On infinite system of resonance and eigenvalues with exponential asymptotics generated by distant perturbations

D. I. Borisov   ; M. N. Konyrkulzhaeva

Ufa mathematical journal, Tome 12 (2020) no. 4, pp. 3-18

Résumé

We consider an one-dimensional Schrödinger operator with four distant potentials separated by large distance. All distances are proportional to a sam large parameter. The initial potentials are of kink shapes, which are glued mutually so that the final potential vanishes at infinity and between the second and third initial potentials and it is equal to one between the first and the second potentials as well as between the third and fourth potentials. The potentials are not supposed to be real and can be complex-valued. We show that under certain, rather natural and easily realizable conditions on the four initial potentials, the considered operator with distant potentials possesses infinitely many resonances and/or eigenvalues of form $\lambda=k_n^2$, $n\in\mathbb{Z}$, which accumulate along a given segment in the essential spectrum. The distance between neighbouring numbers $k_n$ is of order the reciprocal of the distance between the potentials, while the imaginary parts of these quantities are exponentially small. For the numbers $k_n$ we obtain the representations via the limits of some explicitly calculated sequences and the sum of infinite series. We also prove explicit effective estimates for the convergence rates of the sequences and for the remainders of the series.

Détail

Keywords: resonance, exponential asymptotics, non-self-adjoint operator.
Mots-clés : distant perturbations

Bibliographie

[1] D. Borisov, A. Golovina, “On emergence of resonances from multiple eigenvalues of the Schrödinger operator in a cylinder with distant perturbations”, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 163, 2019, 3–14 (in Russian)

[2] D.I. Borisov, A.M. Golovina, “On the resolvents of periodic operators with distant perturbaions”, Ufa Math. J., 4:2 (2012), 65–73 | MR

[3] A.M. Golovina, “Spectrum of periodic elliptic operators with distant perturbations in space”, St. Petersburg Math. J., 25:5 (2014), 735–754 | DOI | MR | Zbl

[4] A.I. Markushevich, The theory of analytic functions, v. 2, Nauka, M., 1967 (in Russian) | MR

[5] T. Aktosun, M. Klaus, Cornelis van der Mee, “On the number of bound states for the one-dimensional Schrödinger equation”, J. Math. Phys, 39:9 (1998), 4249–4259 | DOI | MR

[6] F. Barra, P. Gaspard, “Scattering in periodic systems: from resonances to band structure”, J. Phys. A: Math. Gen, 32:18 (1999), 3357–3375 | DOI | MR | Zbl

[7] D. Borisov, P. Exner, “Exponential splitting of bound states in a waveguide with a pair of distant windows”, J. Phys. A: Math. Gen., 37:10 (2004), 3411–3428 | DOI | MR | Zbl

[8] D. Borisov, P. Exner, “Distant perturbation asymptotics in window-coupled waveguides. I. The non-threshold case”, J. Math. Phys., 47:11 (2006), 113502 | DOI | MR | Zbl

[9] D. Borisov, P. Exner, A. Golovina, “Tunneling resonances in systems without a classical trapping”, J. Math. Phys., 54:1 (2013), 012102 | DOI | MR | Zbl

[10] D.I. Borisov, “Distant perturbations of the Laplacian in a multi-dimensional space”, Ann. H. Poincaré, 8:7 (2007), 1371–1399 | DOI | MR | Zbl

[11] D. Borisov, “Asymptotic behaviour of the spectrum of a waveguide with distant perturbation”, Math. Phys. Anal. Geom., 10:2 (2007), 155–196 | DOI | MR | Zbl

[12] D.I. Borisov, D.A. Zezyulin, “Spacing gain and absorption in a simple $\mathcal{PT}$-symmetric model: spectral singularities and ladders of eigenvalues and resonances”, J. Phys. A. Math. Theor., 52:44 (2019), 445202 | DOI | MR

[13] D.I. Borisov, D.A. Zezyulin, “Sequences of closely spaced resonances and eigenvalues for bipartite complex potentials”, Appl. Math. Lett., 100 (2020), 106049 | DOI | MR | Zbl

[14] D.I. Borisov, A.M. Golovina, “On finitely many resonances emerging under distant perturbations in multi-dimensional cylinders”, J. Math. Anal. Appl., 2021 (to appear) | MR

[15] E.B. Davies, “The twisting trick for double well Hamiltonians”, Comm. Math. Phys., 85:3 (1982), 471–479 | DOI | MR | Zbl

[16] V. Graffi, E.M. Harrell II, H. J. Silverstone, “The $\frac{1}{R}$ expansion for $H_2^+$: analyticity, summability and asymptotics”, Ann. Phys., 165:2 (1985), 441–483 | DOI | MR | Zbl

[17] A.M. Golovina, “On the resolvent of elliptic operators with distant perturbations in the space”, Russ. J. Math. Phys., 19:2 (2012), 182–192 | DOI | MR | Zbl

[18] A.M. Golovina, “Discrete eigenvalues of periodic operators with distant perturbations”, J. Math. Sci., 189:3 (2013), 342–364 | DOI | MR | Zbl

[19] E.M. Harrell, “Double wells”, Comm. Math. Phys., 75:3 (1980), 239–261 | DOI | MR | Zbl

[20] E.M. Harrell, M. Klaus, “On the double-well problem for Dirac operators”, Ann. de l'Inst. H. Poincaré, 38:2 (1983), 153–166 | MR | Zbl

[21] R. Høegh-Krohn, M. Mebkhout, “The $\frac{1}{r}$ expansion for the critical multiple well problem”, Comm. Math. Phys., 91:1 (1983), 65–73 | DOI | MR

[22] M. Klaus, “Some remarks on double-wells in one and three dimensions”, Ann. de l'Inst. H. Poincaré, 34:4 (1981), 405–417 | MR | Zbl

[23] M. Klaus, B. Simon, “Binding of Schrödinger particles through conspiracy of potential wells”, Ann. de l'Inst. H. Poincaré, sect. A, 30:2 (1979), 83–87 | MR

[24] F. Klopp, “Resonances for large one-dimensional “ergodic” systems”, Anal. PDE, 9:2 (2016), 259–352 | DOI | MR | Zbl

[25] M. Zworski, S. Dyatlov, Mathematical theory of scattering resonances, Amer. Math. Soc., Providence, RI, 2019 | MR | Zbl