Feynman path integrals introduced heuristically in the 1940s are a powerful tool used in many areas of physics, but also an intriguing mathematical challenge. In this work we used techniques of infinite dimensional integration (i.e. the infinite dimensional oscillatory integrals) in two different, but strictly connected, directions. On the one hand we construct a functional integral representation for solutions of a general high-order heat-type equations exploiting a recent generalization of infinite dimensional Fresnel integrals; in this framework we prove a a Girsanov-type formula, which is related, in the case of Schrödinger equation, to the Feynman path integral representation for the solution in presence of a magnetic field; eventually a new phase space path integral solution for higher-order heat-type equations is also presented. On the other hand for the three dimensional Schrödinger equation with magnetic field we provide a rigorous mathematical Feynman path integral formula still in the context of infinite dimensional oscillatory integrals; moreover, the requirement of independence of the integral on the approximation procedure forces the introduction of a counterterm, which has to be added to the classical action functional (this is done by the example of a linear vector potential). Thanks to that, it is possible to give a natural explanation for the appearance of the Stratonovich integral in the path integral formula for both the Schrödinger and the heat equation with magnetic field.
Referenze Bibliografiche
[1] Airy, G. B., On the intensity of light in the neighbourhood of a caustic, Transactions of the Cambridge Philosophical Society 6 (1838) 397–402.
[2] Albeverio, S. – Boutet De Monvel-Berthier, A. – Brze{\'{z}}niak, Z., The trace formula for schrödinger operators from infinite dimensional oscillatory integrals, Mathematische Nachrichten 182 (1996) 21–65.
[3] Albeverio, S. – Brzezniak, Z., Finite dimensional approximation approach to oscillatory integrals and stationary phase in infinite dimensions, Journal of Functional Analysis 113 (1993) 177–244.
[4] Albeverio, S. – Brze{\'{z}}niak, Z., Oscillatory integrals on hilbert spaces and schrödinger equation with magnetic fields, Journal of Mathematical Physics 36 (1995) 2135–2156.
[5] Albeverio, S. – Cangiotti, N. – Mazzucchi, S., Generalized feynman path integrals and applications to higher-order heat-type equations, Expositiones Mathematicae 36 (2018) 406–429.
[6] Albeverio, S. – Cangiotti, N. – Mazzucchi, S., A rigorous mathematical construction of feynman path integrals for the schrödinger equation with magnetic field, (2019) Preprint.
[7] Albeverio, S. – Fenstad, J. E. – Hoegh-Krohn, R., Nonstandard methods in stochastic analysis and mathematical physics, DOVER PUBN INC (2009).
[8] Albeverio, S. – Guatteri, G. – Mazzucchi, S., Phase space feynman path integrals, Journal of Mathematical Physics 43 (2002) 2847–2857.
[9] Albeverio, S. – Guatteri, G. – Mazzucchi, S., A representation of the belavkin equation via phase space feynman path integrals, Infinite Dimensional Analysis, Quantum Probability and Related Topics 07 (2004) 507–526.
[10] Albeverio, S. – H{\o}egh-Krohn, R., Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics i, Inventiones Mathematicae 40 (1977) 59–106.
[11] Albeverio, S. – H{\o}egh-Krohn, R. – Mazzucchi, S., Mathematical theory of feynman path integrals, Springer Berlin Heidelberg (2008).
[12] Albeverio, S. – Mazzucchi, S., Generalized fresnel integrals, Bulletin des Sciences Math{\'{e}}matiques 129 (2005) 1–23.
[13] Albeverio, S. – Mazzucchi, S., Feynman path integrals for polynomially growing potentials, Journal of Functional Analysis 221 (2005) 83–121.
[14] Albeverio, S. – Mazzucchi, S., The time-dependent quartic oscillator{\textemdash}a feynman path integral approach, Journal of Functional Analysis 238 (2006) 471–488.
[15] Albeverio, S. – Mazzucchi, S., Infinite dimensional oscillatory integrals as projective systems of functionals, Journal of the Mathematical Society of Japan 67 (2015) 1295–1316.
[16] Albeverio, S. – Mazzucchi, S., An introduction to infinite-dimensional oscillatory and probabilistic integrals, , Stochastic Analysis: A Series of Lectures Springer Basel (2015).
