American Institute of Mathematical Sciences

Advanced
doi: 10.3934/cpaa.2021054

Spectral properties of ordinary differential operators admitting special decompositions

Krzysztof Stempak

Wydziaƚ Matematyki, Politechnika Wrocƚawska, Wyb. Wyspiańskiego 27, 50-370 Wrocƚaw, Poland

Received  July 2020 Revised  January 2021 Published  April 2021

Fund Project: This research was supported by funds of Faculty of Pure and Applied Mathematics, Wrocƚaw University of Science and Technology

We investigate spectral properties of ordinary differential operators related to expressions of the form $ D^{\epsilon}+a $. Here $ a\in \mathbb{R} $ and $ D^{\epsilon} $ denotes a composition of $ \mathfrak{d} $ and $ \mathfrak{d}^+ $ according to the signs in the multi-index $ {\epsilon} $, where $ \mathfrak{d} $ is a first order linear differential expression, called delta-derivative, and $ \mathfrak{d}^+ $ is its formal adjoint in an appropriate $ L^2 $ space. In particular, Sturm-Liouville operators that admit the decomposition of the type $ \mathfrak{d}^+\mathfrak{d}+a $ are considered. We propose an approach, based on weak delta-derivatives and delta-Sobolev spaces, which is particularly useful in the study of the operators $ D^{\epsilon}+a $. Finally we examine a number of examples of operators, which are of the relevant form, naturally arising in analysis of classical orthogonal expansions.

Keywords: Self-adjoint operator, essentially self-adjoint operator, weak derivative, Sobolev space, Sturm-Liouville operator, Friedrichs extension.
Mathematics Subject Classification: Primary: 34B24, 47E05; Secondary: 47B25.
Citation: Krzysztof Stempak. Spectral properties of ordinary differential operators admitting special decompositions. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021054
References:
[1]

A. Arenas, E. Labarga and A. Nowak, Exotic multiplicity functions and heat maximal operators in certain Dunkl settings, Integral Trans. Spec. Funct., 29 (2018), 771-793.  doi: 10.1080/10652469.2018.1498488.  Google Scholar

[2]

N. Ben Salem and T. Samaali, Hilbert transform and related topics associated with the differential Jacobi operator on $(0, +\infty)$, Positivity, 15 (2011), 221-240.  doi: 10.1007/s11117-010-0061-0.  Google Scholar

[3]

J. Betancor and K. Stempak, On Hankel conjugate functions, Studia Sci. Math. Hung., 41 (2004), 59–91. doi: 10.1556/SScMath.41.2004.1.4.  Google Scholar

[4]

J. Betancor and K. Stempak, Relating multipliers and transplantation for Fourier-Bessel expansions and Hankel transform, Tohoku Math. J., 53 (2001), 109-129.  doi: 10.2748/tmj/1178207534.  Google Scholar

[5]

J. Betancor, J. C. Fariña, L. Rodrígues-Mesa, R. Testoni and J. L. Torrea, A choice of Sobolev spaces associated with ultraspherical expansions, Publ. Math., 54 (2010), 221-242.  doi: 10.5565/PUBLMAT_54110_13.  Google Scholar

[6]

B. Bongioanni and J. L. Torrea, Sobolev spaces associated to the harmonic oscillator, Proc. Indian Acad. Sci. Math. Sci., 116 (2006), 337-360.  doi: 10.1007/BF02829750.  Google Scholar

[7]

B. Bongioanni and J. L. Torrea, What is a Sobolev space for the Laguerre function system?, Studia Math., 192 (2009), 147-172.  doi: 10.4064/sm192-2-4.  Google Scholar

[8] E. B. Davies, Spectral Theory and Differential Operators, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511623721.  Google Scholar
[9]

W. N. Everitt, A catalogue of Sturm-Liouville differential equations, in Sturm-Liouville Theory: Past and Present (eds. W. O. Amrein, A. M. Hinz, D. B. Pearson), Birkhäuser Verlag, Basel, 2005,675–711. doi: 10.1007/3-7643-7359-8_12.  Google Scholar

