Abdelsalam, N., Abu-Donia, H., Atia, H. (2023). Separation for Schrodinger-type operators in weighted Hilbert spaces. Bulletin of Faculty of Science, Zagazig University, 2023(3), 98-107. doi: 10.21608/bfszu.2023.182373.1231
Nehal Ahmed Mohamed Abdelsalam; Hassan M. Abu-Donia; Hany Atia. "Separation for Schrodinger-type operators in weighted Hilbert spaces". Bulletin of Faculty of Science, Zagazig University, 2023, 3, 2023, 98-107. doi: 10.21608/bfszu.2023.182373.1231
Abdelsalam, N., Abu-Donia, H., Atia, H. (2023). 'Separation for Schrodinger-type operators in weighted Hilbert spaces', Bulletin of Faculty of Science, Zagazig University, 2023(3), pp. 98-107. doi: 10.21608/bfszu.2023.182373.1231
Abdelsalam, N., Abu-Donia, H., Atia, H. Separation for Schrodinger-type operators in weighted Hilbert spaces. Bulletin of Faculty of Science, Zagazig University, 2023; 2023(3): 98-107. doi: 10.21608/bfszu.2023.182373.1231
Separation for Schrodinger-type operators in weighted Hilbert spaces
1Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
2Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt
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.
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