User:LlewS/drafts/Diff. Eq.

Supplementary Problems 3



What is the standard inner product on L2(a,b)? Provide some examples.

What is the Kronecker delta?

What is orthogonality in L2(a,b)? Provide some examples.

Evaluate the norm of sinx in the space L2(0,1). Normalize this function if its norm is not already one.

Prove or disprove that the set {sinmx} with m=1,2,... is an orthogonal set in L2(0,&pi;)?

What is Parseval's equality? What is completeness of a set of functions? What is the relationship between the completeness and Parseval's equality?

What is the Riesz-Fischer theorem in L2(a,b)?

How is a differential operator and its adjoint related to each other? Explain how the boundary conditions are used in the relationship involving an operator and its adjoint? Provide some examples.

What is a selfadjoint ordinary differential operator? Give some examples of selfadjoint differential operators as well as some examples of nonselfadjoint differential operators.

Prove that the eigenvalues of a selfadjoint operator are real.

Prove that eigenfunctions corresponding to distinct eigenvalues for a selfadjoint operator are orthogonal.

Prove that the eigenvalues of a selfadjoint operator are at most a countable set with no finite cluster point.

What is a Wronskian for a linear differential operator of order n? Provide some examples.

Illustrate the variation of constants method for the differential equation y'+p(x)y+q(x)y=g(x).

What is the differential equation satisfied by the Wronskian associated with an nth order linear differential operator?

<li>Consider the problem i&psi;'=&lambda;&psi; with &psi;(0)=&psi;(1). Identify the differential operator and discuss whether it is selfadjoint. Find the spectrum of this operator and the corresponding eigenfunctions.

<li>Consider the problem i&psi;'=&lambda;&psi; with &psi;(0)=2&psi;(1). Identify the differential operator and discuss whether it is selfadjoint. Find the spectrum of this operator and the corresponding eigenfunctions.

<li>Consider the problem &psi;'=&lambda;&psi; with &psi;(0)=&psi;(1). Identify the differential operator and discuss whether it is selfadjoint. Find the spectrum of this operator and the corresponding eigenfunctions.

<li>Consider the problem -&psi;=&lambda;&psi; with &psi;(0)=&psi;''(1). Identify the differential operator and discuss whether it is selfadjoint. Find the spectrum of this operator and the corresponding eigenfunctions.

<li>Consider the problem -&psi;=&lambda;&psi; with &psi;(0)=''&psi;(1). Identify the differential operator and discuss whether it is selfadjoint. Find the spectrum of this operator and the corresponding eigenfunctions.

<li>Consider the problem -&psi;=&lambda;&psi; with &psi;(0)=0 and &psi;''(1)=0. Identify the differential operator and discuss whether it is selfadjoint. Find the spectrum of this operator and the corresponding eigenfunctions.

<li>Consider the problem -&psi;=&lambda;&psi; with &psi;(0)=0 and ''&psi;(1)=0. Identify the differential operator and discuss whether it is selfadjoint. Find the spectrum of this operator and the corresponding eigenfunctions.

<li>Consider the problem -&psi;=&lambda;&psi; with &psi;(0)+&psi;(0)=0 and ''&psi;(1)=0. Identify the differential operator and discuss whether it is selfadjoint. Find the spectrum of this operator and the corresponding eigenfunctions.

<li>Consider -d2&psi;/dx2+q(x)&psi;=&lambda;&psi; with a real valued potential q belonging to C2(-&infin;,+&infin;) such that q and q' vanish at infinity. Show that this differential operator is selfadjoint.

<li>Consider the differential operator in the previous problem where the relevant functions are defined on (0,+&infin;) instead of (-&infin;,+&infin;). Is the operator selfadjoint if the boundary condition at the origin is &psi;(0)=0? Is it selfadjoint if the boundary condition at the origin is &psi;'(0)=0? Is it selfadjoint if the boundary condition at the origin is (sin&theta;)&psi;'(0)</i>+(cos&theta;)&psi;(0)</i>=0 for any real &theta;?

<li>Consider the first order system &xi;'=-i&lambda;&xi;+q(x)&eta; and

&eta;'=i&lambda;&eta;+r(x)&xi;, where q and r are real valued and vanish at infinity. Discuss some special cases such as q=r and q=-r and determine when the corresponding operator is selfadjoint.

<li>Consider the system in the previous problem where q and r are now allowed to be complex valued.

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