## Spectral $$\zeta$$-functions and $$\zeta$$-regularized functional determinants for regular Sturm-Liouville operators.(English)Zbl 07423435

Summary: The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $$\zeta$$-functions to efficiently compute values of spectral $$\zeta$$-functions at positive integers associated with regular (three-coefficient) self-adjoint Sturm-Liouville differential expressions $$\tau$$. Depending on the underlying boundary conditions, we express the $$\zeta$$-function values in terms of a fundamental system of solutions of $$\tau y=zy$$ and their expansions about the spectral point $$z=0$$. Furthermore, we give the full analytic continuation of the $$\zeta$$-function through a Liouville transformation and provide an explicit expression for the $$\zeta$$-regularized functional determinant in terms of a particular set of this fundamental system of solutions. An array of examples illustrating the applicability of these methods is provided, including regular Schrödinger operators with zero, piecewise constant, and a linear potential on a compact interval.

### MSC:

 47A10 Spectrum, resolvent 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 47G10 Integral operators 34B27 Green’s functions for ordinary differential equations 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 34B24 Sturm-Liouville theory
Full Text:

### References:

 [1] Abramowitz, M.; Stegun, IA, Handbook of Mathematical Functions (1972), New York: Dover, New York · Zbl 0543.33001 [2] Alonso, A., Simon, B.: The Birman-Krein-Vishik theory of self-adjoint extensions of semibounded operators. J. Oper. Theory 4, 251-270 (1980); Addenda: 6, 407 (1981) · Zbl 0467.47017 [3] Amore, P., Spectral sum rules for the Schrödinger equation, Ann. Phys., 423, 168334 (2020) · Zbl 1451.81237 [4] Ashbaugh, M.; Gesztesy, F.; Mitrea, M.; Teschl, G., Spectral theory for perturbed Krein Laplacians in nonsmooth domains, Adv. Math., 223, 1372-1467 (2010) · Zbl 1191.35188 [5] Atkinson, FV, Discrete and Continuous Boundary Value Problems (1964), New York: Academic Press, New York · Zbl 0117.05806 [6] Atkinson, F.V., Mingarelli, A.B.: Asymptotics of the number of zeros and of the eigenvalues of general weighted Sturm-Liouville problems. J. Reine Angew. Math. 375/376, 380-393 (1987) · Zbl 0599.34026 [7] Awonusika, RO, Determinants of the Laplacians on complex projective spaces $${\mathbb{P}}_n() (n \ge 1)$$, J. Number Theory, 190, 131-155 (2018) · Zbl 1404.33005 [8] Ayub, R., Euler and the zeta function, Am. Math. Mon., 81, 1067-1086 (1974) · Zbl 0293.10001 [9] Behrndt, J.; Hassi, S.; De Snoo, H., Boundary Value Problems, Weyl Functions, and Differential Operators, Monographs in Mathematics (2020), Basel: Birkhäuser, Basel · Zbl 1457.47001 [10] Boas, RP, Entire Functions, Pure and Applied Mathematics (1954), New York: Academic Press, New York [11] de Monvel, A. Boutet; Marchenko, V.; Gohberg, I.; Lyubich, Yu, Asymptotic formulas for spectral and Weyl functions of Sturm-Liouville operators with smooth coefficients, New Results in Operator Theory and Its Applications. The Israel M. Glazman Memorial Volume. Operator Theory: Advances and Applications, 102-117 (1997), Boston: Birkhäuser, Boston · Zbl 0890.34020 [12] Burghelea, D.; Friedlander, L.; Kappeler, T., On the determinant of elliptic boundary value problems on a line segment, Proc. Am. Math. Soc., 123, 3027-3038 (1995) · Zbl 0848.34063 [13] Buslaev, VS; Faddeev, LD, Formulas for traces for a singular Sturm-Liouville differential operator, Sov. Math. Dokl., 1, 451-454 (1960) · Zbl 0129.06501 [14] Clark, S.; Gesztesy, F., Weyl-Titchmarsh $$M$$-function asymptotics for matrix-valued Schrödinger operators, Proc. Lond. Math. Soc., 3, 82, 701-724 (2001) · Zbl 1025.34021 [15] Clark, S., Gesztesy, F., Nichols, R., Zinchenko, M.: Boundary data maps and Krein’s resolvent formula for Sturm-Liouville operators on a finite interval. Oper. Matrices 8, 1-71 (2014) · Zbl 1311.34037 [16] Danielyan, AA; Levitan, BM, On the asymptotic behavior of the Weyl-Titchmarsh $$m$$-function, Math. USSR Izv., 36, 487-496 (1991) · Zbl 0723.34038 [17] Demirel, S.; Usman, M., Trace formulas for Schrödinger operators on the half-line, Bull. Math. Sci., 1, 397-427 (2011) · Zbl 1256.35041 [18] Dikki, LA, The zeta function of an ordinary differential equation on a finite interval, Izv. Akad. Nauk SSSR Ser. Mat., 19, 187-200 (1955) [19] Dikiĭ, LA, Trace formulas for Sturm-Liouville differential operators, Am. Math. Soc. Transl., 2, 18, 81-115 (1961) [20] Dreyfus, T.; Dym, H., Product formulas for the eigenvalues of a class of boundary value problems, Duke Math. J., 45, 15-37 (1978) · Zbl 0387.34021 [21] Elizalde, E., Ten Physical Applications of Spectral Zeta Functions. Lecture Notes in Physics (2012), New York: Springer, New York · Zbl 1250.81004 [22] Elizalde, E.; Odintsov, SD; Romeo, A.; Bytsenko, AA; Zerbini, S., Zeta Regularization Techniques with Applications (1994), Singapore: World Scientific, Singapore · Zbl 1050.81500 [23] Epstein, P., Zur Theorie allgemeiner Zetafunktionen, Math. Ann., 56, 615-644 (1903) · JFM 34.0461.02 [24] Epstein, P., Zur Theorie allgemeiner Zetafunktionen. II, Math. Ann., 63, 205-216 (1907) · JFM 37.0433.02 [25] Falco, GM; Fedorenko, AA; Gruzberg, IA, On functional determinants of matrix differential operators with multiple zero modes, J. Phys. A, 50, 485201 (2017) · Zbl 1380.81119 [26] Forman, R.: Functional determinants and geometry. Invent. Math. 88, 447-493 (1987) · Zbl 0602.58044 [27] Forman, R., Determinants, finite-difference operators and boundary value problems, Commun. Math. Phys., 147, 485-526 (1992) · Zbl 0767.58043 [28] Freitas, P.; Lipovský, J., Spectral determinant for the damped wave equation on an interval, Acta Phys. Polon. A, 136, 817-823 (2019) [29] Freitas, P.; Lipovský, J., The determinant of one-dimensional polyharmonic operators of arbitrary order, J. Funct. Anal., 279, 108783 (2020) · Zbl 1453.34107 [30] Fucci, G., Gesztesy, F., Kirsten, K., Littlejohn, L.L., Nichols, R., Stanfill, J.: The Krein-von Neumann extension revisited. Appl. Anal. (2021) 25 pp. doi:10.1080/00036811.2021.1938005 [31] Fucci, G.; Graham, C.; Kirsten, K., Spectral functions for regular Sturm-Liouville problems, J. Math. Phys., 56, 043503 (2015) · Zbl 1317.30060 [32] Gesztesy, F.; Holden, H.; Simon, B.; Zhao, Z., Higher order trace relations for Schrödinger operators, Rev. Math. Phys., 7, 893-922 (1995) · Zbl 0833.34084 [33] Gesztesy, F.; Kirsten, K., Effective computation of traces, determinants, and $$\zeta$$-functions for Sturm-Liouville operators, J. Funct. Anal., 276, 520-562 (2019) · Zbl 06988448 [34] Gesztesy, F., Zinchenko, M.: Sturm-Liouville Operators, Their Spectral Theory, and Some Applications, vol. I (in preparation) · Zbl 1108.34063 [35] Gohberg, I.; Krein, MG, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs (1969), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0181.13504 [36] Gohberg, IC; Krein, MG, Theory and Applications of Volterra Operators in Hilbert Space, Translations of Mathematical Monographs (1970), Providence, RI: American Mathematical Society, Providence, RI [37] Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, Corrected