Nondissipative curves in Hilbert spaces having a limit of the corresponding correlation function

Abstract

In this work a class of nondissipative curves in Hilbert spaces whose correlation functions have a limit as t -+ :t=oo is presented. These curves correspond to a class ~ of nondissipative basic operators that are a coupling of a dissipative operator and an antidissipative one. The wave operators and the scattering operator for the couple (A*, A) (A E ~R) are obtained. The present work is a continuation and a generalization of the investigations of K. Kirchev and V.Zolotarev [1,2,3] on the model representations of curves in Hilbert spaces where the respective semigroup generator is a dissipative operator. This article includes four parts. A new form of the triangular model of M.S.Liv~ic ([4, 5]) for the considered operators is introduced in the first part by the help of a suitable representation of the selfadjoint operator L. This allows us to describe the studied class of nondissipative curves. The second part studies some results concerning the application of the analogue for multiplicative integrals of the well-known Privalov's theorem ([6]) about the limit values in the scalar case. This analogue is a reconstruction of measure by limit values in Stieltjes-Perron's style and it is obtained by L. A.Sakhnovich ([7]). Another problem considered in the second part is the analogue in C m of the classical gamma-function and several properties for further consideration. In the third part the asymptotics of the studied curves corresponding to the nondissipative operators -couplings of a dissipative and an antidissipative operator with absolutely continuous real spectra and the limits of their correlation functions are obtained. In the fourth part a scattering theory of a couple (A*, A) with a nondissipative operator A from ~R is constructed as in the selfadjoint case ([8, 9, 10]) and in the dissipative case ([7]). These results show an interesting new effect: the studied nondissipative case is near to the dissipative one.

Regular Couplings of Dissipative and Anti-Dissipative Unbounded Operators, Asymptotics of the Corresponding Non-Dissipative Processes and the Scattering Theory

“…Livšic. The asymptotic behaviour of the corresponding processes (continuous curves) is similar to the case of bounded non-selfadjoint operators [5,12] and is presented by the authors in [13], using the characteristic function of M.S. Livšic.…”

“…Before continuing with the asymptotics of the non-dissipative processes T t f generated by unbounded K r -operators with a triangular model A, defined by (2.12), it has to mention that in the course of proving of the asymptotics we use the analogue in C m of the classical gamma-function Γ(εI − iT (u)) = +∞ 0 e −x e ((ε−1)I−iT (u)) ln x dx (ε > 0) (4.33) (where T (u) is a matrix function) and its properties, introduced and considered by the authors in [12]. Next we will denote by T 1 (t) ∼ T 2 (t) as t → ±∞ that ||T 1 (t)−T 2 (t)|| L 2 −→ 0 for matrix functions T 1 (t) and T 2 (t).…”

“…The other addends in (4.29) tend to 0 as t → +∞ which follows directly (using the Lebesgue lemma for Fourier transform and Im λ 0 > 0, Im µ 0 > 0). Next we use the methods and ideas as in the bounded case ( [12,6]) and the unbounded case ( [13]) together with the presented inequalities (4.13),(4.14),. .…”

“…From the obtained asymptotics (4.52), the introduced denotations (4.46), (4.47), (4.48) (4.49), (4.50), (4.51) and the properties of the gamma-function ( [12], Lemma6, Lemma7) we have (see [13], proof of Theorem 6.15)…”

“…This model generates semigroups of operators {T t } t≤0 and {T t } t≥0 from the class (C 0 ) with generators iA. The explicit obtaining of the asymptotics of the corresponding nondissipative processes T t f as t → ±∞ allows to construct the scattering theory (as in the bounded case of the model A in [12]) for the couple (A * , A): in other words to obtain the wave operators W ± (A * , A), the scattering operator and the similarity of A and the operator Q of multiplication by the independent variable. All results are obtained explicitly using the multiplicative integrals and matrix generalization of the classical gamma-function.…”