Spectral Attributes of Self-Adjoint Fredholm Operators in Hilbert Space: A Rudimentary Insight

In defining the finiteness or infiniteness conditions of discrete spectrum of the Schrodinger operators, a fundamental understanding on n(1,F()) is crucial, where n(1,F) is the number of eigenvalues of the Fredholm operator F to the right of 1. Driven by this idea, this paper provided the invertibility condition for some class of operators. A sufficient condition for finiteness of the discrete spectrum involving the self-adjoint operator acting on Hilbert space was achieved. A relation was established between the eigenvalue 1 of the self-adjoint Fredholm operator valued function F() defined in the interval of (a,b) and discontinuous points of the function n(1,F()). Besides, the obtained relation allowed us to define the finiteness of the numbers z(a,b) for which 1 is an eigenvalue of F(z) even if F() is not defined at a and b. Results were validated through some examples.