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On the number of bound states of point interactions on hyperbolic manifolds

Fatih Erman

Department of Mathematics, İzmir Institute of Technology, Urla 35430, İzmir, Turkey

https://doi.org/10.1142/S0219887817500116Cited by:4 (Source: Crossref)

Abstract

We study the bound state problem for $N$ attractive point Dirac $δ$ -interactions in two- and three-dimensional Riemannian manifolds. We give a sufficient condition for the Hamiltonian to have $N$ bound states and give an explicit criterion for it in hyperbolic manifolds $ℍ^{2}$ and $ℍ^{3}$ . Furthermore, we study the same spectral problem for a relativistic extension of the model on $ℝ^{2}$ and $ℍ^{2}$ .

Keywords:

AMSC: 47A10, 34L40

References

1. S. Albeverio, F. Gesztesy, R. Hegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, 2nd edn. (AMS Chelsea, RI, 2004). Crossref, Google Scholar
2. S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators Solvable Schrödinger-Type Operators (Cambridge University Press, Cambridge, 2000). Crossref, Google Scholar
3. Yu. N. Demkov and V. N. Ostrovskii, Zero-Range Potentials and Their Applications in Atomic Physics (Plenum Press, New York, 1998). Google Scholar
4. F. A. Berezin and L. D. Faddeev, A remark on Schrödinger’s equation with a singular potential, Soviet Math. Dokl. 2 (1961) 372–375. Google Scholar
5. Z. Rudnick and H. Ueberschär, Statistics of wave functions for a point scatterer on the torus, Comm. Math. Phys. 316 (2012) 763–782. Crossref, Web of Science, Google Scholar
6. P. Exner, R. Gawlista, P. Seba and M. Tater, Point interactions in a strip, Ann. Phys. 252(1) (1996) 133–179. Crossref, Web of Science, Google Scholar
7. J. Brüning, V. Geyler and K. Pankrashkin, Spectra of self-adjoint extensions and applications to solvable Schrödinger operators, Rev. Math. Phys. 20 (2008) 1–70. Link, Web of Science, Google Scholar
8. B. Altunkaynak, F. Erman and O. T. Turgut, Finitely many Dirac-delta interactions on Riemannian manifolds, J. Math. Phys. 47 (2006) 082110. Crossref, Web of Science, Google Scholar
9. F. Erman and O. T. Turgut, Point interactions in two- and three-dimensional Riemannian manifolds, J. Phys. A: Math. Theor. 43 (2010) 335204. Crossref, Web of Science, Google Scholar
10. C. Dogan, F. Erman and O. T. Turgut, Existence of Hamiltonians for some singular interactions on manifolds, J. Math. Phys. 53 (2012) 043511. Crossref, Web of Science, Google Scholar
11. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4 (Academic Press, NewYork, 1978). Google Scholar
12. S. Albeverio and L. Nizhnik, Schrödinger operators with a number of negative eigenvalues equal to the number of point interactions, Methods Funct. Anal. Topology 9(4) (2003) 273–286. Google Scholar
13. S. Albeverio and L. Nizhnik, On the number of negative eigenvalues of one-dimensional Schrödinger operator with point interactions, Lett. Math. Phys. 65 (2003) 27–35. Crossref, Web of Science, Google Scholar
14. O. Ogurisu, On the number of negative eigenvalues of a Schrödinger operator with point interactions, Lett. Math. Phys. 85 (2008) 129–133. Crossref, Web of Science, Google Scholar
15. O. Ogurisu, On the number of negative eigenvalues of a Schrödinger operator with $δ$ interactions, Methods Funct. Anal. Topology 16 (2010) 42–50. Google Scholar
16. N. I. Goloshchapova and L. L. Oridoroga, The one-dimensional Schrödinger operator with point $δ$ - and $δ^{'}$ -interactions, Math. Notes 84(1) (2008) 125–129. Crossref, Web of Science, Google Scholar
17. N. I. Goloshchapova and L. L. Oridoroga, On the negative spectrum of one-dimensional Schrödinger operators with point interactions, Integral Equations Operator Theory 67(1) (2010) 1–14. Crossref, Web of Science, Google Scholar
18. O. Ogurisu, On the number of negative eigenvalues of a multi-dimensional Schrödinger operator with point interactions, Methods Funct. Anal. Topology 16(4) (2010) 383–392. Google Scholar
