Hierarchical Schr\"odinger Operators with Singular Potentials

Alexander Bendikov ; Alexander Grigor'yan ; Stanislav Molchanov

Informatics and Automation, Theory of Functions of Several Real Variables and Its Applications, Tome 323 (2023), pp. 17-52.

Résumé

We consider the operator $H=L+V$ that is a perturbation of the Taibleson–Vladimirov operator $L=\mathfrak {D}^\alpha $ by a potential $V(x)=b\|x\|^{-\alpha }$, where $\alpha >0$ and $b\geq b_*$. We prove that the operator $H$ is closable and its minimal closure is a nonnegative definite self-adjoint operator (where the critical value $b_*$ depends on $\alpha $). While the operator $H$ is nonnegative definite, the potential $V(x)$ may well take negative values as $b_*0$ for all $0\alpha 1$. The equation $Hu=v$ admits a Green function $g_H(x,y)$, that is, the integral kernel of the operator $H^{-1}$. We obtain sharp lower and upper bounds on the ratio of the Green functions $g_H(x,y)$ and $g_L(x,y)$.

Détail

Keywords: ultrametric space, $p$-adic numbers, Dyson model, hierarchical Laplacian, hierarchical Schrödinger operator, Vladimirov operator.

Bibliographie

[1] Aizenman M., Molchanov S., “Localization at large disorder and at extreme energies: An elementary derivation”, Commun. Math. Phys., 157:2 (1993), 245–278 | DOI | MR | Zbl

[2] Albeverio S., Karwowski W., “A random walk on $p$-adics—the generator and its spectrum”, Stochastic Processes Appl., 53:1 (1994), 1–22 | DOI | MR | Zbl

[3] Bendikov A., “Heat kernels for isotropic-like Markov generators on ultrametric spaces: A survey”, p-Adic Numbers, Ultrametric Anal. Appl., 10:1 (2018), 1–11 | DOI | MR | Zbl

[4] Bendikov A., Cygan W., Woess W., “Oscillating heat kernels on ultrametric spaces”, J. Spectr. Theory, 9:1 (2019), 195–226 | DOI | MR | Zbl

[5] Bendikov A., Grigor'yan A., Molchanov S., On the spectrum of the hierarchical Schrödinger type operators, E-print, 2020, arXiv: 2006.02263v1 [math.SP] | MR | Zbl

[6] A. D. Bendikov, A. A. Grigor'yan, S. A. Molchanov, and G. P. Samorodnitsky, “On a class of random perturbations of the hierarchical Laplacian”, Izv. Math., 79:5 (2015), 859–893 | DOI | DOI | MR | Zbl

[7] Bendikov A., Grigor'yan A., Pittet Ch., “On a class of Markov semigroups on discrete ultra-metric spaces”, Potential Anal., 37:2 (2012), 125–169 | DOI | MR | Zbl

[8] A. D. Bendikov, A. A. Grigor'yan, Ch. Pittet, and W. Woess, “Isotropic Markov semigroups on ultra-metric spaces”, Russ. Math. Surv., 69:4 (2014), 589–680 | DOI | DOI | MR | Zbl

[9] Bendikov A., Krupski P., “On the spectrum of the hierarchical Laplacian”, Potential Anal., 41:4 (2014), 1247–1266 | DOI | MR | Zbl

[10] F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Kluwer, Dordrecht, 1991 | MR | Zbl

[11] Berg Ch., Forst G., Potential theory on locally compact abelian groups, Ergeb. Math. Grenzgeb., 87, Springer, Berlin, 1975 | MR | Zbl

[12] Beurling A., Deny J., “Dirichlet spaces”, Proc. Natl. Acad. Sci. USA, 45 (1959), 208–215 | DOI | MR | Zbl

[13] Bovier A., “The density of states in the Anderson model at weak disorder: A renormalization group analysis of the hierarchical model”, J. Stat. Phys, 59 (1990), 745–779 | DOI | MR | Zbl

[14] Davies E.B., Spectral theory and differential operators, Cambridge Stud. Adv. Math., 42, Cambridge Univ. Press, Cambridge, 1995 | MR | Zbl

[15] Del Muto M., Figà-Talamanca A., “Diffusion on locally compact ultrametric spaces”, Expo. math., 22:3 (2004), 197–211 | DOI | MR | Zbl

[16] Derfel G., Grabner P.J., Vogl F., “Laplace operators on fractals and related functional equations”, J. Phys. A: Math. Theor., 45:46 (2012), 463001 | DOI | MR | Zbl

[17] Dyson F.J., “Existence of a phase-transition in a one-dimensional Ising ferromagnet”, Commun. Math. Phys., 12 (1969), 91–107 | DOI | MR | Zbl

[18] Dyson F.J., “An Ising ferromagnet with discontinuous long-range order”, Commun. Math. Phys., 21 (1971), 269–283 ; Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, v. 2, URSS, M., 2010 | DOI | MR | MR

