Skip to main content
SpringerLink
Log in

Hierarchical Schrödinger Operators with Singular Potentials

  • Research Articles
  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

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 \(g_L(x,y)\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.

Notes

  1. A Markov process \(\{X(t),P_x\}\) with state space \( {\mathcal X} \) is called a pure jump process if, starting from any point \(x\in {\mathcal X} \), it has all sample paths constant except for isolated jumps, and is right-continuous. The basic data which define the process are (i) a function \(0<\lambda(x)<\infty\) and (ii) a Markov kernel \( {\mathcal U} (x,dy)\) satisfying the equality \( {\mathcal U} (x,\{x\})=0\). Its Laplacian (i.e., minus Markov generator) has the form

    $$Lf(x)=\intop_{ {\mathcal X} }\bigl(f(x)-f(y)\bigr) \lambda(x) {\mathcal U} (x,dy).$$

    Intuitively a particle starting from \(x\) remains there for an exponentially distributed time with parameter \(\lambda(x)\), after which it “jumps” to a new position \(x'\) according to the distribution \( {\mathcal U} (x,\cdot \kern1pt )\), and so on.

  2. In the case where \(\Phi(\tau)\) is a Bernstein function, the relation \(L_{\Phi}=\Phi(L_{\text{Id}})\) has been studied in the well-known Bochner subordination theory (see [19]).

  3. This relation must be compared with the Green function estimates for Schrödinger operators on Riemannian manifolds (see [22]).

  4. This condition of transience was first introduced by A. Beurling and J. Deny in the unreplaceable paper [12]. It is slightly more restrictive than the definition of transience given by M. Fukushima [20, Sect. 1.5].

  5. The following counterpart of Theorem 4.5 is in order: Let \(X^{ {\mathcal H} }\) and \(X^{ {\mathcal L} }\) be the Hunt processes associated with the Dirichlet forms \(Q_{ {\mathcal H} }\) and \(Q_{ {\mathcal L} }\), respectively. According to [20, Theorem 5.5.2 and Example 5.5.1], their paths are related by the random time change \(X_t^{ {\mathcal H} } =X_{\tau_t}^{ {\mathcal L} }\), where \(\tau_t=\inf\{s>0 \colon\, A_t>t\}\) and \(A_t=\intop_0^th(X_s^{ {\mathcal H} })\,ds\) is a positive continuous additive functional. It follows, in particular, that Dynkin’s characteristic operators for these processes are related by the equation \((- {\mathcal H} u)(x)=(- {\mathcal L} u)(x)/h(x)\). We are going to use this fact in the next subsections to solve the equation \(Hu=v\).

References

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

    Article  ADS  MathSciNet  Google Scholar 

  2. S. Albeverio and W. Karwowski, “A random walk on \(p\)-adics—the generator and its spectrum,” Stochastic Processes Appl. 53 (1), 1–22 (1994).

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. A. Bendikov, W. Cygan, and W. Woess, “Oscillating heat kernels on ultrametric spaces,” J. Spectr. Theory 9 (1), 195–226 (2019).

    Article  MathSciNet  Google Scholar 

  5. A. Bendikov, A. Grigor’yan, and S. Molchanov, “On the spectrum of the hierarchical Schrödinger type operators,” arXiv: 2006.02263v1 [math.SP].

  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), 859–893 (2015) [transl. from Izv. Ross. Akad. Nauk, Ser. Mat. 79 (5), 3–38 (2015)].

    Article  MathSciNet  Google Scholar 

  7. A. Bendikov, A. Grigor’yan, and Ch. Pittet, “On a class of Markov semigroups on discrete ultra-metric spaces,” Potential Anal. 37 (2), 125–169 (2012).

    Article  MathSciNet  Google Scholar 

  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), 589–680 (2014) [transl. from Usp. Mat. Nauk 69 (4), 3–102 (2014)].

    Article  MathSciNet  Google Scholar 

  9. A. Bendikov and P. Krupski, “On the spectrum of the hierarchical Laplacian,” Potential Anal. 41 (4), 1247–1266 (2014).

    Article  MathSciNet  Google Scholar 

  10. F. A. Berezin and M. A. Shubin, The Schrödinger Equation (Kluwer, Dordrecht, 1991) [transl. from Russian (Mosk. Gos. Univ., Moscow, 1983)].

    Book  Google Scholar 

  11. Ch. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups (Springer, Berlin, 1975), Ergeb. Math. Grenzgeb. 87.

    Book  Google Scholar 

  12. A. Beurling and J. Deny, “Dirichlet spaces,” Proc. Natl. Acad. Sci. USA 45, 208–215 (1959).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  14. E. B. Davies, Spectral Theory and Differential Operators (Cambridge Univ. Press, Cambridge, 1995), Cambridge Stud. Adv. Math. 42.

    Book  Google Scholar 

  15. M. Del Muto and A. Figà-Talamanca, “Diffusion on locally compact ultrametric spaces,” Expo. Math. 22 (3), 197–211 (2004).

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  18. F. J. Dyson, “An Ising ferromagnet with discontinuous long-range order,” Commun. Math. Phys. 21, 269–283 (1971).

    Article  ADS  MathSciNet  Google Scholar 

  19. W. Feller, An Introduction to Probability Theory and Its Applications, 2nd ed. (J. Wiley & Sons, New York, 1971), Vol. 2.

    Google Scholar 

  20. M. Fukushima, Dirichlet Forms and Markov Processes (North-Holland, Amsterdam, 1980), North-Holland Math. Libr. 23.

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. A. Grigor’yan, “Heat kernels on weighted manifolds and applications,” in The Ubiquitous Heat Kernel: AMS Spec. Sess., Boulder, 2003 (Am. Math. Soc., Providence, RI, 2006), Contemp. Math. 398, pp. 93–191.

