## Difference equations in the complex plane: quasiclassical asymptotics and Berry phase.(English)Zbl 1484.39018

Summary: We consider the equation $$\Psi (z+h) = M(z) \Psi (z)$$, where $$z \in \mathbb{C}$$, $$h>0$$ is a parameter, and $$M: \mathbb{C} \mapsto \mathrm{SL}(2,\mathbb{C}$$ is a given analytic function. We get asymptotics of its analytic solutions as $$h \rightarrow 0$$. The asymptotic formulas contain an analog of the geometric (Berry) phase well known in the quasiclassical analysis of differential equations.

### MSC:

 39A45 Difference equations in the complex domain

### Keywords:

complex plane; quasiclassical asymptotics; geometric phase
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### References:

 [1] Fedoryuk, MV., Asymptotic analysis. Linear ordinary differential equations (2009), Berlin, Heidelberg GmbH: Springer-Verlag, Berlin, Heidelberg GmbH [2] Wasow, W., Asymptotic expansions for ordinary differential equations (1987), New York: Dover Publications, New York · Zbl 0169.10903 [3] Geronimo, JS; Bruno, O.; Assche, WV., WKB and turning point theory for second-order difference equations, Oper Theory Adv Appl, 69, 269-301 (1992) [4] Helffer, B.; Sjöstrand, J., Analyse semi-classique pour l’équation de Harper (avec application à l’équationde Schrödinger avec champ magnétique), Mémoires de la SMF (nouvelle série), 34, 1-113 (1988) · Zbl 0714.34130 [5] Maslov, VP; Fedoriuk, MV., Semi-classical approximation in quantum mechanics (1981), Amsterdam: Reidel, Amsterdam [6] Dobrokhotov, SY; Tsvetkova, AV., Lagrangian manifolds related to asymptotics of Hermite polynomials, Math Notes, 104, 810-822 (2018) · Zbl 1409.42019 [7] Buslaev, V.; Fedotov, A., The complex WKB method for Harper equation, St Petersburg Math J, 6, 495-517 (1995) [8] Fedotov, AA; Shchetka, EV., The complex WKB method for difference equations in bounded domains, J Math Sci, 224, 157-169 (2017) · Zbl 1476.39025 [9] Fedotov, A.; Shchetka, E., Complex WKB method for a difference Schrödinger equation with the potential being a trigonometric polynomial, St Petersburg Math J, 29, 363-381 (2018) · Zbl 1385.39001 [10] Fedotov, A.; Klopp, F., The complex WKB method for difference equations and Airy functions, SIAM J Math An, 51, 6, 4413-4447 (2019) · Zbl 1425.39011 [11] Fedotov, A.; Klopp, F., WKB asymptotics of meromorphic solutions to difference equations, Appl Anal (2019) · Zbl 1476.39023 [12] Berry, M., Quantal phase factors accompanying adiabatic changes, Proc R Soc Lond A, 392, 45-57 (1984) · Zbl 1113.81306 [13] Simon, B., Holonomy, the quantum adiabatic theorem and Berry’s phase, Phys Rev Lett, 51, 2167-2170 (1983) [14] Wilkinson, M., An exact renormalisation group for Bloch electrons in a magnetic field, J Phys A Math Gen, 20, 4337-4354 (1987) [15] Guillement, JP; Helffer, B.; Treton, P., Walk inside Hofstadter’s butterfly, J Phys France, 50, 2019-2058 (1989) [16] Avila, A.; Jitomirskaya, S., The ten martini problem, Ann Math, 170, 303-342 (2009) · Zbl 1166.47031 [17] Fedotov, AA., Monodromization method in the theory of almost-periodic equations, St Petersburg Math J, 25, 303-325 (2014) · Zbl 1326.39011 [18] Buslaev, V, Fedotov, A.The monodromization and Harper equation. Séminaire sur les Équations aux Dérivées Partielles 1993-1994, Exp. no EXXI, École Polytech Palaiseau; 1994. 23 pp. · Zbl 0880.34082 [19] Lyalinov, MA; Zhu, NY., A solution procedure for second-order difference equations and its application to electromagnetic-wave diffraction in a wedge-shaped region, Proc R Soc Lond A, 459, 3159-3180 (2003) · Zbl 1092.78008 [20] Fedotov, A, Shchetka, E.Berry phase for difference equations. Days on Diffraction. Institute of Electrical and Electronics Engineers Inc.; 2017. p. 113-115. [21] Springer, G., Introduction to Riemann surfaces (1957), New York: Addison-Wesley, New York · Zbl 0078.06602 [22] Fedotov, A.; Klopp, F., Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schrödinger operators, Annales Scientifiques de l’Ecole Normale Superieure, 38, 889-950 (2005) · Zbl 1112.47038
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