Schrödinger operators with singular potentials and magnetic fields

V. N. Kolokoltsov

Sbornik. Mathematics, Tome 194 (2003) no. 6, pp. 897-917

Résumé

A formal Schrödinger operator of the form $$ H=\biggl(-i\frac\partial{\partial x}+A(x)\biggr)^2+V(x), $$ in ${\mathbb R}^d$ is considered, where $A$ is a bounded measurable vector-valued function and both $V(x)$ and $\operatorname{div}A$ are measures satisfying certain additional conditions. It is shown that one can give meaning to such an operator as a lower bounded self-adjoint operator in $L^2({\mathbb R}^d)$. The corresponding heat kernel is constructed and its small-time asymptotics are obtained. A rigorous Feynman path integral representation for the solutions of the heat and Schrödinger's equations with generator $H$ is given.

Bibliographie

[1] Kolokoltsov V., Semiclassical analysis for diffusion and stochastic processes, Lecture Notes in Math., 1724, Springer-Verlag, Berlin, 2000 | MR | Zbl

[2] Kolokoltsov V., “A new path integral representation for the solutions of the Schrödinger, heat and stochastic Schrödinger equations”, Math. Proc. Cambridge Philos. Soc., 132 (2002), 353–375 | DOI | MR | Zbl

[3] Arsenev A. A., Singulyarnye potentsialy i rezonansy, Izd-vo MGU, M., 1974 | MR

[4] Liskevich V., Manavi P., “Dominated semigroups with singular complex potentials”, J. Funct. Anal., 151:2 (1997), 281–305 | DOI | MR | Zbl

[5] Zhang Qi S., “Large time behavior of Schrödinger heat kernels and applications”, Comm. Math. Phys., 210 (2000), 371–398 | DOI | MR | Zbl

[6] Albeverio S., Gesztesy F., Hoegh-Krohn R., Holden H., Solvable models in quantum mechanics, Springer-Verlag, Berlin, 1988 | MR | Zbl

[7] Koshmanenko V., Singular quadratic forms in perturbation theory, Kluwer Acad. Publ., Dordrecht, 1999 | MR | Zbl

[8] Blanchard Ph., Ma Zh., “Semigroups of Schrödinger operators with potentials given by Radon measures”, Stochastic processes, physics and geometry, eds. S. Albeverio et al., World Scientific, Teaneck, NJ, 1990, 160–195 | MR

[9] Simon B., “Schrödinger semigroups”, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447–526 | DOI | MR | Zbl

[10] Brasche J. F., “Generalised Schrödinger operators, an inverse problem in spectral analysis and the Efimov effect”, Stochastic processes, physics and geometry, eds. S. Albeverio et al., World Scientific, Teaneck, NJ, 1990, 207–244 | MR

[11] Albeverio S., Fenstad J. E., Hoegh-Krohn R., Karwowski W., Lindstrom T., “Schrödinger operators with potentials supported by null sets”, Ideas and methods in quantum and statistical physics, V. 2. In memory of R. Hoegh-Krohn, eds. S. Albeverio et al., Cambridge Univ. Press, Cambridge, 1992, 63–95 | MR | Zbl

[12] Brasche J. F., Figari R., Teta A., “Singular Schrödinger operators as limits of point interaction Hamiltonians”, Potential Anal., 8:2 (1998), 163–178 | DOI | MR | Zbl

[13] Sturm K.-T., “Schrödinger semigroups on manifolds”, J. Funct. Anal., 118 (1993), 309–350 | DOI | MR | Zbl

[14] Brasche J. F., “Perturbation of Schrödinger Hamiltonians by measures – self-adjointness and lower semiboundedness”, J. Math. Phys., 26 (1985), 621–626 | DOI | MR | Zbl

[15] Savchuk A. M., Shkalikov A. A., “Operatory Shturma–Liuvillya s singulyarnymi potentsialami”, Matem. zametki, 66:6 (1999), 897–912 | MR | Zbl

[16] Dabrowski L., Stovicek P., “Aharonov–Bohm effect with $\delta$-type interaction”, J. Math. Phys., 39:1 (1998), 47–62 | DOI | MR | Zbl

[17] Broderix K., Hundertmark D., Leschke H., “Continuity properties of Schrödinger semigroups with magnetic fields”, Rev. Math. Phys., 12:2 (2000), 181–225 | DOI | MR | Zbl

[18] Loss M., Thaler B., “Optimal heat kernel estimates for Schrödinger operators with magnetic fields in two dimensions”, Comm. Math. Phys., 186:1 (1997), 95–107 | DOI | MR | Zbl

[19] Albeverio S., Boutet de Monvel-Bertier A., Brzezniak Zd., “Stationary phase method in infinite dimensions by finite approximations: Applications to the Schrödinger equation”, Potential Anal., 4:5 (1995), 469–502 | DOI | MR | Zbl

[20] Kolokoltsov V., “Complex measures on path space: an introduction to the Feynman integral applied to the Schrödinger equation”, Methodol. Comput. Appl. Probab., 1:3 (1999), 349–365 | DOI | MR | Zbl

