Compact Operators and the Schrödinger Equation Page: 3
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CHAPTER 2
COMPACT OPERATORS AND THE SCHRODINGER EQUATION
In this chapter we discuss the development of compact operators which, as we will see,
play a key role in finding eigenvalues and eigenvectors of H. H : X -- Y will be given by
Hu = -u" + qu, where q is continuous on [a, b]. We will specify when q is non-negative.
This section consists of a discussion showing that the Hamiltonian operator with zero
boundary conditions has a compact, symmetric, positive, and continuous inverse T with
domain all of Y.
2.1. Compact Operators
To show that H has an inverse consider the following
DEFINITION 2.1. Let n be a positive integer. Define L(R") to be the space of linear trans-
formations from _ -, q.
LEMMA 2.2. Let Q : [a, b] - L(Rn) be continuous. There exists M : [a, b] - L(R") so that
M' = -MQ and M- (t) exists for all t E [a, b].
PROOF. Let
all(t) ... al,n(t) 51,(t) ... 51,n(t)
Q(t) = . and define M(t) =
\an,,l(t) ... an,n(t) Ybn,l(t) ... bn,n(t)
bi4j(t) will be determined shortly . Then
(t)Q(t) (ta(t (t)
bn,i t)ai,1t) * * i =l1 bn,i(t)ai,n/(t)
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Kazemi, Parimah. Compact Operators and the Schrödinger Equation, thesis, December 2006; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc5453/m1/7/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .