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18.307: Integral Equations M.I.T. Department of Mathematics
Spring 2006
Homework 6 Due: Wednesday, 04/12/06
�−ixy u18. (Prob. 4.7 in text by M. Masujima.) Consider the equation u(x) = � −� dy e (y), −� <
x < �, i.e., with a kernel that is not square integrable. Note that solutions to this equation
are essentially “Fourier transforms of themselves.”
(a) Show that there are only 4 eigenvalues � of the kernel e−ixy . What are they?
(b) Show by an explicit calculation that the functions un(x) = e−x2 /2 Hn(x), where Hn(x) =
2
(−1)n ex2 dn e−xare Hermite polynomials, are eigenfunctions with corresponding eigenvalues
dxn
(−i)n/�2� (n= 0, 1, 2, ...). Hence, the eigenvalues of part (a) are infinitely degenerate, i.e.,
there is a infinite number of independent eigenfunctions for each of them.
(c) Using the result in (b) and the fact that un are known to form a complete set, in some
sense, show that any square integrable solution is of the form u(x) = f(x) + C f
˜(x), where f(x)
is an (arbitrary) odd or even, square integrable function with Fourier transform f
˜(k), and C is
a suitable constant. Evaluate C and relate its value(s) to the eigenvalues found in part (a).
(d) From (c), construct a solution to the original integral equation by taking f(x) = e−ax2 /2
(Gaussian, a > 0).
19. (Prob. 5.11 in text by M. Masujima.) Consider the kernel K of a 2nd-kind Fredholm equation,
which is given by
K(x, y) = 3, 0 � y < x � 1,
2, 0 � x < y � 1.
(a) Find the kernel eigenfunctions un and corresponding eigenvalues �n.
(b) Is K symmetric? Determine the transpose kernel KT , and find its eigenfunctions vn with
corresponding eigenvalues �n.
(c) Show by an explicit calculation that any un is orthogonal to any vm if m= n.≤
(d) Derive the spectral representation of K(x, y) in terms of un and vn
20. (Probs. 5.3 & 5.4, Chap. 6 in text by I. Stakgold.) The energy levels E of an atom that expe-
riences non-local interactions with other atoms in a dilute gas are described by the eigenvalue
problem Au = E u, where A is the integrodifferential operator defined by
� 1
d2u
Au = + dy xy u(y), 0 <x<1,
2
− dx 0
and u(x) is any function that has a second derivative continuous for 0 < x < 1 and satisfies
the boundary conditions u(0) = 0 and u�(1) = 0. We denote the space of such functions as DA.
Beware: The constant E above multiplies the u outside the integral.
(a) Show that all eigenvalues En of this problem are real and positive, and that eigenfunctions
corresponding to different eigenvalues are orthogonal. Justify your answer. Why is it
sufficient to restrict ourselves to real eigenfunctions?
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