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This is the first of three papers dealing with the XX finite quantum chainwith arbitrary, not necessarily hermitian, boundary terms. This extendsprevious work where the periodic or diagonal boundary terms were considered. Inorder to find the spectrum and wave-functions an auxiliary quantum chain isexamined which is quadratic in fermionic creation and annihilation operatorsand hence diagonalizable. The secular equation is in general complicated butseveral cases were found when it can be solved analytically. For these casesthe ground-state energies are given. The appearance of boundary states is alsodiscussed and in view to the applications considered in the next papers, theone and two-point functions are expressed in terms of Pfaffians.Comment: 56 pages, LaTeX, some minor correction

The -model with boundaries: Part I. Diagonalization of the finite chain