Separation for Schrodinger-type operators in weighted Hilbert spaces

Abdelsalam, Nehal Ahmed Mohamed; Abu-Donia, Hassan M.; Atia, Hany

doi:10.21608/bfszu.2023.182373.1231

Separation for Schrodinger-type operators in weighted Hilbert spaces

Document Type : Original Article

Authors

¹ Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

² Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt

10.21608/bfszu.2023.182373.1231

Abstract

The aim of this paper is to study the separation property of the Schrodinger operator L of the form Lf(x)=-L_0 f(x)+V(x)f(x),x∈R^n, in the weighted Hilbert space H^∼=L_(2,k) (R^n,H), the statement that achieve the separation, and the coercive estimate, with the operator potential V(x)∈L(H) for every x∈R^n, where L(H) is the space of all bounded linear operators on the arbitrary Hilbert space H. The operator L_0=∑_(i,j=1)^n ∂/(∂x_i ) a_ij (x)∂/(∂x_j )+∑_(i=1)^n b_i (x)∂/(∂x_i ) is the differential operator with the real-valued continuous functions a_ij (x) and b_i (x). Furthermore, we study the existence and uniqueness of the solution of the second order differential equation -∑_(i,j=1)^n ∂/(∂x_i ) a_ij (x)∂/(∂x_j ) f(x)-∑_(i=1)^n b_i (x)∂/(∂x_i ) f(x)+V(x)f(x)=W(x), where W(x)∈H^∼, in the weighted Hilbert space H^∼=L_(2,k) (R^n,H), such that k∈C^1 (R^n ) is positive weight function.
Keywords: Separation; Schrodinger-type operator; Operator potential; Hilbert space; Laplace operator; Coercive estimate; Existence and uniqueness.
AMS Subject Classification: 47F05, 58J99.

Keywords