and Enlarged Edition, Prepared by A. Jeffery, Academic Press, San Diego (1980) · Zbl 0521.33001 [38] Graham, C.; Kirsten, K.; Morales-Almazan, P.; Quantz Streit, B., Functional determinants for Laplacians on annuli and elliptical regions, J. Math. Phys., 59, 013508 (2018) · Zbl 1390.35056 [39] Hermi, L.; Saito, N., On Rayleigh-type formulas for a non-local boundary value problem associated with an integral operator commuting with the Laplacian, Appl. Comput. Harmon. Anal., 45, 59-83 (2018) · Zbl 1443.47047 [40] Hinton, DB; Klaus, M.; Shaw, JK, Series representation and asymptotics for Titchmarsh-Weyl $$m$$-functions, Differ. Integr. Equ., 2, 419-429 (1989) · Zbl 0715.34044 [41] Jörgens, K., Rellich, F.: Eigenwerttheorie Gewöhnlicher Differentialgleichungen. Springer, Berlin (1976). ((German.)) · Zbl 0347.34013 [42] Jost, R.; Pais, A., On the scattering of a particle by a static potential, Phys. Rev., 82, 840-851 (1951) · Zbl 0042.45206 [43] Kaper, HG; Kong, Man Kam, Asymptotics of the Titchmarsh-Weyl $$m$$-coefficient for integrable potentials, Proc. R. Soc. Edinb., 103A, 347-358 (1986) · Zbl 0618.34019 [44] Kapteyn, W.: Le calcul numérique. Mém. Soc. R. Sci. Liége Ser. 3 VI(9), (1906). (14 pp) · JFM 37.0474.04 [45] Kirsten, K., Generalized multidimensional Epstein zeta functions, J. Math. Phys., 35, 459-470 (1994) · Zbl 0802.11034 [46] Kirsten, K., Spectral Functions in Mathematics and Physics (2002), Boca Raton: CRC Press, Boca Raton · Zbl 1007.58015 [47] Kirsten, K.; McKane, AJ, Functional determinants by contour integration methods, Ann. Phys., 308, 502-527 (2003) · Zbl 1051.58014 [48] Kirsten, K.; McKane, AJ, Functional determinants for general Sturm-Liouville problems, J. Phys. A, 37, 4649-4670 (2004) · Zbl 1064.34016 [49] Lesch, M., Determinants of regular singular Sturm-Liouville operators, Math. Nachr., 194, 139-170 (1998) · Zbl 0924.58107 [50] Lesch, M.; Tolksdorf, J., On the determinant of one-dimensional elliptic boundary value problems, Commun. Math. Phys., 193, 643-660 (1998) · Zbl 0920.47046 [51] Lesch, M.; Vertman, B., Regular singular Sturm-Liouville operators and their zeta-determinants, J. Funct. Anal., 261, 408-450 (2011) · Zbl 1230.34077 [52] Ja, B., Levin, Distribution of Zeros of Entire Functions, rev., ed., Translations of Mathematics Monographs (1980), Providence, RI: American Mathematical Society, Providence, RI [53] Levit, S.; Smilansky, U., A theorem on infinite products of eigenvalues of Sturm-Liouville type operators, Proc. Am. Math. Soc., 65, 299-302 (1977) · Zbl 0374.34016 [54] Levitan, BM; Sargsjan, IS, Introduction to Spectral Theory, Translations of Mathematical Monographs (1975), Providence, RI: American Mathematical Society, Providence, RI [55] Marchenko, V.A.: Sturm-Liouville Operators and Applications, rev. ed., AMS Chelsea Publ., American Mathematical Society, Providence, RI (2011) · Zbl 1298.34001 [56] Mingarelli, A.B.: Some remarks on the order of an entire function associated with a second order differential equation. In: Everitt, W.N., Lewis, R.T. (eds.) Ordinary Differential Equations and Operators. A tribute to F.V. Atkinson, Proceedings of a Symposium held at Dundee, Scotland, March-July 1982. Lecture Notes in Math., vol. 1032, pp. 384-389. Springer, Berlin (1983) [57] Mirzoev, K.A., Safonova, T.A.: Green’s function of ordinary differential operators and an integral representation of sums of certain power series. Dokl. Math. 98, 486-4489 (2018) · Zbl 1406.34059 [58] Müller, W., Relative zeta functions, relative determinants and scattering theory, Commun. Math. Phys., 