[10]

P. Graczyk, J. J. Loeb, I.A. López, A. Nowak and W. Urbina, Higher order Riesz transforms, fractional derivatives, and Sobolev spaces for Laguerre expansions, J. Math. Pures Appl., 84 (2005), 375-405.  doi: 10.1016/j.matpur.2004.09.003.  Google Scholar

[11]

M. Hajmirzaahmad, Jacobi polynomial expansions, J. Math. Anal. Appl., 181 (1994), 35-61.  doi: 10.1006/jmaa.1994.1004.  Google Scholar

[12]

M. Hajmirzaahmad, Laguerre polynomial expansions, J. Comput. Appl. Math., 59 (1995), 25-37.  doi: 10.1016/0377-0427(94)00020-2.  Google Scholar

[13]

M. Hajmirzaahmad and A. M. Krall, Singular second-order operators: the maximal and minimal operators, and selfadjoint operators in between, SIAM Rev., 34 (1992), 614-634.  doi: 10.1137/1034117.  Google Scholar

[14]

H. Hochstadt, The mean convergence of Fourier-Bessel series, SIAM Rev., 9 (1967), 211-218.  doi: 10.1137/1009034.  Google Scholar

[15]

W. G. Kelley and A. C. Peterson, The Theory of Differential Equations. Classical and Qualitative, Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4419-5783-2.  Google Scholar

[16]

T. H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, in Special functions: Group Theoretic Aspects and Applications (eds. R. Askey, T. H Koornwinder, W. Schempp), Reidel, Dordrecht, 1984.  Google Scholar

[17]

B. Langowski, Sobolev spaces associated with Jacobi expansions, J. Math. Anal. Appl., 420 (2014), 1533-1551.  doi: 10.1016/j.jmaa.2014.06.063.  Google Scholar

[18]

N. N. Lebedev, Special Functions and their Applications, Dover Publications, 1972.  Google Scholar

[19]

V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer-Verlag, New York, 2011. doi: 10.1007/978-3-642-15564-2.  Google Scholar

[20]

NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov. Google Scholar

[21]

A. Nowak, P. Sjögren and T. Z. Szarek, Maximal operators of exotic and other semigroups associated with classical orthogonal expansions, Adv. Math., 318 (2017), 307-354.  doi: 10.1016/j.aim.2017.07.026.  Google Scholar

[22]

A. Nowak and K. Stempak, $L^2$-theory of Riesz transforms for orthogonal expansions, J. Fourier Anal. Appl., 12 (2006), 675-711.  doi: 10.1007/s00041-006-6034-9.  Google Scholar

[23]

K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer-Verlag, New York, 2012. doi: 10.1007/978-94-007-4753-1.  Google Scholar

[24]

J. Sun, On the selfadjoint extensions of symmetric differential operators with middle deficiency indices, Acta Math. Sinica, 2 (1986), 152-167.  doi: 10.1007/BF02564877.  Google Scholar

[25]

A. P. Wang and A. Zettl, Ordinary Differential Operators, Mathematical Surveys and Monographs, vol. 245, Amer. Math. Soc., Providence, RI, 2019. doi: 10.1090/surv/245.  Google Scholar

[26]

J. Weidmann, Spectral theory of Sturm-Liouville operators. Approximation by regular problems, in Sturm-Liouville Theory: Past and Present (eds. W. O. Amrein, A. M. Hinz, D. B. Pearson), Birkhäuser Verlag, Basel, 2005, 75–98. doi: 10.1007/3-7643-7359-8_4.  Google Scholar

[27]

S. Q. Yao, J. Sun and A. Zettl, The Sturm-Liouville Friedrichs extension, Appl. Math., 60 (2015), 299-320.  doi: 10.1007/s10492-015-0097-3.  Google Scholar

[28]

A. Zettl, Sturm-Liouville Theory, Mathematical Surveys and Monographs, vol. 121, Amer. Math. Soc., Providence, RI, 2005. doi: 10.1090/surv/121.  Google Scholar

show all references

References:
[1]

