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Resonances of Schr\"odinger Hamiltonians with point interactions areconsidered. The main object under the study is the resonance free region underthe assumption that the centers, where the point interactions are located, areknown and the associated 'strength' parameters are unknown and allowed to bearadditional dissipative effects. To this end we consider the boundary of theresonance free region as a Pareto optimal frontier and study the correspondingoptimization problem for resonances. It is shown that upper logarithmic boundon resonances can be made uniform with respect to the strength parameters. Thenecessary conditions on optimality are obtained in terms of first principalminors of the characteristic determinant. We demonstrate the applicability ofthese optimality conditions on the case of 4 equidistant centers by computingexplicitly the resonances of minimal decay for all frequencies. This exampleshows that a resonance of minimal decay is not necessarily simple, and in somecases it is generated by an infinite family of feasible resonators.

Resonance free regions and non-Hermitian spectral optimization for Schr\"odinger point interactions