The spectrum of a bounded differential equation

AI Thread Summary
The discussion centers on the relationship between the spectrum of a bounded differential operator and the operator itself, particularly in finite versus infinite dimensions. In finite dimensions, knowing the eigenvalues and their multiplicities allows for the identification of the operator up to unitary equivalence, making the spectrum a valuable invariant. However, in infinite dimensions, such as with bounded operators, different operators can share the same spectrum, complicating the identification process. The conversation highlights that additional information, like the action of the operator on a dense subset of the solution space, may be necessary to reconstruct the operator. The context of differential equations emphasizes the importance of self-adjoint operators and their properties.
greentea28a
Messages
11
Reaction score
0
is it possible to work backwards from a spectrum to which operator?
 
Mathematics news on Phys.org
Think about finite dimensions. If I give you all the eigenvalues, you still need to know what the eigenvectors are before you know the linear transformation.
 
homeomorphic said:
Think about finite dimensions. If I give you all the eigenvalues, you still need to know what the eigenvectors are before you know the linear transformation.

But in that case you do know the transformation up to unitary equivalence (well, if you also know the multiplicities of the eigenvalues)! So the spectrum forms a very nice invariant in finite dimensions (called a unitary invariant).

Sadly enough, the same is not true for infinite dimensions, not even for bounded operators. For example, multiplication by ##x## on ##L^2([0,1])## and an operator with a complete set of eigenfunctions whose eigenvalues are dense in ##[0,1]##. These two operators can't be much different, but they do share the same spectrum.
Finding a stronger unitary invariant can be done and yields the commutative multiplicity theorem. I refer to the beautiful book by Reed and Simon for a discussion.
 
But in that case you do know the transformation up to unitary equivalence (well, if you also know the multiplicities of the eigenvalues)! So the spectrum forms a very nice invariant in finite dimensions (called a unitary invariant).

Well, if you assume it's Hermitian, right? Then all your eigenvectors are orthogonal.

The context was that of differential equations. That means the operator will be self-adjoint if your boundary conditions make it so, but not always.

I assume with enough extra information, you can eventually get back the operator. For example, if you knew how the operator acted on dense subset of the solution space, you should be good to go.
 
homeomorphic said:
Well, if you assume it's Hermitian, right? Then all your eigenvectors are orthogonal.

Yes, I thought that was given, but apparently I was mistaken.
 

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Thread 'Non-orthogonal bases'

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
View full post »

Similar threads

I Partial Differential Equation solved using Products
Replies
2
Views
2K
B A differential equation, or an identity?
Replies
6
Views
2K
MHB Solve Differential Eq: xe^-1/(k+e^-1) for x, k, t
Replies
2
Views
2K
A Spectrum of an algebra (Alain Connes paper)
Replies
19
Views
3K
A Spectrum of the Liouville Operator
Replies
2
Views
1K
Blue or green solar thermal collector?
Replies
2
Views
479
I Fourier smoothing and Savitzky-Golay filtering
Replies
11
Views
4K
I Is there a name for this sort of differential equation?
Replies
2
Views
1K
Recommendation Needed for a Spectrum Analyzer
Replies
46
Views
3K
I Electromagnetic Absorption in the Ocean
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top