Distortion analyticity and molecular resonance curves. (English) Zbl 0619.46068

Hunziker, W.

Ann. Inst. Henri Poincaré, Phys. Théor. 45, 339-358 (1986).

Resonance energies of n electrons in the field of N fixed nuclei are defined as discrete eigenvalues of non-selfadjoint operators which arise from the Hamiltonian H by a general class of complex distortions of \({\mathbb{R}}^ 3\) around the fixed nuclei. They are identified with the poles in the analytic continuation of resolvent matrix-elements \((\phi,(z-H)^{-1}\psi)\) between states \(\phi\), \(\psi\) of an explicitely given set A of analytic vectors, and thus shown to be independent of the particular choice of the distortion. Distortions are also used to derive local analyticity properties of bound state- and resonance energies in the nuclear coordinates.
The same techniques also yield existence and uniqueness of solutions to the Schrödinger equation for n electrons in the time-dependent field of classically moving (non-colliding) nuclei.

MSC:

46N99	Miscellaneous applications of functional analysis
81V10	Electromagnetic interaction; quantum electrodynamics
47F05	General theory of partial differential operators

Keywords:

Resonance energies; discrete eigenvalues of non-selfadjoint operators; complex distortions; bound state; resonance; existence and uniqueness of solutions to the Schrödinger equation for n electrons in the time- dependent field of classically moving (non-colliding) nuclei

References:

[1]	P. Aventini , R. Seiler , On the electronic spectrum of the diatomic molecular ion . Commun. Math. Phys. , t. 41 , 1975 , p. 119 - 134 . Article | MR 371301
[2]	E. Balslev , J.M. Combes , Spectral properties of many-body Schrödinger operators with dilation analytic interactions . Commun. Math. Phys. , t. 22 , 1971 , p. 280 - 294 . Article | MR 345552 | Zbl 0219.47005 · Zbl 0219.47005
[3]	J.M. Combes , P. Duclos , R. Seiler , The Born-Oppenheimer approximation , in: Rigorous atomic and molecular physics , Eds. G. Velo and A. S. Wightman. New York , London , Plenum Press , 1981 .
[4]	H.L. Cycon , Resonances defined by modified dilations . Helv. Phys. Acta , t. 58 , 1985 , p. 969 - 981 . MR 821113
[5]	P. Deift , W. Hunziker , B. Simon , E. Vock , Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems. IV . Commun. Math. Phys ,, t. 64 , 1978 , p. 1 - 34 . Article | MR 516993 | Zbl 0419.35079 · Zbl 0419.35079
[6]	R. Froese , I. Herbst , Exponential bounds and absence of positive eigenvalues for N-body Schrödinger operators . Commun. Math. Phys. , t. 87 , 1983 , p. 429 - 447 . Article | MR 682117 | Zbl 0509.35061 · Zbl 0509.35061
[7]	W. Hunziker , C. Günther , Bound states in dipole fields and continuity properties of electronic spectra . Helv. Phys. Acta , t. 53 , 1980 , p. 201 - 208 . MR 597559
[8]	T. Kato , Perturbation theory for linear operators . Berlin , Heidelberg , New York , Springer , 1966 . MR 203473 | Zbl 0148.12601 · Zbl 0148.12601
[9]	M. Klaus , On H2+ for small internuclear separation . J. Phys. A : Math. Gen. , t. 16 , 1983 , p. 2709 - 2720 . MR 715732
[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]	I.M. Sigal , Complex transformation method and resonances in one-body quantum systems . Ann. Inst. Henri Poincaré , t. 41 , 1984 , p. 103 - 114 . Numdam | MR 760129 | Zbl 0568.47008 · Zbl 0568.47008
[12]	B. Simon , The definition of molecular resonance curves by the method of exterior complex scaling . Phys. Lett. , t. 71A , 1979 , p. 211 - 214 .
[13]	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
[14]	U. Wüller , Existence of the time evolution for Schrödinger operators with time dependent singular potentials . Ann. Inst. Henri Poincaré , t. 44 , 1986 , p. 155 - 171 . Numdam | MR 839282 | Zbl 0598.35033 · Zbl 0598.35033
[15]	K. Yajima , Existence of solutions for Schrödinger evolution equations . University of Tokyo , preprint, 1986 . MR 891945 · Zbl 0693.35147
[16]	K. Yosida , Functional analysis . New York , Academic Press , 1965 . MR 180824 | Zbl 0126.11504 · Zbl 0126.11504