192, 309-347 (1998) · Zbl 0947.58025 [59] Muñoz-Castañeda, JM; Kirsten, K.; Bordag, M., QFT over the finite line. Heat kernel coefficients, spectral zeta functions and selfadjoint extensions, Lett. Math. Phys., 105, 523-549 (2015) · Zbl 1325.81114 [60] Naimark, M.A.: Linear Differential Operators. Part I: Elementary Theory of Linear Differential Operators, Transl. by E. R. Dawson, Engl. translation edited by W. N. Everitt. Ungar Publishing, New York (1967) · Zbl 0219.34001 [61] Naimark, M.A.: Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space, Transl. by E. R. Dawson, Engl. translation edited by W. N. Everitt. Ungar Publishing, New York (1968) · Zbl 0227.34020 [62] Östensson, J.; Yafaev, DR; Dym, H.; Kaashoek, MA; Lancaster, P.; Langer, H.; Lerer, L., A trace formula for differential operators of arbitrary order, Panorama of Modern Operator Theory and Related Topics. The Israel Gohberg Memorial Volume, Operator Theory: Advances and Applications, 541-570 (2012), Basel: Birkhäuser, Basel [63] Reed, M.; Simon, B., Methods of Modern Mathematical Physics. IV. Analysis of Operators (1978), New York: Academic Press, New York · Zbl 0401.47001 [64] Robert, D.; Sordoni, V.; Greiner, PC; Ivrii, V.; Seco, LA; Sulem, C., Generalized determinants for Sturm-Liouville problems on the real line, Partial Differential Equations and Their Applications. CRM Proceedings & Lecture Notes, 251-259 (1997), Providence, RI: American Mathematical Society, Providence, RI [65] Rozenblum, GV; Shubin, MA; Solomyak, MZ, Spectral theory of differential operators, Partial Differential Equations VII, Encyclopedia of Mathematical Science (1994), Berlin: Springer, Berlin [66] Rybkin, A., On a complete analysis of high-energy scattering matrix asymptotics for one dimensional Schrödinger operators with integrable potentials, Proc. Am. Math. Soc., 130, 59-67 (2001) · Zbl 0986.34073 [67] Rybkin, A., Some new and old asymptotic representations of the Jost solution and the Weyl $$m$$-function for Schrödinger operators on the line, Bull. Lond. Math. Soc., 34, 61-72 (2002) · Zbl 1048.34140 [68] Simon, B., Notes on infinite determinants of Hilbert space operators, Adv. Math., 24, 244-273 (1977) · Zbl 0353.47008 [69] Simon, B., Trace Ideals and Their Applications, Mathematical Surveys and Monographs (2005), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1074.47001 [70] Simon, B., Operator Theory. A Comprehensive Course in Analysis, Part 4 (2015), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1334.00003 [71] Spreafico, M., Zeta determinants of Sturm-Liouville operators, Funct. Anal. Appl., 54, 149-154 (2020) · Zbl 1454.34048 [72] Takhtajan, LA, Quantum Mechanics for Mathematicians, Graduate Studies in Mathematics (2008), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1156.81004 [73] Titchmarsh, EC, A theorem on infinite products, J. Lond. Math. Soc., 1, 35-37 (1926) · JFM 52.0212.04 [74] Titchmarsh, EC, On integral functions with real negative zeros, Proc. Lond. Math. Soc., 26, 186-200 (1927) · JFM 53.0295.01 [75] Vertman, B., Regularized limit of determinants for discrete tori, Monatsh. Math., 186, 539-557 (2018) · Zbl 1394.58018 [76] Weidmann, J., Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics (1980), New York: Springer, New York [77] Weidmann, J.: Lineare Operatoren in Hilberträumen. Teil II: Anwendungen, Teubner, Stuttgart (2003) · Zbl 1025.47001 [78] Zettl, A., Sturm-Liouville Theory, Mathematical Surveys and Monographs (2005), Providence, RI: American Mathematical Society, Providence, RI
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.