[17] Albeverio, S. – Mazzucchi, S., A unified approach to infinite-dimensional integration, Reviews in Mathematical Physics 28 (2016) 1650005.
[18] Albeverio, S. – Mazzucchi, S., A unified approach to infinite dimensional integrals of probabilistic and oscillatory type with applications to feynman path integrals, , Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis European Mathematical Society Publishing House (2018).
[19] Amour, L. – Jager, L. – Nourrigat, J., On bounded pseudodifferential operators in wiener spaces, Journal of Functional Analysis 269 (2015) 2747–2812.
[20] Andersson, L. – Driver, B. K., Finite dimensional approximations to wiener measure and path integral formulas on manifolds, Journal of Functional Analysis 165 (1999) 430–498.
[21] Arnold, V.I. – Gusein-Zade, S.M. – Varchenko, A.N., Singularities of differentiable maps, volume 1, Birkhäuser Boston (2012).
[22] Beghin, Luisa – Hochberg, Kenneth J. – Orsingher, Enzo, Conditional maximal distributions of processes related to higher-order heat-type equations, Stochastic Processes and their Applications 85 (2000) 209–223.
[23] Berberian, S. K., Measure and integration, A series of advanced mathematics texts Macmillan (1965).
[24] Bochner, Slomon, Harmonic analysis and the theory of probability., Berkeley: University of California Press. (1955).
[25] Bock, W. – Grothaus, M., A white noise approach to phase space feynman path integrals, Theory of Probability and Mathematical Statistics 85 (2013) 7–22.
[26] Bonaccorsi, S. – Calcaterra, C. – Mazzucchi, S., An it{\^{o}} calculus for a class of limit processes arising from random walks on the complex plane, Stochastic Processes and their Applications 127 (2017) 2816–2840.
[27] Bonaccorsi, Stefano – Mazzucchi, Sonia, High order heat-type equations and random walks on the complex plane, Stochastic Processes and their Applications 125 (2015) 797–818.
[28] Broderix, K. – Hundertmark, D. – Leschke, H., Continuity properties of schr\"o- dinger semigroups with magnetic fields, Reviews in Mathematical Physics 12 (2000) 181–225.
[29] Brown, L. M., Feynman{\textquotesingle}s thesis {\textemdash} a new approach to quantum theory, Wolrd Scientific (2005).
[30] Burdzy, K., Some path properties of iterated brownian motion, , Seminar on Stochastic Processes, 1992 Birkhäuser Boston (1993).
[31] Burdzy, K. – Madrecki, A., An asymptotically 4-stable process, , Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993) Special Issue (1995).
[32] Burdzy, K. – Madrecki, A., Itô formula for an asymptotically 4-stable process, The Annals of Applied Probability 6 (1996) 200–217.
[33] Cameron, R. H., A family of integrals serving to connect the wiener and feynman integrals, Journal of Mathematics and Physics 39 (1960) 126–140.
[34] Cangiotti, N. – Mazzucchi, S., Notes on the ogawa integrability and a condition for convergence in the multidimensional case, (2018) Preprint.
[35] Cartier, P. – DeWitt-Morette, C., Functional integration, Journal of Mathematical Physics 41 (2000) 4154–4187.
[36] Cartier, Pierre – DeWitt-Morette, Cecile, Functional integration: action and symmetries, Appendix D (contributed by Alexander Wurm) Cambridge University Press (2006).
[37] Cerdeira, H. A. – Lundqvist, S. – Mugnai, D. – Ranfagni, A. – Sa-yakanit, V. – Schulman, L. S., Lectures on path integration: Trieste 1991, , Lectures on Path Integration: Trieste 1991 World Scientific (1993).
[38] Chatterji, S. D. – Mandrekar, V., Quasi-invariance of measures under translation, Mathematische Zeitschrift 154 (1977) 19–29.
[39] Cycon, H. L. – Froese, R. G. – Kirsch, W. – Simon, B., Schrödinger operators, Springer-Verlag GmbH (2007).
[40] Dalang, Robert C. – Mueller, Carl – Tribe, Roger, A feynman-kac-type formula for the deterministic and stochastic wave equations and other p.d.e.{\textquotesingle}s, Transactions of the American Mathematical Society 360 (2008) 4681–4703.
