Separation for Schrodinger-type operators in weighted Hilbert spaces | ||
| Bulletin of Faculty of Science, Zagazig University | ||
| Article 10, Volume 2023, Issue 3, October 2023, Pages 98-107 PDF (1.05 M) | ||
| Document Type: Original Article | ||
| DOI: 10.21608/bfszu.2023.182373.1231 | ||
| Authors | ||
| Nehal Ahmed Mohamed Abdelsalam* 1; Hassan M. Abu-Donia2; Hany Atia2 | ||
| 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. | ||
| Keywords | ||
| Separation; Schrodinger-type operator; Operator potential; Hilbert space; Existence and uniqueness | ||
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