Functional differential equation with dilated and rotated argument

A boundary value problem in a plane bounded domain for a second-order functional differential equation containing a combination of dilations and rotations of the argument in the leading part is considered. Necessary and sufficient conditions are found in the algebraic form for the fulfillment of the Gårding-type inequality, which ensures the unique (Fredholm) solvability and discreteness and sectorial structure of the spectrum of the Dirichlet problem. The term strongly elliptic equation is customary in this situation in literature. The derivation of the above conditions expressed directly through the coefficients of the equation, is based on a combination of the Fourier and Gel'fand transforms of elements of the commutative $B^*$-algebra generated by the dilatation and rotation operators. The main point here is to clarify the structure of the space of maximal ideals of this algebra. It is proved that the space of maximal ideals is homeomorphic to the direct product of the spectra of the dilatation operator (the circle) and the rotation operator (the whole circle if the rotation angle $\alpha$ is incommensurable with $\pi$, and a finite set of points on the circle if $\alpha$ is commensurable with $\pi$). Such a difference between the two cases for $\alpha$ leads to the fact that, depending on $\alpha$, the conditions for the unique solvability of the boundary value problem may have significantly different forms and, for example, for $\alpha$ commensurable with $\pi$, may depend not only on the absolute value, but also on the sign of the coefficient at the term with rotation.