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- W2743039622 abstract "Background. We consider nonselfadjoint singular rank one perturbation of a self-adjoint operator by nonsymmetric potential, i.e. expression of the form [ tilde A=A+alphaleftlanglecdot,delta_1rightrangledelta_2, ] where A is a selfadjoint semi-bounded operator in the separable Hilbert space [{mathcal H}.] Our investigations consist in the fact that it is unknown whether [tilde{A}] is a spectral operator for vectors [delta_1,delta_2in{mathcal H}_{-1},~delta_1not=delta_2]. Objective. Purpose of the study is to establish the existence of wave operators in [tilde{A}] provided “weak-weak” singular perturbation rank “one-one” [{rm dim}({mathcal H}ominus {mathfrak D})=0, {rm dim}({mathcal H}_{+1}ominus {mathfrak D})=1, {rm dim}({mathcal H}ominus {mathfrak D}_*)=0, {rm dim}({mathcal H}_{+1}ominus {mathfrak D}_*)=1, ] where subsets [{mathfrak D}={fin{mathfrak D}(A)cap{mathfrak D}(tilde A) vert Af=tilde Af}, {mathfrak D}_*={fin{mathfrak D}(A)cap{mathfrak D}(tilde A^*) vert Af=tilde A^*f} ] dense both in [{mathcal H}.] Methods. Known T. Kato theorem is used to the operator [tilde{A}.] Results. Using the explicit description of [tilde{A}] we prove the existence of wave operators with [vertalphavert<infty,] corresponding [tilde{A}.] Wave operators are defined by equality [ (tilde W_{pm}u,v)=(u,v)mpfrac{alpha}{2pi i} int limits_{-infty}^{+infty} langle R_{lambdapm i0}u,omega_1ranglelangleomega_2,tilde R_{lambdamp i0}^*vrangle dlambda, quad u,vin{mathcal H}, ] where [tilde{R}] and [R] are resolvents of perturbed and unperturbed operators. Conclusions. The existence of wave operators is provided and their form of action is given for singular perturbation of rank one nonsymmetric potential." @default.
- W2743039622 created "2017-08-17" @default.
- W2743039622 creator A5000363984 @default.
- W2743039622 date "2015-09-18" @default.
- W2743039622 modified "2023-10-01" @default.
- W2743039622 title "Wave Operators of a Rank One Singular Perturbation by Nonsymmetric Potential" @default.
- W2743039622 doi "https://doi.org/10.20535/1810-0546.2015.4.50461" @default.
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