A. Arenas, E. Labarga and A. Nowak, Exotic multiplicity functions and heat maximal operators in certain Dunkl settings, Integral Trans. Spec. Funct., 29 (2018), 771-793.  doi: 10.1080/10652469.2018.1498488.  Google Scholar

[2]

N. Ben Salem and T. Samaali, Hilbert transform and related topics associated with the differential Jacobi operator on $(0, +\infty)$, Positivity, 15 (2011), 221-240.  doi: 10.1007/s11117-010-0061-0.  Google Scholar

[3]

J. Betancor and K. Stempak, On Hankel conjugate functions, Studia Sci. Math. Hung., 41 (2004), 59–91. doi: 10.1556/SScMath.41.2004.1.4.  Google Scholar

[4]

J. Betancor and K. Stempak, Relating multipliers and transplantation for Fourier-Bessel expansions and Hankel transform, Tohoku Math. J., 53 (2001), 109-129.  doi: 10.2748/tmj/1178207534.  Google Scholar

[5]

J. Betancor, J. C. Fariña, L. Rodrígues-Mesa, R. Testoni and J. L. Torrea, A choice of Sobolev spaces associated with ultraspherical expansions, Publ. Math., 54 (2010), 221-242.  doi: 10.5565/PUBLMAT_54110_13.  Google Scholar

[6]

B. Bongioanni and J. L. Torrea, Sobolev spaces associated to the harmonic oscillator, Proc. Indian Acad. Sci. Math. Sci., 116 (2006), 337-360.  doi: 10.1007/BF02829750.  Google Scholar

[7]

B. Bongioanni and J. L. Torrea, What is a Sobolev space for the Laguerre function system?, Studia Math., 192 (2009), 147-172.  doi: 10.4064/sm192-2-4.  Google Scholar

[8] E. B. Davies, Spectral Theory and Differential Operators, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511623721.  Google Scholar
[9]

W. N. Everitt, A catalogue of Sturm-Liouville differential equations, in Sturm-Liouville Theory: Past and Present (eds. W. O. Amrein, A. M. Hinz, D. B. Pearson), Birkhäuser Verlag, Basel, 2005,675–711. doi: 10.1007/3-7643-7359-8_12.  Google Scholar

[10]

P. Graczyk, J. J. Loeb, I.A. López, A. Nowak and W. Urbina, Higher order Riesz transforms, fractional derivatives, and Sobolev spaces for Laguerre expansions, J. Math. Pures Appl., 84 (2005), 375-405.  doi: 10.1016/j.matpur.2004.09.003.  Google Scholar

[11]

M. Hajmirzaahmad, Jacobi polynomial expansions, J. Math. Anal. Appl., 181 (1994), 35-61.  doi: 10.1006/jmaa.1994.1004.  Google Scholar

[12]

M. Hajmirzaahmad, Laguerre polynomial expansions, J. Comput. Appl. Math., 59 (1995), 25-37.  doi: 10.1016/0377-0427(94)00020-2.  Google Scholar

[13]

M. Hajmirzaahmad and A. M. Krall, Singular second-order operators: the maximal and minimal operators, and selfadjoint operators in between, SIAM Rev., 34 (1992), 614-634.  doi: 10.1137/1034117.  Google Scholar

[14]

H. Hochstadt, The mean convergence of Fourier-Bessel series, SIAM Rev., 9 (1967), 211-218.  doi: 10.1137/1009034.  Google Scholar

[15]

W. G. Kelley and A. C. Peterson, The Theory of Differential Equations. Classical and Qualitative, Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4419-5783-2.  Google Scholar

[16]

T. H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, in Special functions: Group Theoretic Aspects and Applications (eds. R. Askey, T. H Koornwinder, W. Schempp), Reidel, Dordrecht, 1984.  Google Scholar

[17]