[41] Daletskii, Yu. L. – Fomin, S. V., Generalized measures in function spaces, Teor. Veroyatnost. i Primenen 10 (1965) 329–343.
[42] Daubechies, I. – Klauder, J. R. – Paul, T., Wiener measures for path integrals with affine kinematic variables, Journal of Mathematical Physics 28 (1987) 85–102.
[43] Delgado, R., Multiple ogawa, stratonovich and skorohod anticipating integrals, Stochastic Analysis and Applications 16 (1998) 859–872.
[44] Doob, J. L., Stochastic processes, John Wiley & Sons Inc. (1953).
[45] Doss, H., Sur une résolution stochastique de l'équation de schrödinger à coefficients analytiques, Comm. Math. Phys. 73 (1980) 247–264.
[46] Duistermaat, J. J., Oscillatory integrals, lagrange immersions and unfolding of singularities, Communications on Pure and Applied Mathematics 27 (1974) 207–281.
[47] Dynkin, E. B., Theory of markov processes, Dover Publications (2006).
[48] Elworthy, D. – Truman, A., Feynman maps, cameron-martin formulae and anharmonic oscillators, Annales de l'I.H.P. Physique théorique 41 (1984) 115–142.
[49] Exner, P., Open quantum systems and feynman integrals, Springer Netherlands (1985).
[50] Feynman, R. P., Space-time approach to non-relativistic quantum mechanics, Reviews of Modern Physics 20 (1948) 367–387.
[51] Feynman, R. P. – Hibbs, A. R., Quantam mechanics and path integrals, Dover Publications Inc. (2010).
[52] Freidlin, M. I., Functional integration and partial differential equations - volume 109 (annals of mathematics studies), Princeton University Press (1985).
[53] Fresnel, A. J., Mémoire sur la diffraction de la lumière, où l’on examine particulièrement le phénomènes des franges colorées que présentent les ombres des corp éclaireés par un point lumineux., Annales de Chimie et de Physique 1 (1816) 239–281.
[54] Fujiwara, D., Rigorous time slicing approach to feynman path integrals, Springer Japan (2017).
[55] Fujiwara, D. – Tsuchida, T., The time slicing approximation of the fundamental solution for the schrödinger equation with electromagnetic fields, Journal of the Mathematical Society of Japan 49 (1997) 299–327.
[56] Fulling, S. A., Pseudodierential operators, covariant quantization, the inescapable van vleck-morette determinant, and the r/6 controversy, International Journal of Modern Physics D 05 (1996) 597–608.
[57] Funaki, T., Probabilistic construction of the solution of some higher order parabolic differential equation, Proceedings of the Japan Academy, Series A, Mathematical Sciences 55 (1979) 176–179.
[58] Gaveau, B. – Mih{\'{o}}kov{\'{a}}, E. – Roncadelli, M. – Schulman, L. S., Path integral in a magnetic field using the trotter product formula, American Journal of Physics 72 (2004) 385–388.
[59] Gaveau, B. – Schulman, L. S., Sensitive terms in the path integral: Ordering and stochastic options, Journal of Mathematical Physics 30 (1989) 2019–2022.
[60] Gaveau, Bernard – Vauthier, Jacques, Int{\'{e}}grales oscillantes stochastiques: l{\textquotesingle}{\'{e}}quation de pauli, Journal of Functional Analysis 44 (1981) 388–400.
[61] Gerald W. Johnson, Michel L. Lapidus, The feynman integral and feynman's operational calculus, OUP Oxford (2002).
[62] Gross, L., Measurable functions on hilbert space, Transactions of the American Mathematical Society 105 (1962) 372–372.
[63] Gross, L., Abstract wiener spaces, , Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Contributions to Probability Theory, Part 1 University of California PressBerkeley, Calif. (1967).
[64] Grothaus, M. – Riemann, F., A fundamental solution to the schrödinger equation with doss potentials and its smoothness, Journal of Mathematical Physics 58 (2017) 053506.
[65] Gubinelli, M. – L\"orinczi, J., Gibbs measures on brownian currents, Communications on Pure and Applied Mathematics 62 (2009) 1–56.
[66] G\"uneysu, B., The feynman{\textendash}kac formula for schrödinger operators on vector bundles over complete manifolds, Journal of Geometry and Physics 60 (2010) 1997–2010.
