×

The similarity problem for indefinite Sturm-Liouville operators and the HELP inequality. (English) Zbl 1287.34017

Summary: We study two problems. The first one is the similarity problem for the indefinite Sturm-Liouville operator \[ A=-(\mathrm{sgn}\;x)\frac{d}{wdx}\frac{d}{rdx} \] acting in \(L_w^2(-b,b)\). It is assumed that \(w, r\in L_{\mathrm{loc}}^1(-b,b)\) are even and positive a.e. on \((-b,b)\).
The second object is the so-called HELP inequality \[ \left(\int _0^b\frac{1}{\tilde r}|f'|dx\right)^2\leq K^2\int _0^b|f|^2\tilde w \;dx\int _0^b\biggl|\frac{1}{\tilde w}\left(\frac{1}{\tilde r}f'\right)'\biggr|^2\tilde w dx, \] where the coefficients \(\tilde w, \tilde r\in L_{\mathrm{loc}}^1[0,b)\) are positive a.e. on (\(0,b\)).
Both problems are well understood when the corresponding Sturm-Liouville differential expression is regular. The main objective of the present paper is to give criteria for both the validity of the HELP inequality and the similarity to a self-adjoint operator in the singular case. Namely, we establish new criteria formulated in terms of the behavior of the corresponding Weyl-Titchmarsh \(m\)-functions at \(0\) and at \(\infty \). As a biproduct of this result we show that both problems are closely connected. Namely, the operator \(A\) is similar to a self-adjoint one precisely if the HELP inequality with \(\tilde w=r\) and \(\tilde r=w\) is valid.
Next we characterize the behavior of \(m\)-functions in terms of coefficients and then these results enable us to reformulate the obtained criteria in terms of coefficients. Finally, we apply these results for the study of the two-way diffusion equation, also known as the time-independent Fokker-Planck equation.

MSC:

34B24 Sturm-Liouville theory
26D10 Inequalities involving derivatives and differential and integral operators
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
47A10 Spectrum, resolvent
47A75 Eigenvalue problems for linear operators
34B20 Weyl theory and its generalizations for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abasheeva, N. L.; Pyatkov, S. G., Counterexamples in indefinite Sturm-Liouville problems, Siberian Adv. Math., 7, 1-8 (1997) · Zbl 0942.34022
[2] Abasheeva, N. L.; Pyatkov, S. G., Solvability of boundary value problems for operator-differential equations of mixed type, Sib. Math. J., 41, 6, 1174-1187 (2000) · Zbl 0964.35171
[3] Abasheeva, N. L.; Pyatkov, S. G., Solvability of boundary value problems for operator-differential equations of mixed type: the degenerate case, Sib. Math. J., 43, 3, 549-561 (2002) · Zbl 1003.35040
[4] Akopjan, R. V., On the regularity at infinity of the spectral function of a \(J\)-nonnegative operator, Izv. Akad. Nauk Arm. SSR Ser. Mat., 15, 5, 357-364 (1980), (in Russian) · Zbl 0447.47018
[5] Baouendi, M. S.; Grisvard, P., Sur une équation d’évolution changeante de type, J. Funct. Anal., 2, 352-367 (1968) · Zbl 0164.12701
[6] Beals, R., An abstract treatment of some forward-backward problems of transport and scattering, J. Funct. Anal., 34, 1-20 (1979) · Zbl 0425.34067
[7] Beals, R., Partial-range completeness and existence of solutions to two-way diffusion equations, J. Math. Phys., 22, 5, 954-960 (1981) · Zbl 0473.76080
[8] Beals, R., Indefinite Sturm-Liouville problems and half-range completeness, J. Differential Equations, 56, 391-407 (1985) · Zbl 0512.34017
[9] Beals, R.; Protopopescu, V., Half-range completeness for the Fokker-Planck equation, J. Stat. Phys., 32, 565-584 (1983) · Zbl 0571.35088
[10] Bennewitz, C., The HELP inequality in the regular case, Internat. Schriftenreihe Numer. Math., 80, 337-346 (1987)
[11] Bennewitz, C., Spectral asymptotics for Sturm-Liouville equations, Proc. Lond. Math. Soc. (3), 59, 294-338 (1989) · Zbl 0681.34023
[12] Binding, P.; Fleige, A., Conditions for an indefinite Sturm-Liouville Riesz basis property, Oper. Theory Adv. Appl., 198, 87-95 (2009) · Zbl 1200.34105
