Let us consider $A$ a translation invariant lower semi-bounded operator on $L^2(\mathbb{R}^n)$ with domain $D(A)$ and with empty discrete spectrum (I exclude bound states). I have the following intuition : since the operator is invariant by translation, the information should be "at infinity" and the bottom of its (essential) spectrum should not depend on any finite region. More specifically, I'm wondering if the following equality holds :
$\text{inf} \hspace{0.1cm} \sigma(A) = \underset{\psi \in D(A) \\ \| \psi \|=1}{\text{inf}} \langle \psi|A \psi \rangle = \underset{\psi \in D(A) \cap \{\psi = 0 \text{ on } B_R\} \\ \| \psi \|=1}{\text{inf}} \langle \psi|A \psi \rangle$ $\hspace{1cm}$ where $B_R$ is the ball of radius $R>0$.
However, I didn't find such a result in the books I looked through, so I was wondering if anyone knew wether this is true or not. I think it holds for example for the Laplace operator, for instance $\sigma(\frac{-d}{dx^2}) = [0, +\infty)$ on $L^2(\mathbb{R})$ but also on $L^2(\mathbb{R}\setminus [-R,R])$ if we add Neumann or Dirichlet boundary conditions.
