- Ability to perform spectral decompositions of linear operators and matrices.
- Ability to calculate spectral projections and understand applications in functional analysis.
- Understanding spectral properties of integral operators and differential operators.
- Ability to apply spectral theory to concrete problems in quantum mechanics or digital signal processing.
- Developing an understanding of the basics of distribution theory and its various applications.
- Ability to formulate and solve differential equations using distributions.
- Ability to apply distributions in quantum mechanics and other areas of mathematical physics.
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Spectrum of linear operators: eigenvalues, eigenvectors, spectral radius, resolvent.
Spectral properties of compact operators.
Spectral theory of bounded and compact Hermitian operators.
Functional analysis: Hilbert spaces, continuous functionals, duality, projections.
Applications of spectral theory to mathematical physics.
Definition and properties of distributions.
Basic operations with distributions.
Weak derivatives and Antiderivatives of distributions.
Examples of distributions (Dirac sequences) and their applications.
Convergence of distributions.
Tempered distributions, Fourier transform of distributions.
Applications in PDE and physics.
Schwartz space: definition, properties, and applications.
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