19. R. Jackiw, Delta-Function Potentials in Two- and Three-Dimensional Quantum Mechanics, M. A. B. Beg Memorial Volume (World Scientific, Singapore, 1991). Google Scholar
20. M. H. Al-Hashimi, A. M. Shalaby and U. J. Wiese, Asymptotic freedom, dimensional transmutation, and an infrared conformal fixed point for the $δ$ -function potential in one-dimensional relativistic quantum mechanics, Phys. Rev. D 89(12) (2014) 125023. Crossref, Web of Science, Google Scholar
21. B. De Witt, Dynamical theory in curved spaces I. A review of the classical and quantum action principles, Rev. Modern Phys. 29 (1957) 377–397. Crossref, Google Scholar
22. P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem (Publish or Perish, Wilmington DE, 1984). Google Scholar
23. J. Wang, Global heat kernel estimates, Pacific J. Math. 178(2) (1997) 377–398. Crossref, Web of Science, Google Scholar
24. A. Grigoryan, Heat Kernel and Analysis on Manifolds, AMS/IP Studies in Advanced Mathematics, ed. S.-T. Yau, Vol. 47 (American Mathematical Society, Rhode Island, 2009). Google Scholar
25. A. Grigoryan, Spectral Theory and Geometry, London Mathematical Society Lecture Notes, eds. E. B. Davies and Y. Safarov, Vol. 273 (Cambridge University Press, Cambridge, 1999), pp. 140–225. Crossref, Google Scholar
26. J. Cheeger and S.-T. Yau, A lower bound for the heat kernel, Comm. Pure Appl. Math. 34 (1981) 465–480. Crossref, Web of Science, Google Scholar
27. F. W. J. Olver, Asymptotics and Special Functions (Academic Press, New York, 1974). Google Scholar
28. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (Oxford University Press, London, 1935). Google Scholar
29. N. N. Lebedev, Special Functions and Their Applications (Prentice Hall, NJ Englewood Cliffs, 1965). Crossref, Google Scholar
30. T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics (Springer-Verlag, corrected printing of the second edition, Berlin, 1995). Crossref, Google Scholar
31. H. Donnelly, Negative curvature and embedded eigenvalues, Math. Z. 203 (1990) 301–308. Crossref, Web of Science, Google Scholar
32. N. Charalambous and Z. Lu, On the spectrum of the Laplacian, Math. Ann. 359 (2014) 211–238. Crossref, Web of Science, Google Scholar
33. H. Donnelly, On the essential spectrum of a complete Riemannian manifold, Topology 20 (1981) 1–14. Crossref, Web of Science, Google Scholar
34. R. P. Feynman, Forces in molecules, Phys. Rev. 56 (1939) 340–343. Crossref, Google Scholar
35. H. G. A. Hellmann, Importance of kinetic energy of electrons for interatomic forces (in German), Z. Phys. 85 (1933) 180–190. Crossref, Google Scholar
36. C. Cohen-Tannoudji, B. Diu and F. Laloe, Quantum Mechanics, Vol. 1 (Wiley-Interscience, New York, 2006). Google Scholar
37. E. B. Davies and N. Mandouvalos, Heat kernel bounds on hyperbolic space and Kleinian groups, Proc. London Math. Soc. 3(57) (1988) 182–208. Crossref, Web of Science, Google Scholar
38. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th edn. (Academic Press, 2007). Google Scholar
39. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1992). Google Scholar
40. C. Dogan and O. T. Turgut, Renormalized interaction of relativistic bosons with delta function potentials, J. Math. Phys. 51(8) (2000) 082305. Crossref, Web of Science, Google Scholar
41. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer-Verlag, New York, 1983). Crossref, Google Scholar
42. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1 (Academic Press, Revised and Enlarged Edition, NewYork, 1980). Google Scholar
43. J. Dimock and S. G. Rajeev, Multi-particle Schrödinger operators with point interactions in the plane, J. Phys. A: Math. Gen. 37 (2004) 9157–9173. Crossref, Google Scholar
44. I. Chavel, Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, Vol. 115 (Academic Press, Orlando, 1984). Google Scholar