# Article

Full entry | Fulltext not available (moving wall 24 months)
Keywords:
Kreĭn--Feller operator; spectral asymptotics; harmonic analysis
Summary:
Motivated by the fundamental theorem of calculus, and based on the works of W. Feller as well as M. Kac and M.\,G. Kreĭn, given an atomless Borel probability measure $\eta$ supported on a compact subset of $\mathbb R$ U. Freiberg and M. Zähle introduced a measure-geometric approach to define a first order differential operator $\nabla_{\eta}$ and a second order differential operator $\Delta_{\eta}$, with respect to $\eta$. We generalize this approach to measures of the form $\eta := \nu + \delta$, where $\nu$ is non-atomic and $\delta$ is finitely supported. We determine analytic properties of $\nabla_{\eta}$ and $\Delta_{\eta}$ and show that $\Delta_{\eta}$ is a densely defined, unbounded, linear, self-adjoint operator with compact resolvent. Moreover, we give a systematic way to calculate the eigenvalues and eigenfunctions of $\Delta_{\eta}$. For two leading examples, we determine the eigenvalues and the eigenfunctions, as well as the asymptotic growth rates of the eigenvalue counting function.
References:
[1] Arzt P.: Measure theoretic trigonometric functions. J. Fractal Geom. 2 (2015), no. 2, 115–169. DOI 10.4171/JFG/18 | MR 3353090
[2] Beals R., Greiner P. C.: Strings, waves, drums: spectra and inverse problems. Anal. Appl. (Singap.) 7 (2009), no. 2, 131–183. DOI 10.1142/S0219530509001335 | MR 2513598
[3] Berry M. V.: Distribution of modes in fractal resonators. Structural Stability in Physics, Proc. Internat. Symposia Appl. Catastrophe Theory and Topological Concepts in Phys., Inst. Inform. Sci., Univ. Tübingen, 1978, Springer Ser. Synergetics, 4, Springer, Berlin, 1979, pages 51–53. DOI 10.1007/978-3-642-67363-4_7 | MR 0556688
[4] Berry M. V.: Some geometric aspects of wave motion: wavefront dislocations, diffraction catastrophes, diffractals. Geometry of the Laplace operator, Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, 1980, pages 13–28. MR 0573427
[5] Biggs N.: Algebraic Graph Theory. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993. MR 1271140 | Zbl 0797.05032
[6] Ehnes T.: Stochastic heat equations defined by fractal Laplacians on Cantor-like sets. available at arXiv: 1902.02175v2 [math.PR] (2019), 27 pages.
[7] Feller W.: Generalized second order differential operators and their lateral conditions. Illinois J. Math. 1 (1957), 459–504. DOI 10.1215/ijm/1255380673 | MR 0092046
[8] Freiberg U.: A survey on measure geometric Laplacians on Cantor like sets. Wavelet and fractal methods in science and engineering, Part I., Arab. J. Sci. Eng. Sect. C Theme Issues 28 (2003), no. 1, 189–198. MR 2030736
[9] Freiberg U.: Analytical properties of measure geometric Krein–Feller-operators on the real line. Math. Nachr. 260 (2003), 34–47. DOI 10.1002/mana.200310102 | MR 2017701
[10] Freiberg U.: Spectral asymptotics of generalized measure geometric Laplacians on Cantor like sets. Forum Math. 17 (2005), no. 1, 87–104. DOI 10.1515/form.2005.17.1.87 | MR 2110540
[11] Freiberg U., Zähle M.: Harmonic calculus on fractals---a measure geometric approach. I. Potential Anal. 16 (2002), no. 3, 265–277. DOI 10.1023/A:1014085203265 | MR 1885763
[12] Fujita T.: A fractional dimension, self-similarity and a generalized diffusion operator. Probabilistic Methods in Mathematical Physics, Katata/Kyoto, 1985, Academic Press, Boston, 1987, pages 83–90. MR 0933819
[13] Gordon C., Webb D., Wolpert S.: Isospectral plane domains and surfaces via Riemannian orbifolds. Invent. Math. 110 (1992), no. 1, 1–22. DOI 10.1007/BF01231320 | MR 1181812
[14] Halmos P. R.: Measure Theory. D. Van Nostrand Company, New York, 1950. MR 0033869 | Zbl 0283.28001
[15] Jin X.: Spectral representation of one-dimensional Liouville Brownian motion and Liouville Brownian excursion. available at arXiv: 1705.01726v1 [math.PR] (2017), 23 pages.
