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May  2021, 20(5): 1961-1986. 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  May 2021 Early access  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 and Applied Analysis, 2021, 20 (5) : 1961-1986. doi: 10.3934/cpaa.2021054
References:
[1]

A. ArenasE. 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.

[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.

[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.

[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.

[5]

J. BetancorJ. C. FariñaL. Rodrígues-MesaR. 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.

[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.

[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.

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

[10]

P. GraczykJ. J. LoebI.A. LópezA. 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.

[11]

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

[12]

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

[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.

[14]

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

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[21]

A. NowakP. 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.

[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.

[23]

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

[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.

[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.

[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.

[27]

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

[28]

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

show all references

References:
[1]

A. ArenasE. 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.

[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.

[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.

[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.

[5]

J. BetancorJ. C. FariñaL. Rodrígues-MesaR. 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.

[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.

[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.

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

[10]

P. GraczykJ. J. LoebI.A. LópezA. 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.

[11]

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

[12]

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

[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.

[14]

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

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[21]

A. NowakP. 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.

[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.

[23]

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

[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.

[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.

[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.

[27]

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

[28]

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

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