[21] Smolyanov O. G., Shavgulidze E. T., Kontinualnye integraly, Izd-vo MGU, M., 1990 | MR | Zbl

[22] Ichinose W., “On convergence of the Feynman path integral formulated through broken line paths”, Rev. Math. Phys., 11 (1999), 1001–1025 | DOI | MR

[23] Loo K., “A rigorous real time Feynman path integral”, J. Math. Phys., 40:1 (1999), 64–70 | DOI | MR | Zbl

[24] Albeverio S., Kolokoltsov V. N., Smolyanov O. G., “Continuous quantum measurement: Local and global approaches”, Rev. Math. Phys., 9:8 (1997), 907–920 | DOI | MR | Zbl

[25] Truman A., Zhao H., “Stochastic Hamilton–Jacobi equation, stochastic heat equation and Schrödinger equations”, Stochastic analysis and application, ed. A. M. Etheridge, World Scientific Press, Singapore, 1996, 441–464 | MR | Zbl

[26] Zastawniak T., “Path integral for the Dirac equation – some recent developments in mathematical theory”, Stochastic analysis, path integration and dynamics, Pitman Research Notes in Mathematics Series, 200, eds. K. D. Elworthy, J.-C. Zambrini, 1989, 243–264 | MR

[27] Asai N., Kubo I., Kuo H.-H., “Feynman integrals associated with Albeverio–Hoegh-Krohn and Laplace transform potentials”, Stochastics in finite and infinite dimensions, Birkhäuser, Boston, 2000, 29–48 | MR | Zbl

[28] Gelfand I. M., Yaglom A. M., “Integrirovanie v funktsionalnykh prostranstvakh s prilozheniyami k kvantovoi fizike”, UMN, 11:1 (1956), 77–114 | MR | Zbl

[29] Reed M., Simon B., Methods of mathematical physics. II: Fourier analysis, self-adjointness, Academic Press, New York, 1975 | MR | Zbl

[30] Doss H., “Sur une resolution stochastique de l'equation de Schrödinger a coefficient analytiques”, Comm. Math. Phys., 73 (1980), 243–264 | DOI | MR

[31] Haba Z., “Stochastic interpretation of Feynman path integral”, J. Math. Phys., 35:2 (1994), 6344–6359 | DOI | MR | Zbl

[32] Haba Z., “Feynman integral in regularized nonrelativistic quantum electrodynamics”, J. Math. Phys., 39:4 (1998), 1766–1787 | DOI | MR | Zbl

[33] Maslov V. P., Chebotarev A. M., “Processus à sauts et leur application dans la mécanique quantique”, Feynman path integrals, Lecture Notes in Phys., 106, 1979, 58–72 | MR | Zbl

[34] Maslov V. P., Kompleksnye markovskie tsepi i kontinualnyi integral Feinmana, Nauka, M., 1976 | MR

[35] Gaveau B., Kac M., “A probabilistic formula for the quantum $N$-body problem and the nonlinear Schrödinger equation in operator algebra”, J. Funct. Anal., 66 (1986), 308–322 | DOI | MR | Zbl

[36] Serva M., “Probabilistic solutions of generalized birth and death equations and applications to field theory”, J. Phys. A, 20 (1987), 435–446 | DOI | MR

[37] Chebotarev A. M., Quezada R. B., “Stochastic approach to time-dependent quantum tunnelling”, Russian J. Math. Phys., 4:3 (1998), 275–286 | MR

[38] Gaveau B., “Representation formulas of the Cauchy problem for hyperbolic systems generalizing Dirac system”, J. Funct. Anal., 58 (1984), 310–319 | DOI | MR | Zbl

[39] Kitarev D. V., Yegikian R. S., “Feynman path integral for Dirac system with analytic potential”, J. Math. Phys., 34:7 (1993), 2821–2826 | DOI | MR

[40] Menskii M. V., “Trudnosti v matematicheskom opredelenii kontinualnykh integralov preodolevayutsya v teorii nepreryvnykh kvantovykh izmerenii”, TMF, 93:2 (1992), 264–272 | MR

[41] Kolokoltsov V., Singular Schrödinger equations with magnetic fields: self-adjointness and path integral representation, Preprint 7/00, Dep. Math. Stat. and O. R. Nottingham Trent University, 2000

[42] Albeverio S., Brzezniak Z., Dabrowski L., “Fundamental solution of the heat and Schrödinger equations with point interaction”, J. Funct. Anal., 130 (1995), 220–254 | DOI | MR | Zbl

[43] Yajima K., “Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations”, Comm. Math. Phys., 181 (1996), 605–629 | DOI | MR | Zbl

[44] Yajima K., “Boundedness and continuity of the fundamental solution of the time dependent Schrödinger equation with singular potentials”, Tôhoku Math. J., 50 (1998), 577–595 | DOI | MR | Zbl

[45] Yajima K., “Schrödinger evolution equations with magnetic fields”, J. Anal. Math., 56 (1991), 29–76 | DOI | MR | Zbl

[46] Leinfelder H., Simader C., “Sshrödinger operators with singular magnetic vector potentials”, Math. Z., 176 (1981), 1–19 | DOI | MR | Zbl