American Institute of Mathematical Sciences

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

Higher order eigenvalues for non-local Schrödinger operators

Niels Jacob 1, and  Feng-Yu Wang 2,3,,

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.
Keywords: Eigenvalues, non-local Schrödinger operator, jump process, pseudo differential operator, intrinsic super Poincaré inequality.
Mathematics Subject Classification: Primary:35P15, 47G30;Secondary:60G52.
Citation: Niels Jacob, Feng-Yu Wang. Higher order eigenvalues for non-local Schrödinger operators. Communications on Pure & 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. Google Scholar

[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. Google Scholar

[3]

K. Bogdan and T. Byczkowski, Potential theory for α-stable Schrödinger operators on bounded Lipschitz domains, Studia Math., 133 (1999), 53-92. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

H. Cycon, R. Froese, W. Kirsch and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry Springer Verlag, Berlin, 1987. Google Scholar

[11]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics Ergebnisse der Mathematik und ihrer Grenzgebiete, Ser. 3, Vol. 19, Springer Verlag, 1990.Google Scholar

[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. Google Scholar

[13]

P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem 2nd ed. CRC Press, Boca Raton FL, 1995. Google Scholar

[14]

P. B. Gilkey, Asymptotic Formulae in Spectral Geometry Chapman & Hall /CRC Press, Boca Baton, Fl, 2004. Google Scholar

[15]

L. Hörmander, The Analysis of Linear Partial Differential Operators Ⅳ Grundlehren der mathematischen Wissenschaften, Vol. 275, Springer Veralg, 1985. Google Scholar

[16]

V. Ya. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics Springer Verlag, Berlin 1998. doi: 10.1007/978-3-662-12496-3. Google Scholar

[17]

V. Ya. Ivrii, 100 years of Weyl's law, Bull. Math. Sci., 6 (2016), 379-452. doi: 10.1007/s13373-016-0089-y. Google Scholar

[18]

N. Jacob, Pseudo-Differential Operators and Markov Processes, Vol Ⅰ-Ⅲ, Imperial College Press, 2001-2005. doi: 10.1142/9781860947155. Google Scholar

[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. Google Scholar

[20]

N. Jacob and E. Rhind, Aspects of micro-local analysis in the study of Lévy-type generators, (in preparation).Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[26]

M. Kwaśnicki, Spectral gap estimate for stable processes on arbitrary bounded open sets, Probab. Math. Stat., 28 (2008), 163-167. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

M. Reed and B. Simon, Methods in Modern Mathematical Physics Vol. Ⅳ. Analysis of Operators, Academic Press, New York 1978. Google Scholar

[31]

E. Rhind, Forthcoming PhD-thesis Swansea University.Google Scholar

[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. Google Scholar

[33]

B. Simon, Schrödinger operators with purely discrete spectrum, Meth. Funct. Anal. Top., 15 (2009), 61-66. Google Scholar

[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.Google Scholar

[35]

F.-Y. Wang, Functional inequalities for empty essential spectrum, J. Funct. Anal., 170 (2000), 219-245. doi: 10.1006/jfan.1999.3516. Google Scholar

[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. Google Scholar

[37]

F. -Y. Wang, Functional Inequalities, Markov Semigroups and Spectral Theory Science Press, Beijing, 2005.Google Scholar

[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. Google Scholar

[39]

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenschwingungeneines beliebig gestalteten elastischen Körpers, Rend. Circ. Mat. Palermo, 39 (1915), 1-49. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[3]

K. Bogdan and T. Byczkowski, Potential theory for α-stable Schrödinger operators on bounded Lipschitz domains, Studia Math., 133 (1999), 53-92. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

H. Cycon, R. Froese, W. Kirsch and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry Springer Verlag, Berlin, 1987. Google Scholar

[11]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics Ergebnisse der Mathematik und ihrer Grenzgebiete, Ser. 3, Vol. 19, Springer Verlag, 1990.Google Scholar

[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. Google Scholar

[13]

P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem 2nd ed. CRC Press, Boca Raton FL, 1995. Google Scholar

[14]

P. B. Gilkey, Asymptotic Formulae in Spectral Geometry Chapman & Hall /CRC Press, Boca Baton, Fl, 2004. Google Scholar

[15]

L. Hörmander, The Analysis of Linear Partial Differential Operators Ⅳ Grundlehren der mathematischen Wissenschaften, Vol. 275, Springer Veralg, 1985. Google Scholar

[16]

V. Ya. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics Springer Verlag, Berlin 1998. doi: 10.1007/978-3-662-12496-3. Google Scholar

[17]

V. Ya. Ivrii, 100 years of Weyl's law, Bull. Math. Sci., 6 (2016), 379-452. doi: 10.1007/s13373-016-0089-y. Google Scholar

[18]

N. Jacob, Pseudo-Differential Operators and Markov Processes, Vol Ⅰ-Ⅲ, Imperial College Press, 2001-2005. doi: 10.1142/9781860947155. Google Scholar

[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. Google Scholar

[20]

N. Jacob and E. Rhind, Aspects of micro-local analysis in the study of Lévy-type generators, (in preparation).Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[26]

M. Kwaśnicki, Spectral gap estimate for stable processes on arbitrary bounded open sets, Probab. Math. Stat., 28 (2008), 163-167. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

M. Reed and B. Simon, Methods in Modern Mathematical Physics Vol. Ⅳ. Analysis of Operators, Academic Press, New York 1978. Google Scholar

[31]

E. Rhind, Forthcoming PhD-thesis Swansea University.Google Scholar

[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. Google Scholar

[33]

B. Simon, Schrödinger operators with purely discrete spectrum, Meth. Funct. Anal. Top., 15 (2009), 61-66. Google Scholar

[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.Google Scholar

[35]

F.-Y. Wang, Functional inequalities for empty essential spectrum, J. Funct. Anal., 170 (2000), 219-245. doi: 10.1006/jfan.1999.3516. Google Scholar

[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. Google Scholar

[37]

F. -Y. Wang, Functional Inequalities, Markov Semigroups and Spectral Theory Science Press, Beijing, 2005.Google Scholar

[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. Google Scholar

[39]

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenschwingungeneines beliebig gestalteten elastischen Körpers, Rend. Circ. Mat. Palermo, 39 (1915), 1-49. Google Scholar

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