    Chapter  Google Scholar 

  23. E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1: Structure of Topological Groups. Integration Theory. Group Representations (Springer, Berlin, 1963), Grundl. Math. Wiss. 115.

    Book  Google Scholar 

  24. T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1966), Grundl. Math. Wiss. 132.

    Book  Google Scholar 

  25. N. Koblitz, \(p\)-Adic Numbers, \(\,p\)-Adic Analysis, and Zeta-Functions (Springer, New York, 1977), Grad. Texts Math. 58.

    Book  Google Scholar 

  26. A. N. Kochubei, Pseudo-differential Equations and Stochastics over Non-Archimedean Fields (M. Dekker, New York, 2001), Pure Appl. Math. 244.

    Book  Google Scholar 

  27. S. V. Kozyrev, “Wavelets and spectral analysis of ultrametric pseudodifferential operators,” Sb. Math. 198 (1), 97–116 (2007) [transl. from Mat. Sb. 198 (1), 103–126 (2007)].

    Article  MathSciNet  Google Scholar 

  28. E. Kritchevski, “Hierarchical Anderson Model,” in Probability and Mathematical Physics (Am. Math. Soc., Providence, RI, 2007), CRM Proc. Lect. Notes 42, pp. 309–322.

    Chapter  Google Scholar 

  29. E. Kritchevski, “Spectral localization in the hierarchical Anderson model,” Proc. Am. Math. Soc. 135 (5), 1431–1440 (2007).

    Article  MathSciNet  Google Scholar 

  30. E. Kritchevski, “Poisson statistics of eigenvalues in the hierarchical Anderson model,” Ann. Henri Poincaré 9 (4), 685–709 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  31. S. Molchanov, “Lectures on random media,” in Lectures on Probability Theory: Summer Sch., Saint-Flour, 1992 (Springer, Berlin, 1994), Lect. Notes Math. 1581, pp. 242–411.

    Chapter  Google Scholar 

  32. S. Molchanov, “Hierarchical random matrices and operators. Application to Anderson model,” in Multidimensional Statistical Analysis and Theory of Random Matrices: Proc. 6th Eugene Lukacs Symp., Bowling Green, USA, 1996 (VSP, Utrecht, 1996), pp. 179–194.

    Chapter  Google Scholar 

  33. S. Molchanov and B. Vainberg, “On general Cwikel–Lieb–Rozenblum and Lieb–Thirring inequalities,” in Around the Research of Vladimir Maz’ya. III: Analysis and Applications, Ed. by A. Laptev (Springer, Dordrecht, 2010), Int. Math. Ser. 13, pp. 201–246.

    Chapter  Google Scholar 

  34. S. Molchanov and B. Vainberg, “On the negative spectrum of the hierarchical Schrödinger operator,” J. Funct. Anal. 263 (9), 2676–2688 (2012).

    Article  MathSciNet  Google Scholar 

  35. J. J. Rodríguez-Vega and W. A. Zúñiga-Galindo, “Taibleson operators, \(p\)-adic parabolic equations and ultrametric diffusion,” Pac. J. Math. 237 (2), 327–347 (2008).

    Article  MathSciNet  Google Scholar 

  36. M. H. Taibleson, Fourier Analysis on Local Fields (Princeton Univ. Press, Princeton, NJ, 1975), Math. Notes 15.

    Google Scholar 

  37. V. S. Vladimirov, “Generalized functions over the field of \(p\)-adic numbers,” Russ. Math. Surv. 43 (5), 19–64 (1988) [transl. from Usp. Mat. Nauk 43 (5), 17–53 (1988)].

    Article  MathSciNet  Google Scholar 

  38. V. S. Vladimirov and I. V. Volovich, “\(p\)-Adic Schrödinger-type equation,” Lett. Math. Phys. 18 (1), 43–53 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  39. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, \(p\)-Adic Analysis and Mathematical Physics (World Scientific, Singapore, 1994), Ser. Sov. East Europ. Math. 1.

    Book  Google Scholar 

  40. W. A. Zúñiga-Galindo, “Parabolic equations and Markov processes over \(p\)-adic fields,” Potential Anal. 28 (2), 185–200 (2008).

    Article  MathSciNet  Google Scholar 

Download references

Funding

A.B. and A.G. were funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant no. SFB 1283/2 2021 – 317210226. The work of S.M. was supported by the Russian Science Foundation under grant no. 17-11-01098, https://rscf.ru/en/project/17-11-01098/.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Wroclaw University, 50-384, Wroclaw, Poland

    Alexander Bendikov

  2. Department of Mathematics, University of Bielefeld, 33501, Bielefeld, Germany

    Alexander Grigor’yan

  3. Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC, 28223, USA

    Stanislav Molchanov

  4. National Research University Higher School of Economics, Moscow, 109028, Russia

    Stanislav Molchanov

Authors
  1. Alexander Bendikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Grigor’yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stanislav Molchanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Alexander Bendikov, Alexander Grigor’yan or Stanislav Molchanov.

Ethics declarations

The authors of this work declare that they have no conflicts of interest.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Vol. 323, pp. 17–52 https://doi.org/10.4213/tm4356.

Publisher’s note. Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bendikov, A., Grigor’yan, A. & Molchanov, S. Hierarchical Schrödinger Operators with Singular Potentials. Proc. Steklov Inst. Math. 323, 12–46 (2023). https://doi.org/10.1134/S0081543823050024

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543823050024

Keywords

Search

Navigation