What are Quasi Eigenvalues associates with an Quantum Operator?

Question

I know all the basic maths to do with Eigenvalues and operators and such. I recently came across the concept of Quasi Eigenfunction/ Eigenvalue and I cannot find a definition for it, So any contribution is appreciated thanks

Answer 1

Try: -

http://en.wikipedia.org/wiki/Eigenvalue_...

http://en.wikipedia.org/wiki/Nonlinear_e...

Wikipedia, adds, 'For transformations on real vector spaces, the coefficients of the characteristic polynomial are all real. However, the roots are not necessarily real; they may well be complex numbers, or a mixture of real and complex numbers. For example, a matrix representing a planar rotation of 45 degrees will not leave any non-zero vector pointing in the same direction. Over a complex vector space, the fundamental theorem of algebra guarantees that the characteristic polynomial has at least one root, and thus the linear transformation has at least one eigenvalue.

As well as distinct roots, the characteristic equation may also have repeated roots. However, having repeated roots does not imply there are multiple distinct (i.e. linearly independent) eigenvectors with that eigenvalue. The algebraic multiplicity of an eigenvalue is defined as the multiplicity of the corresponding root of the characteristic polynomial. The geometric multiplicity of an eigenvalue is defined as the dimension of the associated eigenspace, i.e. number of linearly independent eigenvectors with that eigenvalue.

Over a complex space, the sum of the algebraic multiplicities will equal the dimension of the vector space, but the sum of the geometric multiplicities may be smaller. In a sense, then it is possible that there may not be sufficient eigenvectors to span the entire space. This is intimately related to the question of whether a given matrix may be diagonalized by a suitable choice of coordinates.'

http://en.wikipedia.org/wiki/Eigenvalue

Answer 2

Huh, new one to me also.  I taught linear algebra at the Masters level, too.

I can only speculate.  Eigen means same or similar in deutsch.  The eigenvalue supposedly satisfies Lx = Ax (sorry can't do lambda).  I can only infer from "quasi" that a quasi eigenvalue only partially satisfies that relationship.  That is a quasi eigenvalue is only partially the same as Ax, or, maybe, Lx ~ Ax and not equal at all.

I'd start a little browsing on the web; it contains a wealth of info.