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Spectral properties of ordinary differential operators admitting special decompositions
Wydziaƚ Matematyki, Politechnika Wrocƚawska, Wyb. Wyspiańskiego 27, 50-370 Wrocƚaw, Poland |
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.
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. |
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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. |
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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. |
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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. 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. |
[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. 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. |
[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,
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[18] |
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[19] |
V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer-Verlag, New York, 2011.
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[20] |
NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov. |
[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. |
[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. Yao, J. 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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. Yao, J. 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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