Runge property and approximation by complete systems of solutions for strongly elliptic equations

Abstract

We give an overview of some concepts related to the approximation of solutions of a strongly elliptic operator. A definition of a complete system of classical solutions is given, and we show how using the Runge property, it is possible to extend the completeness of these systems onto strictly internal domains in the $L_2$, $H^1$ and $H^2$ norms. This result is applied to Schrödinger equations with potentials possessing some symmetries.

Keywords:

• Complete system of solutions
• Runge property
• Schrödinger equation
• transmutation operator

AMS Subject Classifications:

• 35A35
• 35D30
• 35A09
• 35J10

Acknowledgments

Research was supported by CONACYT, Mexico via the project 284470 as well as by the Regional mathematical center of the Southern Federal University, Russia.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 $\text{Int}(K_{n+1})$ denotes the interior of the set $K_{n+1}$.

2 That is, there exist constants $c_1,c_2>0$ such that $\mathrm{\Psi }[u,u]⩾c_1\Vert u{\Vert }_{H^1(\mathrm{\Omega })}-c_2\Vert u{\Vert }_{L_2(\mathrm{\Omega })}$.

3 If $V\subset X$, the annihilator of V is defined as $V^a:=\{f\in X^{\star }|(f,x{)}_X=0\mathrm{\forall }x\in V\}$

4 In fact, the intersection of star-shaped domains with respect to x = 0 is a star-shaped domain, and $\text{Star}(A)$ is the intersection of all that contain A.

Additional information

Funding

This work was supported by the National Council of Science and Technology, Mexico [grant number 284470].