# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021054

## Spectral properties of ordinary differential operators admitting special decompositions

 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.

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. 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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. 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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

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##### 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
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