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Essential self-adjointness for covariant tri-harmonic operators on manifolds and the separation problem

Year 2022, Volume: 51 Issue: 5, 1321 - 1332, 01.10.2022
https://doi.org/10.15672/hujms.1021920

Abstract

Consider the tri-harmonic differential expression $L_{V}^{\nabla}u=\left(\nabla^{+}\nabla\right)^{3}u+Vu$, on sections of a hermitian vector bundle over a complete Riemannian manifold $\left(M,g\right)$ with metric $g$, where $\nabla$ is a metric covariant derivative on bundle E over complete Riemannian manifold, $\nabla^{+}$ is the formal adjoint of $\nabla$ and $V$ is a self adjoint bundle on $E$. We will give conditions for $L_{V}^{\nabla}$ to be essential self-adjoint in $L^{2}\left(E\right).$ Additionally, we provide sufficient conditions for $L_{V}^{\nabla}$ to be separated in $L^{2}\left( E\right)$. According to Everitt and Giertz, the differential operator $L_{V}^{\nabla}$ is said to be separated in $L^{2}\left( E\right) $ if for all $u$ $\in L^{2}\left( E\right)$ such that $L_{V}^{\nabla}u\in L^{2}\left( E\right) $, we have $Vu\in L^{2}\left( E\right)$.

References

  • [1] H.A. Atia, Separation problem for second order elliptic differential operators on Riemannian manifolds, J. Computat. Anal. Appl. 19, 229-240, 2015.
  • [2] H.A. Atia, R.S. Alsaedi and A. Ramady, Separation of bi-harmonic differential operators on Riemannian manifolds, forum Math. 26(3), 953-966, 2014.
  • [3] H.A. Atia, Magnetic bi-Harmonic differential operators on Riemannian manifolds and the separation problem, J. Contemporary Math. Anal. 51, 222-226, 2016.
  • [4] H.A. Atia, Separation problem for bi-harmonic differential operators in L$_{p}$- spaces on manifolds, J. Egyptian Math. Soc. 27, Article number:24, 2019. https://doi.org/10.1186/s42787-019-0029-6.
  • [5] L. Bandara and H. Saratchandran, Essential self-adjointness of powers of first order differential operators on non-compact manifolds with low-regularity metrics, J. Funct. Anal. 273, 3719-3758, 2017.
  • [6] K.Kh. Boimatov, Coercive estimates and separation for second order elliptic differential equations, Soviet Math. Dokl. 38, 157-160, 1989.
  • [7] K.Kh. Boimatov, On the Everitt and Giertz method for Banach spaces, Dokl. Akad. Nauk 356, 10-12, 1997.
  • [8] M. Braverman, O. Milatovic and M. Shubin, Essential self-adjointness of Schrodinger type operators on manifolds, Russian Math. Surveys, 57, 641-692, 2002.
  • [9] R.C. Brown, D.B. Hinton and M.F. Shaw, Some separation criteria and inequalities associated with linear second order differential operators, in: Function spaces and applications, 7-35, Narosa Publishing House, New Delhi, 2000.
  • [10] M. Braverman and S. Cecchini, Spectral theory of von Neumann algebra valued differential operators over non-compact manifolds, J. Noncommut. Geom. 10, 1589-1609, 2016.
  • [11] H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, Schrodinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer, Berlin, 1987.
  • [12] K.J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, in: Graduate Texts in Mathematics 194, Springer, Berlin, 2000.
  • [13] W.D. Evans and A. Zettle, Dirichlet and separation results for Schrodinger type operators, Proc. Roy. Soc. Edinburgh Sect. A 80, 151-162, 1978.
  • [14] W.N. Everitt and M. Giertz, Inequalities and separation for Schrodinger type operators in L$^{2}$(R$^{n}$), Proc. Roy. Soc. Edinburgh Sect. A 79, 257-265, 1977.
  • [15] M. Gaffney, A special Stokes’s theorem for complete Riemannian manifolds, Ann. Math. 60, 140-145, 1954.
  • [16] A. Grigoryan, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics 47, American Mathematical Society, Providence, RI, International Press, Boston, MA, 2009.
  • [17] R. Grummt and M. Kolb, Essential selfadjointness of singular magnetic Schrodinger operators on Riemannian manifolds, J. Math. Anal. Appl. 388, 480-489, 2012.
  • [18] B. Guneysu, Kato’s inequality and form boundedness of Kato potentials on arbitrary Riemannian manifolds, Proc. Am. Math. Soc. 142, 1289-1300, 2014.
  • [19] B. Guneysu, Sequences of Laplacian cut-off functions, J. Geom. Anal. 26, 171-184, 2016.
  • [20] B. Guneysu, Covariant Schrodinger semigroups on Riemannian manifolds, Operator Theory, Advances and Applications 264, Birkhauser, Basel, 2017.
  • [21] B. Guneysu and O. Post, Path integrals and the essential self-adjointness of differential operators on noncompact manifolds, Math. Z. 275, 331-348, 2013.
  • [22] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New york, 1980.
  • [23] O. Milatovic, Separation property for Schrodinger operators on Riemannian manifolds, J. Geom. Phys. 56, 1283-1293, 2006.
  • [24] O. Milatovic, Separation property for Schrodinger operators l$^{p}$-spaces on non compact manifolds, Complex Var. Elliptic Equ. 58, 853-864, 2013.
  • [25] O. Milatovic, Self-adjoitness of perturbed biharmonic operators on Riemannian manifolds, Math. Nachr. 290, 2948-2960, 2017.
  • [26] O. Milatovic, Self-adjointness, m-accretivity, and separability for perturbations of laplacian and bi-laplacian on Riemannian manifolds, Integral Equations Operator Theory, 90, Art. 22, 2018.
  • [27] X.D. Nguyen, Essential self-adjointness and self-adjointness for even order elliptic operators, Proc. Roy. Soc. Edinburgh Sect. A, 93, 161-179, 1982.
  • [28] N. Okazawa, An l$^{p}$ theory for Schrodinger operators with nonnegative potentials, J. Math. Soc. Japan 36, 675-688, 1984.
  • [29] N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan 34, 677-701, 1982.
  • [30] M. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis, self-adjointness, Academic Press, New York, 1975.
  • [31] R. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52, 48-79, 1983.
Year 2022, Volume: 51 Issue: 5, 1321 - 1332, 01.10.2022
https://doi.org/10.15672/hujms.1021920

Abstract

References

  • [1] H.A. Atia, Separation problem for second order elliptic differential operators on Riemannian manifolds, J. Computat. Anal. Appl. 19, 229-240, 2015.
  • [2] H.A. Atia, R.S. Alsaedi and A. Ramady, Separation of bi-harmonic differential operators on Riemannian manifolds, forum Math. 26(3), 953-966, 2014.
  • [3] H.A. Atia, Magnetic bi-Harmonic differential operators on Riemannian manifolds and the separation problem, J. Contemporary Math. Anal. 51, 222-226, 2016.
  • [4] H.A. Atia, Separation problem for bi-harmonic differential operators in L$_{p}$- spaces on manifolds, J. Egyptian Math. Soc. 27, Article number:24, 2019. https://doi.org/10.1186/s42787-019-0029-6.
  • [5] L. Bandara and H. Saratchandran, Essential self-adjointness of powers of first order differential operators on non-compact manifolds with low-regularity metrics, J. Funct. Anal. 273, 3719-3758, 2017.
  • [6] K.Kh. Boimatov, Coercive estimates and separation for second order elliptic differential equations, Soviet Math. Dokl. 38, 157-160, 1989.
  • [7] K.Kh. Boimatov, On the Everitt and Giertz method for Banach spaces, Dokl. Akad. Nauk 356, 10-12, 1997.
  • [8] M. Braverman, O. Milatovic and M. Shubin, Essential self-adjointness of Schrodinger type operators on manifolds, Russian Math. Surveys, 57, 641-692, 2002.
  • [9] R.C. Brown, D.B. Hinton and M.F. Shaw, Some separation criteria and inequalities associated with linear second order differential operators, in: Function spaces and applications, 7-35, Narosa Publishing House, New Delhi, 2000.
  • [10] M. Braverman and S. Cecchini, Spectral theory of von Neumann algebra valued differential operators over non-compact manifolds, J. Noncommut. Geom. 10, 1589-1609, 2016.
  • [11] H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, Schrodinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer, Berlin, 1987.
  • [12] K.J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, in: Graduate Texts in Mathematics 194, Springer, Berlin, 2000.
  • [13] W.D. Evans and A. Zettle, Dirichlet and separation results for Schrodinger type operators, Proc. Roy. Soc. Edinburgh Sect. A 80, 151-162, 1978.
  • [14] W.N. Everitt and M. Giertz, Inequalities and separation for Schrodinger type operators in L$^{2}$(R$^{n}$), Proc. Roy. Soc. Edinburgh Sect. A 79, 257-265, 1977.
  • [15] M. Gaffney, A special Stokes’s theorem for complete Riemannian manifolds, Ann. Math. 60, 140-145, 1954.
  • [16] A. Grigoryan, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics 47, American Mathematical Society, Providence, RI, International Press, Boston, MA, 2009.
  • [17] R. Grummt and M. Kolb, Essential selfadjointness of singular magnetic Schrodinger operators on Riemannian manifolds, J. Math. Anal. Appl. 388, 480-489, 2012.
  • [18] B. Guneysu, Kato’s inequality and form boundedness of Kato potentials on arbitrary Riemannian manifolds, Proc. Am. Math. Soc. 142, 1289-1300, 2014.
  • [19] B. Guneysu, Sequences of Laplacian cut-off functions, J. Geom. Anal. 26, 171-184, 2016.
  • [20] B. Guneysu, Covariant Schrodinger semigroups on Riemannian manifolds, Operator Theory, Advances and Applications 264, Birkhauser, Basel, 2017.
  • [21] B. Guneysu and O. Post, Path integrals and the essential self-adjointness of differential operators on noncompact manifolds, Math. Z. 275, 331-348, 2013.
  • [22] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New york, 1980.
  • [23] O. Milatovic, Separation property for Schrodinger operators on Riemannian manifolds, J. Geom. Phys. 56, 1283-1293, 2006.
  • [24] O. Milatovic, Separation property for Schrodinger operators l$^{p}$-spaces on non compact manifolds, Complex Var. Elliptic Equ. 58, 853-864, 2013.
  • [25] O. Milatovic, Self-adjoitness of perturbed biharmonic operators on Riemannian manifolds, Math. Nachr. 290, 2948-2960, 2017.
  • [26] O. Milatovic, Self-adjointness, m-accretivity, and separability for perturbations of laplacian and bi-laplacian on Riemannian manifolds, Integral Equations Operator Theory, 90, Art. 22, 2018.
  • [27] X.D. Nguyen, Essential self-adjointness and self-adjointness for even order elliptic operators, Proc. Roy. Soc. Edinburgh Sect. A, 93, 161-179, 1982.
  • [28] N. Okazawa, An l$^{p}$ theory for Schrodinger operators with nonnegative potentials, J. Math. Soc. Japan 36, 675-688, 1984.
  • [29] N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan 34, 677-701, 1982.
  • [30] M. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis, self-adjointness, Academic Press, New York, 1975.
  • [31] R. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52, 48-79, 1983.
There are 31 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Hany Atia 0000-0001-7747-6400

Hala H. Emam 0000-0002-7764-8070

Publication Date October 1, 2022
Published in Issue Year 2022 Volume: 51 Issue: 5

Cite

APA Atia, H., & Emam, H. H. (2022). Essential self-adjointness for covariant tri-harmonic operators on manifolds and the separation problem. Hacettepe Journal of Mathematics and Statistics, 51(5), 1321-1332. https://doi.org/10.15672/hujms.1021920
AMA Atia H, Emam HH. Essential self-adjointness for covariant tri-harmonic operators on manifolds and the separation problem. Hacettepe Journal of Mathematics and Statistics. October 2022;51(5):1321-1332. doi:10.15672/hujms.1021920
Chicago Atia, Hany, and Hala H. Emam. “Essential Self-Adjointness for Covariant Tri-Harmonic Operators on Manifolds and the Separation Problem”. Hacettepe Journal of Mathematics and Statistics 51, no. 5 (October 2022): 1321-32. https://doi.org/10.15672/hujms.1021920.
EndNote Atia H, Emam HH (October 1, 2022) Essential self-adjointness for covariant tri-harmonic operators on manifolds and the separation problem. Hacettepe Journal of Mathematics and Statistics 51 5 1321–1332.
IEEE H. Atia and H. H. Emam, “Essential self-adjointness for covariant tri-harmonic operators on manifolds and the separation problem”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 5, pp. 1321–1332, 2022, doi: 10.15672/hujms.1021920.
ISNAD Atia, Hany - Emam, Hala H. “Essential Self-Adjointness for Covariant Tri-Harmonic Operators on Manifolds and the Separation Problem”. Hacettepe Journal of Mathematics and Statistics 51/5 (October 2022), 1321-1332. https://doi.org/10.15672/hujms.1021920.
JAMA Atia H, Emam HH. Essential self-adjointness for covariant tri-harmonic operators on manifolds and the separation problem. Hacettepe Journal of Mathematics and Statistics. 2022;51:1321–1332.
MLA Atia, Hany and Hala H. Emam. “Essential Self-Adjointness for Covariant Tri-Harmonic Operators on Manifolds and the Separation Problem”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 5, 2022, pp. 1321-32, doi:10.15672/hujms.1021920.
Vancouver Atia H, Emam HH. Essential self-adjointness for covariant tri-harmonic operators on manifolds and the separation problem. Hacettepe Journal of Mathematics and Statistics. 2022;51(5):1321-32.