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28 May 2024: the solution of rapidly oscillating ordinary differential equations, computation of spectra of Fredholm operators with highly oscillatory kernels and rapid approximation of functions by Birkhoff series.

30 Aug 2019: b) Show that an operator is Fredholm if and only if it is invertible modulo thecompact operators. ... c) Show that the Fredholm operators are open in B(H) and that the index isnorm continuous.

17 Jun 2019: Define a Fredholm operator T , on a Hilbert space H, and define the index of T. ... not using anytheorems about Fredholm operators) that, for each k Z, there is a Fredholm operator Son H with ind S = k.

30 Aug 2019: PAPER 10. ANALYSIS OF OPERATORS. Attempt THREE questions. There are EIGHT questions in total. ... 5 Write an essay on the index of Fredholm operators. You should include in youraccount a discussion of Toeplitz operators on the circle.

17 Jun 2019: Define what it meansto say that T is (i) compact, and (ii) Fredholm. ... H =p2. 2m+. 12mω2x2,. in terms of creation and annihilation operators, defined by.

17 Jun 2019: eiu2du =. πeiπ/4.]. Part II 2002. 26. A3/17 Mathematical Methods. (i) State the Fredholm alternative for Fredholm integral equations of the second kind.

17 Jun 2019: ii) The Lie algebra su(2) of generators of SU(2) is spanned by the operators{J+,J,J3} satisfying the commutator algebra [J+,J] = 2J3 and [J3,J] = J.