Matrix-valued Schrödinger operators over finite adeles. (English) Zbl 1534.81048
Summary: Let \(K\) be an algebraic number field. With \(K\) we associate the ring of finite adeles \(\mathbb{A}_K\). In this paper we give a path integral formula for the propagator of a quantum mechanical system over the abelian group \(\mathbb{A}_K^n\). Specifically, we consider matrix-valued Hamiltonian operators \(H_{\mathbb{A}_K^n}= \Delta_{\mathbb{A}_K^n}\otimes\mathrm{Id}+V\), where \(\Delta_{\mathbb{A}_K^n}\) is the Vladimirov operator and \(V\) is a non-negative definite potential. The free part of the Hamiltonian gives rise to a measure on the Skorokhod space of paths which allows us to prove the Feynman-Kac formula for the Schrödinger semigroup generated by \(-H_{\mathbb{A}_K^n}\) This formula is given in terms of the ordered time exponentials.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |
| 11R56 | Adèle rings and groups |
| 43A25 | Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups |
| 60J65 | Brownian motion |
| 70H05 | Hamilton’s equations |
| 47D08 | Schrödinger and Feynman-Kac semigroups |
| 37M10 | Time series analysis of dynamical systems |
Keywords:
adeles; Vladimirov operator; Hamiltonian; Schrödinger equation; Feynman-Kac formula; path integrals; semigroup of operators; Brownian motion; ordered time exponentialsReferences:
| [1] | Aguilar-Arteaga, V. A., Cruz-López, M. and Estala-Arias, S., Non-Archimedean analysis and a wave-type pseudodifferential equation on finite adèles, J. Pseudo-Differ. Oper. Appl.11 (2020) 1139-1181. · Zbl 1466.35380 |
| [2] | Bakken, E. M. and Digernes, T., Finite approximations of physical models over local fields, p-Adic Numbers Ultrametric Anal. Appl.7 (2015) 245-258. · Zbl 1345.47067 |
| [3] | Dragovich, B., Khrennikov, A. Yu., Kozyrev, S. V. and Volovich, I. V., On \(p\)-adic mathematical physics, p-Adic Numbers Ultrametric Anal. Appl.1 (2009) 1-17. · Zbl 1187.81004 |
| [4] | Digernes, T., Varadarajan, V. S. and Weisbart, D. E., Schrödinger operators on local fields: Self-adjointness and path integral representations for propagators, Infin. Dimens. Anal. Quantum Probab. Relat. Top.11 (2008) 495-512. · Zbl 1162.47038 |
| [5] | Digernes, T. and Weisbart, D., Matrix-valued Schröinger operators over local fields, p-Adic Numbers Ultrametric Anal. Appl.1 (2009) 136-144. · Zbl 1187.47037 |
| [6] | Halmos, P. R., Measure Theory, , Vol. 18 (Springer-Verlag, 2014). |
| [7] | Karwowski, W. and Mendes, R. V., Hierarchical structures and asymmetric stochastic processes on \(p\)-adics and adeles, J. Math. Phys.35 (1994) 4637-4650. · Zbl 0814.60103 |
| [8] | Kochubei, A. N., Pseudo-differential Equations and Stochastics over non-Archimedean Fields, , Vol. 244 (Marcel Dekker, 2001). · Zbl 0984.11063 |
| [9] | Manin, Yu. I., Reflections on arithmetical physics, in Conformal Invariance and String Theory (Poiana Braşov, 1987), (Academic Press, 1989), pp. 293-303. |
| [10] | Meurice, Y., A path integral formulation of \(p\)-adic quantum mechanics, Phys. Lett. B245 (1990) 99-104. |
| [11] | Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers, , 3rd edn. (Springer-Verlag, 2004). · Zbl 1159.11039 |
| [12] | Neukirch, J., Algebraic Number Theory (, Vol. 322 (Springer-Verlag, 1999). Translated from the 1992 German original and with a note by Norbert Schappacher. · Zbl 0956.11021 |
| [13] | Parisi, G., On \(p\)-adic functional integrals, Mod. Phys. Lett. A3 (1988) 639-643. |
| [14] | Ramakrishnan, D. and Valenza, R. J., Fourier Analysis on Number Fields, , Vol. 186 (Springer-Verlag, 1999). · Zbl 0916.11058 |
| [15] | Rammal, R., Toulouse, G. and Virasoro, M. A., Ultrametricity for physicistsRev. Mod. Phys.58 (1986) 765-788. |
| [16] | Schmidt, K., Dynamical Systems of Algebraic Origin, , Vol. 128 (Birkhäuser Verlag, 1995). · Zbl 0833.28001 |
| [17] | Urban, R., Markov processes on the adeles and Dedekind’s zeta function, Statist. Probab. Lett.82 (2012) 1583-1589. · Zbl 1248.60090 |
| [18] | Urban, R., On a diffusion on finite adeles and the Feynman-Kac integral, J. Math. Phys.63 (2022) 122101. · Zbl 1514.47069 |
| [19] | Varadarajan, V. S., Path integrals for a class of \(p\)-adic Schrödinger equations, Lett. Math. Phys.39 (1997) 97-106. · Zbl 0868.47047 |
| [20] | Varadarajan, V. S. and Weisbart, D., Convergence of quantum systems on grids, J. Math. Anal. Appl.336 (2007) 608-624. · Zbl 1119.81059 |
| [21] | V. S. Vladimirov, Generalized functions over the field of \(p\)-adic numbers, Usp. Mat. Nauk.43 (1988) 17-53, translation in Russ. Math. Surv.43 (1988) 19-64 (in Russian). · Zbl 0678.46053 |
| [22] | V. S. Vladimirov, On the spectrum of some pseudodifferential operators over the field of \(p\)-adic numbers, Algebra i Analiz2 (1990) 107-124; translation in Leningrad Math. J.2 (1991) 1261-1278 (in Russian). · Zbl 0744.47046 |
| [23] | Vladimirov, V. S. and Volovich, I. V., \(p\)-adic quantum mechanics, Commun. Math. Phys.123 (1989) 659-676. · Zbl 0688.22004 |
| [24] | Vladimirov, V. S. and Volovich, I. V., \(p\)-adic Schrödinger-type equation, Lett. Math. Phys.18 (1989) 43-53. · Zbl 0702.35219 |
| [25] | V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, it p-adic Analysis and Mathematical Physics (Nauka, 1994), translation in Series on Soviet and East European Mathematics, Vol. 1. (World Scientific Publishing, 1994) (in Russian). · Zbl 0812.46076 |
| [26] | Volovich, I. V., Number theory as the ultimate physical theory, \(p\)-Adic Numbers, Ultrametric Anal. Appl.2 (2010) 77-87. · Zbl 1258.81074 |
| [27] | Weil, A., Basic Number Theory (Springer-Verlag, 1974). · Zbl 0326.12001 |
| [28] | Zú”ñiga-Galindo, W. A., Pseudodifferential equations over non-Archimedean spaces, , 2174 (Springer, Cham, 2016). · Zbl 1367.35006 |
| [29] | Yasuda, K., Markov processes on the adeles and Chebyshev function, Statist. Probab. Lett.83 (2013) 238-244. · Zbl 1264.60059 |
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