KatoRellichteoremet

KatoRellichteoremet, also known as the Kato-Rellich theorem, is a fundamental result in functional analysis concerning the self-adjointness of differential operators. It provides crucial conditions under which a differential operator, defined on a Hilbert space, remains self-adjoint after a perturbation. The theorem is named after Tosio Kato and Franz Rellich, who made significant contributions to its development.

The theorem typically deals with an operator $A_0$ which is already self-adjoint and a bounded operator $B$ on a Hilbert space $H$. If $A = A_0 + B$ is a densely defined operator, the Kato-Rellich theorem states that if $B$ is relatively bounded with respect to $A_0$, then the self-adjointness of $A_0$ implies the self-adjointness of $A$ on the domain of $A_0$.

More formally, if $A_0$ is a self-adjoint operator and $B$ is a symmetric operator such that for every $\epsilon > 0$, there exists a constant $c_\epsilon > 0$ where $\|Bx\|^2 \le \epsilon \|A_0x\|^2 + c_\epsilon \|x\|^2$ for all $x$ in the domain of $A_0$, then $A_0 + B$ is self-adjoint on the domain of $A_0$. The condition $\|Bx\|^2 \le \epsilon \|A_0x\|^2 + c_\epsilon \|x\|^2$ is known as relative boundedness of $B$ with respect to $A_0$.

The Kato-Rellich theorem has broad applications in quantum mechanics and partial differential equations. It is instrumental in establishing the self-adjointness of Hamiltonian operators, which is a prerequisite for their spectral properties to be physically meaningful, such as the reality of energy eigenvalues. It also plays a role in the theory of elliptic and parabolic partial differential equations where differential operators are often analyzed through their spectral properties.