[19] W. Feller, An Introduction to Probability Theory and Its Applications, v. 2, 2nd ed., J. Wiley Sons, New York, 1971 | MR | MR | Zbl

[20] Fukushima M., Dirichlet forms and Markov processes, North-Holland Math. Libr., 23, North-Holland, Amsterdam, 1980 | MR | Zbl

[21] Grabner P.J., Woess W., “Functional iterations and periodic oscillations for simple random walk on the Sierpiński graph”, Stochastic Processes Appl., 69:1 (1997), 127–138 | DOI | MR | Zbl

[22] Grigor'yan A., “Heat kernels on weighted manifolds and applications”, The ubiquitous heat kernel: AMS spec. sess. (Boulder, 2003), Contemp. Math., 398, Amer. Math. Soc., Providence, RI, 2006, 93–191 | DOI | MR | Zbl

[23] Hewitt E., Ross K.A., Abstract harmonic analysis, v. 1, Grundl. Math. Wiss., 115, Structure of topological groups. Integration theory. Group representations, Springer, Berlin, 1963 | MR | Zbl

[24] Kato T., Perturbation theory for linear operators, Grundl. Math. Wiss., 132, Springer, Berlin, 1966 ; Koblits N., $p$-Adicheskie chisla, $p$-adicheskii analiz i dzeta-funktsii, Mir, M., 1982 | MR | Zbl | MR

[25] N. Koblitz, $p$-Adic Numbers, $p$-Adic Analysis, and Zeta-Functions, Grad. Texts Math., 58, Springer, New York, 1977 | DOI | MR | MR | Zbl

[26] Kochubei A.N., Pseudo-differential equations and stochastics over non-Archimedean fields, Pure Appl. Math., 244, M. Dekker, New York, 2001 | MR | Zbl

[27] S. V. Kozyrev, “Wavelets and spectral analysis of ultrametric pseudodifferential operators”, Sb. Math., 198:1 (2007), 97–116 | DOI | DOI | MR | MR | Zbl

[28] Kritchevski E., “Hierarchical Anderson Model”, Probability and mathematical physics, CRM Proc. Lect. Notes, 42, Amer. Math. Soc., Providence, RI, 2007, 309–322 | DOI | MR | Zbl

[29] Kritchevski E., “Spectral localization in the hierarchical Anderson model”, Proc. Amer. Math. Soc., 135:5 (2007), 1431–1440 | DOI | MR | Zbl

[30] Kritchevski E., “Poisson statistics of eigenvalues in the hierarchical Anderson model”, Ann. Henri Poincaré, 9:4 (2008), 685–709 | DOI | MR | Zbl

[31] Molchanov S., “Lectures on random media”, Lectures on probability theory: Summer sch. (Saint-Flour, 1992), Lect. Notes Math., 1581, Springer, Berlin, 1994, 242–411 | DOI | MR | Zbl

[32] Molchanov S., “Hierarchical random matrices and operators. Application to Anderson model”, Multidimensional statistical analysis and theory of random matrices: Proc. 6th Eugene Lukacs Symp. (Bowling Green, USA, 1996), VSP, Utrecht, 1996, 179–194 | DOI | MR | Zbl

[33] Molchanov S., Vainberg B., “On general Cwikel–Lieb–Rozenblum and Lieb–Thirring inequalities”, Around the research of Vladimir Maz'ya. III: Analysis and applications, Int. Math. Ser., 13, ed. by A. Laptev, Springer, Dordrecht, 2010, 201–246 | DOI | MR | Zbl

[34] Molchanov S., Vainberg B., “On the negative spectrum of the hierarchical Schrödinger operator”, J. Funct. Anal., 263:9 (2012), 2676–2688 | DOI | MR | Zbl

[35] Rodríguez-Vega J.J., Zúñiga-Galindo W.A., “Taibleson operators, $p$-adic parabolic equations and ultrametric diffusion”, Pac. J. Math., 237:2 (2008), 327–347 | DOI | MR | Zbl

[36] Taibleson M.H., Fourier analysis on local fields, Math. Notes, 15, Princeton Univ. Press, Princeton, NJ, 1975 | MR | Zbl

[37] V. S. Vladimirov, “Generalized functions over the field of $p$-adic numbers”, Russ. Math. Surv., 43:5 (1988), 19–64 | DOI | MR | Zbl

[38] Vladimirov V.S., Volovich I.V., “$p$-Adic Schrödinger-type equation”, Lett. Math. Phys., 18:1 (1989), 43–53 | DOI | MR | Zbl

[39] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, $p$-Adic Analysis and Mathematical Physics, Ser. Sov. East Europ. Math., 1, World Scientific, Singapore, 1994 | MR | MR | Zbl

[40] Zúñiga-Galindo W.A., “Parabolic equations and Markov processes over $p$-adic fields”, Potential Anal., 28:2 (2008), 185–200 | DOI | MR | Zbl