Laplacian eigenvalue on a domain with singularities

Question

Suppose $\mathbb{B}$ is the compact unit disc with boundary with center at orgin, now we consider $\Omega=\mathbb{B}\setminus \{z: \operatorname{Re}(z)=0\}$. Does following eigenvalue problem make sense? $$\Delta u=\lambda u \text{ in }\Omega;\ u=0 \text{ on} \ \partial \Omega$$ Is there any difference with the eigenvalue problem on the disc?