751 a what is the determinant of the translation operator of problem 7

Question

7.51 (a) What is the determinant of the translation operator of problem 7.27? 7.27 On the vector space of functions of x, define the translation operator Taf = g means g(x) = f(x - a) This picks up a function and moves it by a to the right. (a) Pick a simple example function f and test this definition graphically to verify that it does what I said. (b) On the space of cubic polynomials and using a basis of your choice, find the components of this operator. (c) Square the resulting matrix and verify that the result is as it should be. (d) What is the inverse of the matrix? (You should be able to guess the answer and then verify it. Or you can work out the inverse the traditional way.) (e) What if the parameter a is huge? Interpret some of the components of this first matrix and show why they are clearly correct. (If they are.) (f) What is the determinant of this operator? (g) What are the eigenvectors and eigenvalues of this operator? (b) What is the determinant of d/dx on the vector space of problem 7.26? 7.26 The set of Hermite polynomials starts out as H_{0}=\mathbf{1}, \quad H_{1}=2 x_{3} \quad H_{2}=4 x^{2}-2, \quad H_{2}=8 x^{2}-12 x_{2} \quad H_{4}=16 x^{4}-44 x^{2}+12 (a) For the vector space of cubic polynomials in x, choose a basis of Hermite polynomials and compute the matrix of components of the differentiation operator, d/dx. (b) Compute the components of the operator d^2/dx^2 and show the relation between this matrix and the preceding one.