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Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

 
 

 

Sturm-Liouville operators


Author: K. A. Mirzoev
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 75 (2014), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2014, 281-299
MSC (2010): Primary 34B24; Secondary 34B20, 34B40
DOI: https://doi.org/10.1090/S0077-1554-2014-00234-X
Published electronically: November 5, 2014
MathSciNet review: 3308613
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Abstract: Let $ (a,b)\subset \mathbb{R}$ be a finite or infinite interval, let $ p_0(x)$, $ q_0(x)$, and $ p_1(x)$, $ x\in (a,b)$, be real-valued measurable functions such that $ p_0,p^{-1}_0$, $ p^2_1p^{-1}_0$, and $ q^2_0p^{-1}_0$ are locally Lebesgue integrable (i.e., lie in the space $ L^1_{\operatorname {loc}}(a,b)$), and let $ w(x)$, $ x\in (a,b)$, be an almost everywhere positive function. This paper gives an introduction to the spectral theory of operators generated in the space $ \mathcal {L}^2_w(a,b)$ by formal expressions of the form

$\displaystyle l[f]:=w^{-1}\{-(p_0f')'+i[(q_0f)'+q_0f']+p'_1f\},$    

where all derivatives are understood in the sense of distributions. The construction described in the paper permits one to give a sound definition of the minimal operator $ L_0$ generated by the expression $ l[f]$ in $ \mathcal {L}^2_w(a,b)$ and include $ L_0$ in the class of operators generated by symmetric (formally self-adjoint) second-order quasi-differential expressions with locally integrable coefficients. In what follows, we refer to these operators as Sturm-Liouville operators. Thus, the well-developed spectral theory of second-order quasi-differential operators is used to study Sturm-Liouville operators with distributional coefficients. The main aim of the paper is to construct a Titchmarsh-Weyl theory for these operators. Here the problem on the deficiency indices of $ L_0$, i.e., on the conditions on $ p_0$, $ q_0$, and $ p_1$ under which Weyl's limit point or limit circle case is realized, is a key problem. We verify the efficiency of our results for the example of a Hamiltonian with $ \delta $-interactions of intensities $ h_k$ centered at some points $ x_k$, where