[16] Kac I. S., Kreĭn M. G.: Criteria for the discreteness of the spectrum of a singular string. Izv. Vysš. Učebn. Zaved. Matematika 1958 (1958), no. 2 (3), 136–153. MR 0139804
[17] Kac M.: Can one hear the shape of a drum?. Amer. Math. Monthly 73 (1966), no. 4, part II, 1–23. DOI 10.1080/00029890.1966.11970915 | MR 0201237
[18] Kesseböhmer M., Niemann A., Samuel T., Weyer H.: Generalised Kreĭn–Feller operators and Liouville Brownian motion via transformations of measure spaces. available at arXiv:1909.08832v2 [math.FA], (2019), 13 pages.
[19] Kesseböhmer M., Samuel T., Weyer H.: A note on measure-geometric Laplacians. Monatsh. Math. 181 (2016), no. 3, 643–655. DOI 10.1007/s00605-016-0906-0 | MR 3552804
[20] Kesseböhmer M., Samuel T., Weyer H.: Measure-geometric Laplacians for discrete distributions. Horizons of Fractal Geometry and Complex Dimensions, Contemp. Math., 731, Amer. Math. Soc., Providence, 2019, pages 133–142. DOI 10.1090/conm/731/14676 | MR 3989819
[21] Kigami J., Lapidus M. L.: Self-similarity of volume measures for Laplacians on p.c.f. self-similar fractals. Comm. Math. Phys. 217 (2001), no. 1, 165–180. DOI 10.1007/s002200000326 | MR 1815029
[22] Kotani S., Watanabe S.: Kreĭn's spectral theory of strings and generalized diffusion processes. Functional analysis in Markov Processes, Katata/Kyoto, 1981, Lecture Notes in Math., 923, Springer, Berlin, 1982, pages 235–259. MR 0661628
[23] Lapidus M. L.: Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl–Berry conjecture. Trans. Amer. Math. Soc. 325 (1991), no. 2, 465–529. DOI 10.1090/S0002-9947-1991-0994168-5 | MR 0994168
[24] Lapidus M. L.: Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media and the Weyl–Berry conjecture. Ordinary and Partial Differential Equations, Vol. IV, Dundee, 1992, Pitman Res. Notes Math. Ser., 289, Longman Sci. Tech., Harlow, 1993, pages 126–209. MR 1234502
[25] Lapidus M. L., Pomerance C.: Fonction zêta de Riemann et conjecture de Weyl–Berry pour les tambours fractals. C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 6, 343–348 (French. English summary). MR 1046509
[26] Lapidus M. L., Pomerance C.: The Riemann zeta-function and the one-dimensional Weyl–Berry conjecture for fractal drums. Proc. London Math. Soc. (3) 66 (1993), no. 1, 41–69. MR 1189091
[27] Lapidus M. L., Pomerance C.: Counterexamples to the modified Weyl–Berry conjecture on fractal drums. Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 1, 167–178. DOI 10.1017/S0305004100074053 | MR 1356166
[28] Milnor J.: Eigenvalues of the Laplace operator on certain manifolds. Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542. DOI 10.1073/pnas.51.4.542 | MR 0162204
[29] Reed M., Simon B.: Methods of Modern Mathematical Physics. I. Functional analysis, Academic Press, Harcourt Brace Jovanovich Publishers, New York, 1980. MR 0493421
[30] Rhodes R., Vargas V.: Spectral dimension of Liouville quantum gravity. Ann. Henri Poincaré 15 (2014), no. 12, 2281–2298. DOI 10.1007/s00023-013-0308-y | MR 3272822
[31] Urakawa H.: Bounded domains which are isospectral but not congruent. Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 441–456. DOI 10.24033/asens.1433 | MR 0690649
[32] Weyl H.: Über die Abhängigkeit der Eigenschwingungen einer Membran und deren Begrenzung. J. Reine Angew. Math. 141 (1912), 1–11 (German). MR 1580843

Partner of