[13] Binding, P.; Fleige, A., A review of a Riesz basis property for indefinite Sturm-Liouville problems, Oper. Matrices, 5, 4, 735-755 (2011) · Zbl 1244.34042
[14] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1987), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0617.26001
[15] Bothe, W., Die Streuabsorption der elektronenstrahlen, Z. Phys., 54, 161-178 (1929) · JFM 55.0531.03
[16] Brown, B. M.; Evans, W. D., On an extension of Copson’s inequality for infinite series, Proc. Roy. Soc. Edinburgh Sect. A, 121, 169-183 (1992) · Zbl 0754.26010
[17] Brown, B. M.; Langer, M.; Schmidt, K. M., The HELP inequality on trees, (Analysis on Graphs and its Applications. Analysis on Graphs and its Applications, Proc. Sympos. Pure Math., vol. 77 (2008), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 337-354 · Zbl 1165.26315
[18] Buldygin, V. V.; Klesov, O. I.; Steinebach, J. S., On some properties of asymptotic quasi-inverse functions, Theory Probab. Math. Statist., 77, 15-30 (2008) · Zbl 1068.26002
[20] Ćurgus, B.; Langer, H., A Krein space approach to symmetric ordinary differential operators with an indefinite weight function, J. Differential Equations, 79, 31-61 (1989) · Zbl 0693.34020
[21] Ćurgus, B.; Najman, B., The operator \((sgn x) \frac{d^2}{d x^2}\) is similar to a selfadjoint operator in \(L^2(R)\), Proc. Amer. Math. Soc., 123, 1125-1128 (1995) · Zbl 0835.47021
[22] Evans, W. D.; Everitt, W. N., A return to the Hardy-Littlewood integral inequality, Proc. R. Soc. Lond. Ser. A, 380, 447-486 (1982) · Zbl 0487.26005
[23] Evans, W. D.; Everitt, W. N., HELP inequalities for limit-circle and regular problems, Proc. R. Soc. Lond. Ser. A, 432, 367-390 (1991) · Zbl 0724.34086
[24] Evans, W. D.; Zettl, A., Norm inequalities involving derivatives, Proc. Roy. Soc. Edinburgh Sect. A, 82, 51-70 (1978) · Zbl 0409.26008
[25] Everitt, W. N., On an extension to an integro-differential inequality of Hardy, Littlewood and Polya, Proc. Roy. Soc. Edinburgh Sect. A, 69, 295-333 (1972) · Zbl 0268.26013
[26] Everitt, W. N., A note on the Dirichlet condition for second-order differential expression, Canad. J. Math., 27, 312-320 (1976) · Zbl 0338.34011
[27] Faddeev, M. M.; Shterenberg, R. G., On similarity of differential operators to a selfadjoint one, Math. Notes, 72, 292-303 (2002)
[28] Fisch, N. J.; Kruskal, M. D., Separating variables in two-way diffusion equations, J. Math. Phys., 21, 4, 740-750 (1980) · Zbl 0455.35110
[29] Fleige, A., (Spectral Theory of Indefinite Krein-Feller Differential Operators. Spectral Theory of Indefinite Krein-Feller Differential Operators, Mathematical Research, vol. 98 (1996), Akademie Verlag: Akademie Verlag Berlin) · Zbl 0856.34085
[30] Fleige, A., A counterexample to completeness properties for indefinite Sturm-Liouville problems, Math. Nachr., 190, 123-128 (1998) · Zbl 0898.34018
[31] Fleige, A.; Najman, B., Nonsingularity of critical points of some differential and difference operators, Oper. Theory Adv. Appl., 102, 85-95 (1998) · Zbl 0908.47029
[32] Greenberg, W.; van der Mee, C. V.M.; Protopopescu, V., (Boundary Value Problems in Abstract Kinetic Theory. Boundary Value Problems in Abstract Kinetic Theory, Oper. Theory: Adv. Appl., vol. 23 (1987), Birkhäuser) · Zbl 0624.35003
[33] Hardy, G. H.; Littlewood, J. L.; Polya, G., Inequalities (1934), Cambridge University Press · JFM 60.0169.01
[34] Hartman, P., Ordinary Differential Equations (1964), John Wiley and Sons: John Wiley and Sons New York, London, Sidney · Zbl 0125.32102
[35] Kac, I. S., Integral characteristics of the growth of spectral functions for generalized second order boundary problems with boundary conditions at a regular end, Izv. Akad. Nauk SSSR Ser. Mat.. Izv. Akad. Nauk SSSR Ser. Mat., Math. Ussr-Izv., 5, 161-191 (1973), English transl. · Zbl 0285.34017
[36] Kac, I. S., A generalization of the asymptotic formula of V.A. Marchenko for the spectral function of a second order boundary value problem, Izv. Akad. Nauk SSSR Ser. Mat.. Izv. Akad. Nauk SSSR Ser. Mat., Math. Ussr-Izv., 7, 424-436 (1973), English transl.
