Separation for Schrodinger-type operators with operator potentials in Banach spaces

Abstract

Let H be a Banach space of all complex sequences h=(h ₁,h ₂, …), h_j ∈ C, equipped with the norm

We introduce the Banach space H ₁ which is the space of all vector functions u(x)=(u ₁(x), u ₂(x), …), x∈ Rⁿ (u_j ∈ L ₁(Rⁿ ), j=1,2, …) that has a range in H, equipped with the norm

In this article we study the separation of the Schrodinger-type operator A= − P +V(x), x∈ Rⁿ , in the Banach space H ₁, with the linear operator potential V(x). Moreover, we study the separation of the nonlinear Schrodinger-type operator B= − P+V(x,h), x∈ Rⁿ , h∈ H, in the Banach space H ₁, with the nonlinear operator potential V(x,h), where

with real valued functions b _k (x) and a_kr (x) in C ¹(Rⁿ ), and

.

Keywords:

• Separation
• Elliptic differential operators
• Banach space
• Coercive estimate
• AMS Subject Classifications: 47F05
• 58J99

Notes

Email: [email protected]

Email: [email protected]

Email: [email protected]

Additional information

Notes on contributors

H. A. Atia ‡

Email: [email protected] Email: [email protected]