[67] Güneysu, Batu, Heat kernels in the context of kato potentials on arbitrary manifolds, Potential Analysis 46 (2016) 119–134.
[68] G\"uneysu, B. – Keller, M. – Schmidt, M., A feynman{\textendash}kac{\textendash}it{\^{o}} formula for magnetic schr\"odinger operators on graphs, Probability Theory and Related Fields 165 (2015) 365–399.
[69] Haba, Z., Stochastic interpretation of feynman path integral, Journal of Mathematical Physics 35 (1994) 6344–6359.
[70] Herzberg, F., Stochastic calculus with infinitesimals, Springer Berlin Heidelberg (2013).
[71] Hida, T. – Kuo, H. H. – Potthoff, J. – Streit, L., White noise, Springer Netherlands (1993).
[72] Hille, E. – Phillips, R. S., Functional analysis and semi-groups (colloquium publications (amer mathematical soc)), American Mathematical Society (1996).
[73] Hinz, M. – R\"ockner, M. – Teplyaev, A., Vector analysis for dirichlet forms and quasilinear {PDE} and {SPDE} on metric measure spaces, Stochastic Processes and their Applications 123 (2013) 4373–4406.
[74] Hochberg, K. J., A signed measure on path space related to wiener measure, The Annals of Probability 6 (1978) 433–458.
[75] Hochberg, K. J. – Orsingher, E., Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations, Journal of Theoretical Probability 9 (1996) 511–532.
[76] H\"ormander, L., The analysis of linear partial differential operators i, Springer Berlin Heidelberg (2003).
[77] I. M. Gel'fand, A. M. Yaglom, Integration in function spaces and its application to quantum physics, Uspekhi Mat. Nauk, 11 (1956) 77–114.
[78] Ichinose, W., On the formulation of the feynman path integral through broken line paths, Communications in Mathematical Physics 189 (1997) 17–33.
[79] Ichinose, W., On the feynman path integral for the magnetic schrödinger equation with a polynomially growing electromagnetic potential, Reviews in Mathematical Physics (2019).
[80] Ichinose, W. – Aoki, T., Notes on the cauchy problem for the self-adjoint and non-self-adjoint schrödinger equations with polynomially growing potentials, Journal of Pseudo-Differential Operators and Applications (2019).
[81] Ikeda, N. – Manabe, S., Van vleck-pauli formula for wiener integrals and jacobi fields, , It{\^{o}}'s Stochastic Calculus and Probability Theory Springer Japan (1996).
[82] It{\^o}, K., Wiener integral and feynman integral, , Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Contributions to Probability Theory University of California PressBerkeley, Calif. (1961).
[83] It{\^o}, K., Generalized uniform complex measures in the hilbertian metric space with their application to the feynman integral, , Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Contributions to Probability Theory, Part 1 University of California PressBerkeley, Calif. (1967).
[84] It{\^o}, K. – Nisio, M., On the convergence of sums of independent banach space valued random variables, Osaka J. Math. 5 (1968) 35–48.
[85] Jefferies, Brain, Advances and applications of the feynman integral, , Real and Stochastic Analysis Birkhäuser Boston (2004).
[86] Johnson, G. W. – Lapidus, M. L., The feynman integral and feynman's operational calculus, OUP Oxford (2002).
[87] Kac, M., Enigmas of chance: An autobiography (alfred p. sloan foundation series), HarperCollins (1985).
[88] Karatzas, I. – Shreve, S. E., Brownian motion and stochastic calculus, Springer New York (1998).
[89] Kluv{\'a}nek, I., Integration structures, Proceedings of the Centre for Mathematical Analysis, Australian National University Centre for Mathematical Analysis, Australian National University (1988).
[90] Kolokoltsov, V. N., Semiclassical analysis for diffusions and stochastic processes, Springer Berlin Heidelberg (2000).
[91] Kolokoltsov, V. N., Mathematics of the feynman path integral, Filomat 15 (2001) 293–311.
[92] Kolokoltsov, V. N., Schr\"odinger operators with singular potentials and magnetic fields, Sbornik: Mathematics 194 (2003) 897–917.