$\displaystyle l[f]=-f''+\sum _{j}h_j\delta (x-x_j)f.$    


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  • 1. N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, vol. 2, Vyshcha shkola, Kharkov, 1978; English transl., Pitman, Boston, MA-London, 1981. MR 509335 (82g:47001b)
  • 2. R. S. Ismagilov, Conditions for self-adjointness of differential operators of higher order, Dokl. Akad. Nauk SSSR 142 (1962), no. 6, 1239-1242; English transl., Sov. Math. Dokl. 3 (1962), 279-283. MR 0131594 (24:A1443)
  • 3. R. S. Ismagilov, On the self-adjointness of the Sturm-Liouville operator, Uspekhi Mat. Nauk 18 (1963), no. 5 (113), 161-166. (Russian) MR 0155037 (27:4979)
  • 4. N. N. Konechnaya, Asymptotic integration of symmetric second-order quasidifferential equations, Mat. Zametki 90 (2011), no. 6, 875-884; English transl., Math. Notes 90 (2011), no. 5-6, 850-858. MR 2962962
  • 5. A. S. Kostenko and M. M. Malamud, On the one-dimensional Schrödinger operator with $ \delta $-interactions, Funktsional. Anal. i Prilozhen. 44 (2010), no. 2, 87-91; English transl., Funct. Anal. Appl. 44 (2010), no. 2, 151-155. MR 2681961 (2011j:47135)
  • 6. M. G. Krein, On the indeterminate case of the Sturm-Liouville boundary problem in the interval $ (0,\infty )$, Izv. Akad. Nauk SSSR Ser. Mat. 16 (1952) no. 4, 293-324. (Russian) MR 0052004 (14:558g)
  • 7. B. M. Levitan, Expansions in eigenfunctions of second-order differential equations, Gos. Izdat. Tekhn.-Teor. Lit., Moscow-Leningrad, 1950. (Russian) MR 0036918 (12:183e)
  • 8. B. M. Levitan, Proof of the theorem on the expansion in eigenfunctions of self-adjoint differential equations, Dokl. Akad. Nauk SSSR 73 (1950), 651-654. (Russian) MR 0039152 (12:502e)
  • 9. K. A. Mirzoev, The Cauchy function and $ \mathcal {L}^p_w$-properties of solutions of quasidifferential equations, Uspekhi Mat. Nauk 46 (1991), no. 4, 161-162; English transl., Russ. Math. Surveys 46 (1991), no. 4, 190-191. MR 1138966 (93d:34012)
  • 10. K. A. Mirzoev, Analogs of limit-point theorems, Mat. Zametki 57 (1995), no. 3, 394-414; English transl., Math. Notes 57 (1995), no. 3, 275-287. MR 1346439 (97c:34045)
  • 11. V. A. Naimark, Linear differential operators, 2d ed., Nauka, Moscow, 1969; English transl. of the 1st ed., Frederick Ungar, New York, 1967 (part 1) and 1968 (part 2). MR 0353061 (50:5547)
  • 12. Yu. B. Orochko, Sufficient conditions for the self-adjointness of the Sturm-Liouville operator, Mat. Zametki 15 (1974), no. 2, 271-280; English transl., Math. Notes, 15 (1974), no. 2, 155-160. MR 0375002 (51:11198)
  • 13. Yu. V. Pokornyi et al., Differential equations on geometric graphs, Fizmatlit, Moscow, 2004. (Russian)
  • 14. A. M. Savchuk and A. A. Shkalikov, Sturm-Liouville operators with singular potentials, Mat. Zametki 66 (1999), no. 6, 897-912; English transl., Math. Notes 66 (1999), no. 6, 741-753. MR 1756602 (2000m:34061)
  • 15. A. M. Savchuk and A. A. Shkalikov, Sturm-Liouville operators with distribution potentials, Trudy Moskov. Matem. Obshch. 64 (2003), 159-212; English transl., Trans. Moscow Math. Soc. 64 (2003), 143-192 MR 2030189 (2004j:34198)
  • 16. P. Hartman, Ordinary differential equations, Wiley, New York-London-Sydney, 1964. MR 0171038 (30:1270)
  • 17. S. Albeverio, A. Kostenko, and M. Malamud, Spectral theory of semibounded Sturm-Liouville operators with local interactions on a discrete set, J. Math. Phys. 51 (2010), no. 10, 102102. MR 2761288 (2012a:47105)
  • 18. W. O. Amrein, A. M. Hinz, and D. B. Pearson, Sturm-Liouville theory. Past and present, Birkhäuser, Basel, 2005. MR 2132131 (2006a:34003)
  • 19. C. Bennewitz and W. N. Everitt, Some remarks on the Titchmarsh-Weyl $ m$-coefficient, Proc. of the Pleijel Conference, University of Uppsala, Amer. Math. Soc., Providence, RI, 1980, pp. 49-108.
  • 20. D. Buschmann, G. Stolz, and J. Weidmann, One-dimensional Schrödinger operators with local point interactions, J. Reine Angew. Math. 467 (1995), 169-186. MR 1355927 (97c:47055)
