  doi: 10.1016/j.bulsci.2008.06.004. [39] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenschwingungeneines beliebig gestalteten elastischen Körpers, Rend. Circ. Mat. Palermo, 39 (1915), 1-49.
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