[37] Kac, I. S.; Krein, M. G., A discreteness criterion for the spectrum of a singular string, Izv. Vuzov, Mat., 3, 2, 136-153 (1958), (in Russian) · Zbl 1469.34111
[38] Kac, I. S.; Krein, M. G., \(R\)-functions-analytic functions mapping the upper halfplane into itself, Amer. Math. Soc. Transl. Ser. (2), 103, 1-18 (1974) · Zbl 0291.34016
[39] Kac, I. S.; Krein, M. G., On the spectral function of the string, Amer. Math. Soc. Transl. Ser. (2), 103, 19-102 (1974) · Zbl 0291.34017
[40] Kalf, H., Remarks on some Dirichlet type results for semibounded Sturm-Liouville operators, Math. Ann., 210, 197-205 (1974) · Zbl 0297.34019
[41] Karabash, I. M., Abstract kinetic equations with positive collision operators, Oper. Theory Adv. Appl., 188, 175-195 (2008) · Zbl 1189.47079
[42] Karabash, I. M., A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators, Oper. Theory Adv. Appl., 203, 247-287 (2009) · Zbl 1206.47040
[43] Karabash, I. M.; Kostenko, A. S., Indefinite Sturm-Liouville operators with the singular critical point zero, Proc. Roy. Soc. Edinburgh Sect. A, 138, 801-820 (2008) · Zbl 1152.34018
[44] Karabash, I. M.; Kostenko, A. S.; Malamud, M. M., The similarity problem for \(J\)-nonnegative Sturm-Liouville operators, J. Differential Equations, 246, 964-997 (2009) · Zbl 1165.34012
[45] Karabash, I. M.; Malamud, M. M., Indefinite Sturm-Liouville operators \((sgn x)(- d^2 / d x^2 + q(x))\) with finite-zone potentials, Oper. Matrices, 1, 3, 301-368 (2007) · Zbl 1146.47032
[46] Kasahara, Y., Spectral theory of generalized second order differential operators and its applications to Markov processes, Jpn. J. Math., 1, 1, 67-84 (1975) · Zbl 0348.60113
[47] Kato, T., Perturbation Theory for Linear Operators (1966), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0148.12601
[48] Kohn, J. J.; Nirenberg, L., Degenerate elliptic-parabolic equations of second order, Comm. Pure Appl. Math., 20, 797-872 (1967) · Zbl 0153.14503
[49] Korevaar, J., (Tauberian Theory: A Century of Developments. Tauberian Theory: A Century of Developments, Grundlehren der Mathematischen Wissenschaften (2004), Springer) · Zbl 1056.40002
[50] Kostenko, A., On the similarity to a self-adjoint of some \(J\)-positive Sturm-Liouville operators, Math. Notes, 80, 1, 135-138 (2006) · Zbl 1128.47037
[51] Kostenko, A., The similarity problem for indefinite Sturm-Liouville operators with periodic coefficients, Oper. Matrices, 5, 4, 707-722 (2011) · Zbl 1314.47058
[53] Langer, H., Spectral functions of definitizable operators in Krein spaces, (Lecture Notes in Math., vol. 948 (1982)), 1-46
[54] Langer, M., A general HELP inequality connected with symmetric operators, Proc. R. Soc. Lond. Ser. A, 462, 587-606 (2006) · Zbl 1149.26301
[55] Oleinik, O. A.; Radkevic, E. V., Second Order Equations with Nonnegative Characteristic Form (1973), Plenum Press
[56] Pagani, C. D., On the parabolic equation \((sgn x) | x |^p u_y - u_{x x} = 0\) and a related one, Ann. Mat. Pura Appl., 99, 4, 333-399 (1974) · Zbl 0274.35038
[57] Pagani, C. D., On an initial-boundary value problem for the equation \(w_t = w_{x x} - x w_y\), Ann. Sc. Norm. Super. Pisa, 2, 219-263 (1975)
[58] Parfenov, A. I., On an embedding criterion for interpolation spaces and application to indefinite spectral problems, Sib. Math. J., 44, 4, 638-644 (2003) · Zbl 1033.46059
[59] Parfenov, A. I., The Ćurgus condition in indefinite Sturm-Liouville problems, Siberian Adv. Math., 15, 2, 68-103 (2005) · Zbl 1089.34025
[60] Pyatkov, S. G., Some properties of eigenfunctions of linear pencils, Sib. Math. J., 30, 587-597 (1989) · Zbl 0759.47013
[61] Pyatkov, S. G., Operator Theory. Nonclassical Problems (2002), VSP: VSP Utrecht · Zbl 1031.47001
[62] Pyatkov, S. G., Maximal semidefinite invariant subspaces for \(J\)-dissipative operators, Oper. Theory Adv. Appl., 221, 549-570 (2012) · Zbl 1277.47052
[63] Rogozin, B. A., A Tauberian theorem for increasing functions of dominated variation, Sib. Math. J., 43, 2, 353-356 (2002) · Zbl 1012.60039
[64] Seneta, E., (Regularly Varying Functions. Regularly Varying Functions, Lecture Notes in Math., vol. 508 (1976), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York) · Zbl 0324.26002
[65] van der Mee, C., (Exponentially Dichotomous Operators and Applications. Exponentially Dichotomous Operators and Applications, Oper. Theory: Adv. Appl., vol. 182 (2008), Birkhäuser) · Zbl 1158.47001
[66] Veselić, K., On spectral properties of a class of \(J\)-selfadjoint operators. I, Glas. Mat. Ser. III, 7, 2, 229-248 (1972) · Zbl 0249.47027
[67] Volkmer, H., Sturm-Liouville problems with indefinite weights and Everitt’s inequality, Proc. Roy. Soc. Edinburgh Sect. A, 126, 5, 1097-1112 (1996) · Zbl 0865.34017
[68] Wang, M. C.; Uhlenbeck, G. E., On the theory of Brownian motion II, Rev. Modern Phys., 17, 323-342 (1947) · Zbl 0063.08172
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.