  • 21. A. C. Dixon, On the series of Sturm and Liouville, as derived from a pair of fundamental integral equations instead of a differential equation, Philos. Trans. Roy. Soc. London. Ser. A 211 (1912), 411-432.
  • 22. M. S. P. Eastham, On a limit-point method of Hartman, Bull. London Math. Soc. 4 (1972), 340-344. MR 0316801 (47:5349)
  • 23. M. S. P. Eastham and M. L. Thompson, On the limit-point, limit-circle classification of second-order ordinary differential equations, Quart. J. Math. Oxford Ser. (2) 24 (1973), 531-535. MR 0417481 (54:5531)
  • 24. J. Eckhardt, F. Gesztesy, R. Nichols, and G. Teschl, Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials, arXiv:1208.4677v3[math.SP]. MR 3046408
  • 25. W. N. Everitt, On the deficiency index problem for ordinary differential operators 1910-1976, Differential Equations, Proc. Internat. Conf. Uppsala, 1977, pp. 62-81. MR 0477247 (57:16788)
  • 26. W. N. Everitt, I. W. Knowles, and T. T. Read, Limit-point and limit-circle criteria for Sturm-Liouville equations with intermittenly negative principal coefficients, Proc. Roy. Soc. Edinburgh. Sect. A 103 (1986), 215-228. MR 866835 (87k:34028)
  • 27. W. N. Everitt and L. Marcus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, Mathematical Surveys and Monographs, vol. 61, Amer. Math. Soc., Providence, RI, 1999. MR 1647856 (2000c:34030)
  • 28. W. N. Everitt and D. Race, On necessary and sufficient conditions for the existence of Carathéodory solutions of ordinary differential equations, Quaestiones Math. 2 (1977/78), no. 4, 507-512. MR 0477222 (57:16763)
  • 29. W. N. Everitt and D. Race, The regular representation of singular second-order differential expressions using quasi-derivatives, Proc. London Math. Soc. (3) 65 (1992), no. 2, 383-404. MR 1168193 (93m:34140)
  • 30. Ch. T. Fulton, Parametrizations of Titchmarsh's $ m(\lambda )$-functions in the limit circle case, Trans. Amer. Math. Soc. 229 (1977), 51-63. MR 0450657 (56:8950)
  • 31. A. Goriunov and V. Mikhailets, Regularization of singular Sturm-Liouville equations, Methods Funct. Anal. Topology 16 (2010), no. 2, 120-130. MR 2667807 (2011e:34060)
  • 32. A. S. Kostenko and M. M. Malamud, 1-D Schrödinger operators with local point interactions on a discrete set, J. Differential Equations 249 (2010), no. 2, 253-304. MR 2644117 (2011g:34206)
  • 33. T. T. Read, A limit-point criterion for expressions with intermittenly positive coefficients, J. London Math. Soc. (2) 15 (1977), no. 2, 271-276. MR 0437844 (55:10765)
  • 34. C. S. Christ and G. Stolz, Spectral theory of one-dimensional Schrödinger operators with point interactions, J. Math. Anal. Appl. 184 (1994), no. 3, 491-516. MR 1281525 (95k:47072)
  • 35. Ch. Sturm and J. Liouville, Extrait d'un Mémoire sur le developpement des fonctions en séries dont les différents termes sont assujettis à satisfaire à une même équation différentielle linéaire, contenant un paramètre variable, J. Math. Pures Appl. II (1837), no. 2, 220-223.
  • 36. H. Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), 220-269. MR 1511560
  • 37. R. E. White, Weak solutions of $ (p(x)u'(x))'+g(x)u'(x)+qu(x)=f$ with  $ q,f\in H_{-1}[a,b]$ $ 0<p(x)\in L_{\infty }[a,b]$, $ g(x)\in L_{\infty }[a,b]$ and $ u\in H_1[a,b]$, SIAM J. Math. Anal. 10 (1979), no. 6, 1313-1325. MR 547816 (80k:34022)
  • 38. A. Zettl, Sturm-Liouville theory, Math. Surveys and Monographs, vol. 121, Amer. Math. Soc., Providence, RI, 2005. MR 2170950 (2007a:34005)

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Additional Information

K. A. Mirzoev
Affiliation: Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia
Email: mirzoev.karahan@mail.ru

DOI: https://doi.org/10.1090/S0077-1554-2014-00234-X
Keywords: Second-order quasi-differential operator, minimal operator, maximal operator, Sturm--Liouville theory, deficiency index, limit point--limit circle, linear differential equation with distributional coefficients, finite-difference equation, Jacobi matrix
Published electronically: November 5, 2014
Additional Notes: Supported by RFBR Grant No. 14-01-00349
Article copyright: © Copyright 2014 American Mathematical Society

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