[93] Konstantinov, A. A. – Maslov, V. P. – Chebotarev, A. M., Probability representations of solutions of the cauchy problem for quantum mechanical equations, Russian Mathematical Surveys 45 (1990) 1–26.
[94] Krylov, V. Yu., Some properties of the distribution corresponding to the equation $\partial u / \partial t = (-1)^{q+1} \partial^{2q} u / \partial x^{2q}$, Sov. Math., Dokl. 1 (1960) 760–763.
[95] Kumano-go, N., Phase space feynman path integrals with smooth functional derivatives by time slicing approximation, Bulletin des Sciences Math{\'{e}}matiques 135 (2011) 936–987.
[96] Kumano-go, N. – Fujiwara, D., Phase space feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations, Bulletin des Sciences Math{\'{e}}matiques 132 (2008) 313–357.
[97] Kumano-go, N. – Murthy, A. S. Vasudeva, Phase space feynman path integrals of higher order parabolic type with general functional as integrand, Bulletin des Sciences Math{\'{e}}matiques 139 (2015) 495–537.
[98] Kuo, H. H., Gaussian measures in banach spaces, Springer Berlin Heidelberg (1975).
[99] L{\'{e}}andre, R{\'{e}}mi, Stochastic analysis without probability: study of some basic tools, Journal of Pseudo-Differential Operators and Applications 1 (2010) 389–400.
[100] Leinfelder, H. – Simader, C. G., Schr\"odingers operators with singular magnetic vector potentials, Mathematische Zeitschrift 176 (1981) 1–19.
[101] Loss, M. – Thaller, B., Optimal heat kernel estimates for schrödinger operators with magnetic fields in two dimensions, Communications in Mathematical Physics 186 (1997) 95–107.
[102] Lyons, T. – Levin, D., A signed measure on rough paths associated to a {PDE} of high order: results and conjectures, Revista Matem{\'{a}}tica Iberoamericana (2009) 971–994.
[103] M{\k{a}}drecki, A. – Rybaczuk, M., New feynman-kac type formula, Reports on Mathematical Physics 32 (1993) 301–327.
[104] Majer, P. – Mancino, M. E., A counter-example concerning a condition of ogawa integrability, S\'eminaire de probabilit\'es de Strasbourg 31 (1997) 198–206.
[105] Marsden, J. E., Countable and net convergence, Amer. Math. Monthly 75 (1968).
[106] Mazzucchi, S., Feynman path integrals for the inverse quartic oscillator, Journal of Mathematical Physics 49 (2008) 093502.
[107] Mazzucchi, S., Mathematical feynman path integrals and their applications, World Scientific Publishing Company (2009).
[108] Mazzucchi, S., Functional-integral solution for the schrdinger equation with polynomial potential: A white noise approach, Infinite Dimensional Analysis, Quantum Probability and Related Topics 14 (2011) 675–688.
[109] Mazzucchi, S., Infinite dimensional oscillatory integrals with polynomial phase and applications to higher-order heat-type equations, Potential Analysis 49 (2017) 209–223.
[110] Moretti, V., Spectral theory and quantum mechanics, Springer International Publishing (2017).
[111] Murray, J. D., Asymptotic analysis, Springer New York (1984).
[112] Neidhardt, H. – Stephan, A. – Zagrebnov, V. A., Remarks on the operator-norm convergence of the trotter product formula, Integral Equations and Operator Theory 90 (2018).
[113] Nelson, E., Feynman integrals and the schrödinger equation, Journal of Mathematical Physics 5 (1964) 332–343.
[114] Nicola, F., Convergence in $L^p$ for feynman path integrals, Advances in Mathematics 294 (2016) 384–409.
[115] Nualart, D., Nonclausal stochastic integrals and calculus, , Lecture Notes in Mathematics 1316 Springer Berlin Heidelberg (2006).
[116] Nualart, D. – Zakai, M., Generalized stochastic integrals and the malliavin calculus, Probability Theory and Related Fields 73 (1986) 255–280.
[117] Nualart, D. – Zakai, M., Generalized multiple stochastic integrals and the representation of wiener functionals, Stochastics 23 (1988) 311–330.
[118] Nualart, D. – Zakai, M., On the relation between the stratonovich and ogawa integrals, The Annals of Probability 17 (1989) 1536–1540.
