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Applied Mathematical Methods in Theoretical Physics

Michio Masujima

Chapter 5

Hilbert–Schmidt Theory of Symmetric Kernel - all with Video Answers

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Chapter Questions

Problem 1

Find an upper bound and a lower bound for the first eigenvalue of
$$
K(x, y)= \begin{cases}(1-x) y, & 0 \leq y \leq x \leq 1, \\ (1-y) x, & 0 \leq x \leq y \leq 1 .\end{cases}
$$

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Problem 2

Consider the Bessel equation
$$
\left(x u^{\prime}\right)^{\prime}+\lambda x u=0,
$$
with the boundary conditions
$$
u^{\prime}(0)=u(1)=0 .
$$
Transform this differential equation into an integral equation and find approximately the lowest eigenvalue.

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Problem 3

Obtain an upper limit for the lowest eigenvalue of
$$
\nabla^2 u+\lambda r u=0,
$$
where, in three dimensions,
$$
0<r<a \text {, and } u=0 \text { on } r=a .
$$

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Problem 4

Consider the Gaussian kernel $K(x, y)$ given by
$$
K(x, y)=e^{-x^2-y^2}, \quad-\infty<x, y<+\infty .
$$
a) Find the eigenvalues and the eigenfunctions of this kernel.
b) Verify the Hilbert-Schmidt expansion of this kernel.
c) By calculating $A_2$ and $A_4$, obtain the exact lowest eigenvalue.
d) Solve the integro-differential equation
$$
\frac{\partial}{\partial t} \phi(x, t)=\int_{-\infty}^{+\infty} K(x, y) \phi(y, t) d y, \quad \text { with } \quad \phi(x, 0)=f(x) .
$$

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Problem 5

Show that the boundary condition (5.5.2) of the Sturm-Liouville system can be replaced by
$$
\alpha_1 \phi(0)+\alpha_2 \phi^{\prime}(0)=0 \quad \text { and } \quad \beta_1 \phi(h)+\beta_2 \phi^{\prime}(h)=0
$$
where $\alpha_1, \alpha_2, \beta_1$ and $\beta_2$ are some constants and the corresponding boundary condition (5.5.5) on $G(x, y)$ is replaced accordingly.

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Problem 6

Verify the Hilbert-Schmidt expansion for the case of Direction 1 in Section 5.6, when the kernel $K(x, y)$ is self-adjoint and square-integrable.

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01:00

Problem 7

Reproduce all the results of Section 5.7 with the Green's function $G(x, y ; \lambda)$ defined by
$$
\left(L_x-\lambda r(x)\right) G(x, y ; \lambda)=\delta(x-y),
$$
by the weight function,
$$
r(x)>0 \quad \text { on } \quad x \in[0, h] .
$$

Raj Bala
Raj Bala
Numerade Educator

Problem 8

Consider the eigenvalue problem of the fourth-order ordinary differential equation of the form,
$$
\left(\frac{d^4}{d x^4}+1\right) \phi(x)=-\lambda x \phi(x), \quad 0<x<1,
$$
with the boundary conditions,
$$
\begin{aligned}
& \phi(0)=\phi^{\prime}(0)=0, \\
& \phi(1)=\phi^{\prime}(1)=0 .
\end{aligned}
$$
Do the eigenfunctions form a complete set?

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Problem 9

Show that, if $\bar{\lambda}$ is an eigenvalue of the symmetric kernel $K(x, y)$, the inhomogeneous Fredholm integral equation of the second kind,
$$
\phi(x)=f(x)+\tilde{\lambda} \int_a^b K(x, y) \phi(y) d y, \quad a \leq x \leq b,
$$
has no solution, unless the inhomogeneous term $f(x)$ is orthogonal to all of the eigenfunctions $\phi(x)$ corresponding to the eigenvalue $\tilde{\lambda}$.

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Problem 10

Consider the integral equation,
$$
\phi(x)=f(x)+\lambda \int_0^1 \sin ^2[\pi(x-y)] \phi(y) d y, \quad 0 \leq x \leq 1 .
$$
a) Solve the homogeneous equation by setting
$$
f(x)=0 .
$$
Determine all the eigenfunctions and the eigenvalues. What is the spectral representation of the kernel?
b) Find the resolvent kernel of this equation.
c) Is there a solution to the given inhomogeneous integral equation when
$$
f(x)=\exp [i m \pi x], \quad m \text { integer, }
$$
and $\lambda=2$ ?

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Problem 11

Consider the kernel of the Fredholm integral equation of the second kind, which is given by
$$
K(x, y)= \begin{cases}3, & 0 \leq y<x \leq 1, \\ 2, & 0 \leq x<y \leq 1 .\end{cases}
$$
a) Find the eigenfunctions $\phi_n(x)$ and the corresponding eigenvalues $\lambda_n$ of the kernel.
b) Is $K(x, y)$ symmetric? Determine the transposed kernel $K^T(x, y)$, and find its eigenfunctions $\psi_n(x)$ and the corresponding eigenvalues $\lambda_n$.
c) Show by an explicit calculation that any $\phi_n(x)$ is orthogonal to any $\psi_m(x)$ if $m \neq n$.
d) Derive the spectral representation of $K(x, y)$ in terms of $\phi_n(x)$ and $\psi_n(x)$.

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Problem 12

Consider the Fredholm integral equation of the second kind,
$$
\phi(x)=f(x)+\lambda \int_0^1\left[\frac{1}{2}(x+y)-\frac{1}{2}|x-y|\right] \phi(y) d y, \quad 0 \leq x \leq 1 .
$$
a) Find all non-trivial solutions $\phi_n(x)$ and corresponding eigenvalues $\lambda_n$ for $f(x) \equiv 0$.
b) For the original inhomogeneous equation $(f(x) \neq 0)$, will the iteration series converge?
c) Evaluate the series $\sum_n \lambda_n^{-2}$ by using an appropriate integral.

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Problem 13

If $|h|<1$, find the non-trivial solutions of the homogeneous integral equation,
$$
\phi(x)=\frac{\lambda}{2 \pi} \int_{-\pi}^\pi \frac{1-h^2}{1-2 h \cos (x-y)+h^2} \phi(y) d y .
$$
Evaluate the corresponding values of the parameter $\lambda$.

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Problem 14

If $|h|<1$, find the solution of the integral equation,
$$
f(x)=\frac{\lambda}{2 \pi} \int_{-\pi}^\pi \frac{1-h^2}{1-2 h \cos (x-y)+h^2} \phi(y) d y,
$$
where $f(x)$ is the periodic and square-integrable known function.

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