Given the Cauchy process \((X_{t})\) on \(\mathbb R^d\), \(d\geq 1\), the authors investigate the spectral properties of the semigroup \((P_{t}^D)\) obtained from \((X_{t})\) by killing the process upon leaving the bounded open set \(D\) whose boundary has to satisfy some (Lipschitz) regularity property. The basic step is the fact that the functions \(u_{n}(x,t):= P_{t}^D \varphi_{n}(x)\) satisfy a mixed Steklov problem where \(\varphi_{n}\) are the eigenfunctions of the semigroup \(P_{t}^D\) with eigenvalues \(e^{-\lambda_{n}t}\), \(00\) on \(D_{\pm} = \{x\in D\;| \;\pm x_{1} >0\}\) and show that \(\varphi_{\ast}\) has the lowest eigenvalue among all antisymmetric eigenfunctions. The last section is devoted to the one-dimensional problem with \(D=(-1,1)\). Estimates for \(\lambda_{1},\;\lambda_{2}\) and \(\lambda_{3}\) are given and symmetry properties of the corresponding eigenfunctions are proven.