B. Langowski, Sobolev spaces associated with Jacobi expansions, J. Math. Anal. Appl., 420 (2014), 1533-1551.  doi: 10.1016/j.jmaa.2014.06.063.  Google Scholar

[18]

N. N. Lebedev, Special Functions and their Applications, Dover Publications, 1972.  Google Scholar

[19]

V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer-Verlag, New York, 2011. doi: 10.1007/978-3-642-15564-2.  Google Scholar

[20]

NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov. Google Scholar

[21]

A. Nowak, P. Sjögren and T. Z. Szarek, Maximal operators of exotic and other semigroups associated with classical orthogonal expansions, Adv. Math., 318 (2017), 307-354.  doi: 10.1016/j.aim.2017.07.026.  Google Scholar

[22]

A. Nowak and K. Stempak, $L^2$-theory of Riesz transforms for orthogonal expansions, J. Fourier Anal. Appl., 12 (2006), 675-711.  doi: 10.1007/s00041-006-6034-9.  Google Scholar

[23]

K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer-Verlag, New York, 2012. doi: 10.1007/978-94-007-4753-1.  Google Scholar

[24]

J. Sun, On the selfadjoint extensions of symmetric differential operators with middle deficiency indices, Acta Math. Sinica, 2 (1986), 152-167.  doi: 10.1007/BF02564877.  Google Scholar

[25]

A. P. Wang and A. Zettl, Ordinary Differential Operators, Mathematical Surveys and Monographs, vol. 245, Amer. Math. Soc., Providence, RI, 2019. doi: 10.1090/surv/245.  Google Scholar

[26]

J. Weidmann, Spectral theory of Sturm-Liouville operators. Approximation by regular problems, in Sturm-Liouville Theory: Past and Present (eds. W. O. Amrein, A. M. Hinz, D. B. Pearson), Birkhäuser Verlag, Basel, 2005, 75–98. doi: 10.1007/3-7643-7359-8_4.  Google Scholar

[27]

S. Q. Yao, J. Sun and A. Zettl, The Sturm-Liouville Friedrichs extension, Appl. Math., 60 (2015), 299-320.  doi: 10.1007/s10492-015-0097-3.  Google Scholar

[28]

A. Zettl, Sturm-Liouville Theory, Mathematical Surveys and Monographs, vol. 121, Amer. Math. Soc., Providence, RI, 2005. doi: 10.1090/surv/121.  Google Scholar

[1]

Zhang Chao, Minghua Yang. BMO type space associated with Neumann operator and application to a class of parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021104
[2]

Craig Cowan. Supercritical elliptic problems involving a Cordes like operator. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021037
[3]

John Villavert. On problems with weighted elliptic operator and general growth nonlinearities. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021023
[4]

Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2907-2946. doi: 10.3934/dcds.2020391
[5]

Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021026
[6]

Joel Coacalle, Andrew Raich. Compactness of the complex Green operator on non-pseudoconvex CR manifolds. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021061
[7]

Rama Ayoub, Aziz Hamdouni, Dina Razafindralandy. A new Hodge operator in discrete exterior calculus. Application to fluid mechanics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021062
[8]

Zhaoqiang Ge. Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021009
[9]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247
[10]

Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021017
[11]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020
[12]

Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021014
[13]

Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, 2021, 15 (3) : 475-498. doi: 10.3934/ipi.2021001
[14]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
[15]

Masashi Wakaiki, Hideki Sano. Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021021
[16]

Francis Hounkpe, Gregory Seregin. An approximation of forward self-similar solutions to the 3D Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021059
[17]

Joe Gildea, Adrian Korban, Abidin Kaya, Bahattin Yildiz. Constructing self-dual codes from group rings and reverse circulant matrices. Advances in Mathematics of Communications, 2021, 15 (3) : 471-485. doi: 10.3934/amc.2020077
[18]

Wei Wang, Yang Shen, Linyi Qian, Zhixin Yang. Hedging strategy for unit-linked life insurance contracts with self-exciting jump clustering. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021072
[19]

Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021016
[20]

Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021091

2019 Impact Factor: 1.105

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences