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Random banded matrices with linearly increasing diagonal elements arerecently considered as an attractive model for complex nuclei and atoms. Apartfrom early papers by Wigner \cite{Wig} there were no analytical studies on thesubject. In this letter we present analytical and numerical results for localspectral density of states (LDOS) for more general case of matrices with asparsity inside the band. The crossover from the semicircle form of LDOS tothat given by the Breit-Wigner formula is studied in detail.Comment: Misprints are corrected and stylistic changes are made. To be  published in PR

Wigner Random Banded Matrices with Sparse Structure: Local Spectral Density of States