Existence of the time evolution for Schrödinger operators with time dependent singular potentials. (English) Zbl 0598.35033

Wüller, Ulrich

Ann. Inst. Henri Poincaré, Phys. Théor. 44, 155-171 (1986).

From the author’s abstract: ”Let N particles move along fixed paths \(y_ j(t)\) and cause a time dependent potential. The quantum mechanical description for an additional particle is given by the Schrödinger equation \[ i(\partial /\partial t)\psi =-(1/2m)\Delta \psi +\sum^{N}_{j=1}q_ j v(x-y_ j(t))\psi. \] For a class of trajectories \(y_ j(t)\) the propagator is constructed for the Coulomb potential \(v(x)=1/| x|\) \((x\in {\mathbb{R}}^ 3)\) and stronger singularities.”

Reviewer: R.Racke

MSC:

35J10	Schrödinger operator, Schrödinger equation
81U10	\(n\)-body potential quantum scattering theory
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Keywords:

Schrödinger equation; propagator; Coulomb potential

References:

[1]	J.B. Delos , Theory of electronic transitions in slow atomic collisions . Rev. Mod. Phys. , t. 53 , 1981 , p. 287 - 357 .
[2]	G.A. Hagedorn , An analog of the RAGE Theorem for the impact parameter approximation to three particle scattering . Ann. Inst. H. Poincaré , t. A 38 , 1983 , p. 59 - 68 . Numdam | MR 700700 | Zbl 0517.47009 · Zbl 0517.47009
[3]	G.C. Hegerfeld , Remarks on causality, localization and spreading of wave packets . Phys. Rev. D. , t. 22 , 1980 , p. 377 - 384 . MR 578960
[4]	T. Kato , Integration of the equation of evolution in a Banach space . J. Math. Soc. Japan , t. 5 , 1953 , p. 208 - 234 . Article | MR 58861 | Zbl 0052.12601 · Zbl 0052.12601
[5]	T. Kato , On linear differential equations in Banach spaces . Comm. Pure Appl. Math. , t. 9 , 1956 , p. 479 - 486 . MR 86986 | Zbl 0070.34602 · Zbl 0070.34602
[6]	T. Kato , Linear evolution equations of hyperbolic type . J. Fac. Sci. Univ. Tokyo , Sec. 1, t. 17 , 1970 , p. 241 - 258 . MR 279626 | Zbl 0222.47011 · Zbl 0222.47011
[7]	T. Kato , Linear evolution equations of hyperbolic type . II. J. Math. Soc. Japan , t. 25 , 1973 , 648 - 666 . Article | MR 326483 | Zbl 0262.34048 · Zbl 0262.34048
[8]	M. Reed , B. Simon , Methods of modern mathematical physics . I. Functional analysis . New York , Academic Press , 1972 . MR 493419 | Zbl 0242.46001 · Zbl 0242.46001
[9]	M. Reed , B. Simon , Methods of modern mathematical physics . II. Fourier analysis, self-adjointness , New York , Academic Press , 1975 . MR 493420 | Zbl 0308.47002 · Zbl 0308.47002
[10]	M. Reed , B. Simon , Methods of modern mathematical physics. IV . Analysis of operators , New York , Academic Press , 1978 . MR 493421 | Zbl 0401.47001 · Zbl 0401.47001
[11]	B. Simon , Quantum Mechanicsfor Hamiltonians defined as Quadratic Forms , Princeton University Press , 1971 . MR 455975 | Zbl 0232.47053 · Zbl 0232.47053
[12]	H. Tanabe , Equations of Evolution , Pitman , 1979 . MR 533824
[13]	U. Wüller , Existenz von Propagatoren für Schrödinger Operatoren mit zeitabhängigen singulären Potentialen , Diplomarbeit, Ruhr-Universität Bochum , 1984 .
[14]	K. Yajima , A multi-channel scattering theory for some time dependent hamiltonians, Charge transfer problem . Comm. Math. Phys. , t. 75 , 1980 , p. 153 - 178 . Article | MR 582506 | Zbl 0437.47008 · Zbl 0437.47008
[15]	K. Yosida , On the integration of the equation of evolution . J. Fac. Sci. Univ. Tokyo , Sec. 1, t. 9 , 1963 , p. 397 - 402 . MR 150616 | Zbl 0201.18203 · Zbl 0201.18203
[16]	K. Yosida , Time dependent evolution equations in a locally compact space . Math. Ann. , t. 162 , 1966 , p. 83 - 86 . MR 190498 | Zbl 0138.08301 · Zbl 0138.08301
[17]	K. Yosida , Functional Analysis , Berlin , Heidelberg , New York , Springer , 2 nd. ed., 1968 . MR 500055 · Zbl 0152.32102