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

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## Abstract

In defining the finiteness or infiniteness conditions of discrete spectrum of the Schrodinger operators, a fundamental understanding on \(n(1, F(\cdot ))\) 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 \(F(\cdot )\) defined in the interval of (*a*, *b*) and discontinuous points of the function \(n(1, F(\cdot ))\). Besides, the obtained relation allowed us to define the finiteness of the numbers \(z\in (a,b)\) for which 1 is an eigenvalue of *F*(*z*) even if \(F(\cdot )\) is not defined at *a* and *b*. Results were validated through some examples.

## Keywords

Fredholm analytic theorem Self-adjoint operator Discrete spectrum Essential spectrum## Mathematics Subject Classification

47Axx 35Pxx## References

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