Isospectral Property
It can then be shown that the eigenvalues and more generally the spectrum of L are independent of t. The matrices/operators L are said to be isospectral as varies.
The core observation is that the matrices are all similar by virtue of
where is the solution of the Cauchy problem
where I denotes the identity matrix. Note that if L(t) is self-adjoint and P(t) is skew-adjoint, then U(t,s) will be unitary.
In other words, to solve the eigenvalue problem Lψ = λψ at time t, it is possible to solve the same problem at time 0 where L is generally known better, and to propagate the solution with the following formulas:
- (no change in spectrum)
Read more about this topic: Lax Pair
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