Question

Find the eigenvalues and eigenvectors of the operator −id/dx acting in the vector space of differentiable functions c^1(−[infinity],[infinity]).

A) λ = 1, v = e^x
B) λ = -i, v = e^(-ix)
C) λ = 0, v = cos(x)
D) λ = 2i, v = sin(2x)

Answer 1

The eigenvalues are λ = -i and the eigenvector is v(x) = e^(-ix).

Step-by-step explanation:

To find the eigenvalues and eigenvectors of the operator −id/dx acting in the vector space of differentiable functions c^1(−∞,∞), we can start by looking for solutions to the equation −id/dx f(x) = λf(x). Let's denote the eigenvalues as λ and the corresponding eigenvectors as v(x).

If we differentiate the eigenvector v(x) with respect to x and substitute into the equation, we get -iv'(x) = λv(x). This is a first-order ordinary differential equation, and its solutions can be written as v(x) = Ce^(-ix), where C is a constant and i is the imaginary unit.

So the eigenvalues are given by λ = -i, and the corresponding eigenvector is v(x) = e^(-ix). Therefore, the correct answer is B) λ = -i, v = e^(-ix).