On a family of differential operators with the coupling parameter in the boundary condition

We study a family of differential operators $L_α$ in two variables, depending on the coupling parameter $α\ge0$ that appears only in the boundary conditions. Our main concern is the spectral properties of $L_α$, which turn out to be quite different for $α\le 1$ and for $α>1$. In particular, $L_α$ has a unique self-adjoint realization for $α\le 1$ and many such realizations for $α>1$. In the more difficult case $α>1$ an analysis of non-elliptic pseudodifferential operators in dimension one is involved.

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