To every eigenvaluebof a sturm liouvillesystem there corresponds only one linearlyindependent eigenfunction.

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Answered: To every eigenvaluebof a sturm…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4. For the given matrix 1 3 1 3 -1 3 1 2 a) Use the direct method to find all the eigenvalues and corresponding eigen- vectors. (b) Use the Power Method to find the most dominant eigenvalue and the cor- responding eigenvector. (c) Use the Inverse Power method to find an eigenvalue. (d) Use the Wielandt Deflation to approximate second most dominant eigen- value. -2
Consider a four-state system whose Hamiltonian is given by Vo +A 8Vo H 3Vo -2) 7Vo -2) where Vo is a real-valued constant and A is a real-valued parameter. (c) Now decompose this Hamiltonian into an unperturbed Hamiltonian, Ho, and a perturbing piece, H'. Solve the unperturbed system, that is, find the (four) eigenvalues and (four) eigenvectors of your unperturbed Hamiltonion, Ho. (d) Using nondegenerate perturbation theory, find the first- and second-order corrections to the unperturbed energies in the presence of the perturbing Hamiltonian, H'. (e) Solve the system exactly, by diagonalising the full Hamiltonian, H, to find the exact eigen- values of the full Hamiltonian, H. (f) Expand your solutions to part (e) to the appropriate order in A and compare these results to your results from nondegenerate perturbation theory in part (d). What condition must you impose on A to ensure that the leading energy correction from perturbation theory is a good approximation to the exact result?
If v1,v2, and v3 are linearly independent eigenvectors of A corresponding to the eigenvalue λ, and c1,c2, and c3 are scalars (not all zero), show that c1v1+c2v2+c3v3 is also an eigenvector of A corresponding to the eigenvalue λ.
(b) Let C³ - - 1²: B = 1 c² C-3 0 0 с -C² ³+1 " where ce R. How many eigenvalues does B have and with what ge- ometric multiplicities? Without knowing c, is it possible to determine whether B is diagonalizable? Explain your answers.
Mark as TRUE or FALSE. You will need to give an explanation. (a) If v1, v2 ..., Vn are linearly independent, then v1, V2 are linearly independent. (b) If v and w are two eigenvectors for the same eigenvalue c, then the sum v + w is also an eigenvector for the eigenvalue c. (c) Any 2 x 2 matrix has a real eigenvalue. (d) If v is an eigenvector for a matrix A and a matrix B then v is an eigenvector for A+ B.
9. Determine which of the following vectors 1 -1 is an eigenvector of the matrix 1 1 1 1 -1 1 -1 W = -1 -1 1 -1 -1 1 and find the corresponding eigenvalue.
1 3 is diagonalizable. Its eigenvectors are: 4 2 The matrix A = 3. 2 4 3 4
In quantum mechanics, when two states with energies ₁ and E2 mix, the resulting states have different energies E and E, which are the eigenvalues of the matrix H = V (EK). V Choosing to be the state with higher energy, derive expressions for E + E, and E - E in terms of E1, E2 and V.

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To every eigenvaluebof a sturm liouville system there corresponds only one linearly independent eigenfunction.