[119] Ogawa, S., Sur le pruduit direct du bruit blancpar lui-même, Acad. Sci., Paris, Serie A 288 (1979) 359–362.
[120] Ogawa, S., The stochastic integral of noncausal type as an extension of the symmetric integrals, Japan Journal of Applied Mathematics 2 (1985) 229–240.
[121] Ogawa, S., Stochastic integral equations for the random fields, , Lecture Notes in Mathematics Springer Berlin Heidelberg (1991).
[122] Ogawa, S., Noncausal stochastic calculus revisited - around the so-called ogawa integral, , Advances in Deterministic and Stochastic Analysis World Scientific (2007).
[123] Ogawa, S., Noncausal stochastic calculus, Springer Japan (2017).
[124] Orsingher, Enzo – Zhao, Xuelei, Iterated processes and their applications to higher order differential equations, Acta Mathematica Sinica 15 (1999) 173–180.
[125] Osborn, T. A. – Papiez, L. – Corns, R., Constructive representations of propagators for quantum systems with electromagnetic fields, Journal of Mathematical Physics 28 (1987) 103–123.
[126] Ramer, R., On nonlinear transformations of gaussian measures, Journal of Functional Analysis 15 (1974) 166–187.
[127] Reed, Michael – Simon, Barry, Fourier analysis, self-adjointness (methods of modern mathematical physics, vol. 2), Academic Press [1 Edition] (1975).
[128] Rezende, J., The method of stationary phase for oscillatory integrals on hilbert spaces, Comm. Math. Phys. 101 (1985) 187–206.
[129] Rudin, W., Functional analysis, International series in pure and applied mathematics McGraw-Hill (1991).
[130] Sainty, Philippe, Construction of a complex-valued fractional brownian motion of {orderN}, Journal of Mathematical Physics 33 (1992) 3128–3149.
[131] Schulman, L. S., Techniques and applications of path integration, Dover Publications (2005).
[132] Simon, B., Functional integration and quantum physics, American Mathematical Society (2004).
[133] Skorokhod, A. V., On a generalization of a stochastic integral, Theory of Probability & Its Applications 20 (1976) 219–233.
[134] Smolyanov, O. G. – Tokarev, A. G. – Truman, A., Hamiltonian feynman path integrals via the chernoff formula, Journal of Mathematical Physics 43 (2002) 5161.
[135] Streater, R. F., Euclidean quantum mechanics and stochastic integrals, , Lecture Notes in Mathematics Springer Berlin Heidelberg (1981).
[136] Streit, L. – Hida, T., Generalized brownian functionals and the feynman integral, Stochastic Processes and their Applications 16 (1984) 55–69.
[137] Sunada, T., A discrete analogue of periodic magnetic schrödinger operators, , Geometry of the spectrum (Seattle, WA, 1993) Amer. Math. Soc., Providence, RI (1994).
[138] Thaler, H., The doss trick on symmetric spaces, Letters in Mathematical Physics 72 (2005) 115–127.
[139] Thomas, E. G. F., Projective limits of complex measures and martingale convergence, Probability Theory and Related Fields 119 (2001) 579–588.
[140] Trotter, H. F., On the product of semi-groups of operators, Proceedings of the American Mathematical Society 10 (1959) 545–545.
[141] Truman, A., Feynman path integrals and quantum mechanics as $h \rightarrow 0$, Journal of Mathematical Physics 17 (1976) 1852–1862.
[142] Truman, A., The feynman maps and the wiener integral, Journal of Mathematical Physics 19 (1978) 1742–1750.
[143] Truman, A., The polygonal path formulation of the feynman path integral, , Feynman Path Integrals Springer Berlin Heidelberg (1979).
[144] Tsuchida, T., Remarks on fujiwara's stationary phase method on a space of large dimension with a phase function involving electromagnetic fields, Nagoya Mathematical Journal 136 (1994) 157–189.
[145] Watanabe, S. – Ikeda, N., Stochastic differential equations and diffusion processes, volume 24 (north-holland mathematical library), North Holland (1981).
[146] Yajima, K., Schrödinger evolution equations with magnetic fields, Journal d{\textquotesingle}Analyse Math{\'{e}}matique 56 (1991) 29–76.
Con il contributo del Ministero per i Beni e le Attività Culturali