Introduction to Mathematical Physics/Some mathematical problems and their solution/Linear evolution problems, spectral method
Spectral point of view
[edit  edit source]Spectral method is used to solve linear evolution problems of type Problem probevollin. Quantum mechanics (see chapters chapmq and chapproncorps ) supplies beautiful spectral problems {\it via} Schr\"odinger equation. Eigenvalues of the linear operator considered (the hamiltonian) are interpreted as energies associated to states (the eigenfunctions of the hamiltonian). Electromagnetism leads also to spectral problems (cavity modes).
Spectral methods consists in defining first the space where the operator of problem probevollin acts and in providing it with the Hilbert space structure. Functions that verify:
are then seeked. Once eigenfunctions are found, the problem is reduced to integration of an ordinary differential equations (diagonal) system.
The following problem is a particular case of linear evolution problem \index{response (linear)} (one speaks about linear response problem)
Problem:
Find such that:
where is a linear diagonalisable operator and is a linear operator "small" with respect to .
This problem can be tackled by using a spectral method. Section secreplinmq presents an example of linear response in quantum mechanics.
Some spectral analysis theorems
[edit  edit source]In this section, some results on the spectral analysis of a linear operator are presented. Demonstration are given when is a linear operator acting from a finite dimension space to itself. Infinite dimension case is treated in specialized books (see for instance ([ma:equad:Dautray5])). let be an operator acting on . The spectral problem associated to is:
Problem:
Find non zero vectors (called eigenvectors) and numbers (called eigenvalues) such that:
Here is a fundamental theorem:
Theorem:
Following conditions are equivalent:
 matrix is singular
A matrix is said diagonalisable if it exists a basis in which it has a diagonal form ([ma:algeb:Strang76]).
Theorem:
If a squared matrix of dimension has eigenvectors linearly independent, then is diagonalisable. Moreover, if those vectors are chosen as columns of a matrix , then:
Proof:
Let us write vectors as column of matrix and let let us calculate :
Matrix is invertible since vectors are supposed linearly independent, thus:
Remark: LABEL remmatrindep If a matrix has distinct eigenvalues then its eigenvectors are linearly independent.
Let us assume that space is a Hilbert space equipped by the scalar product .
Definition:
Operator adjoint of is by definition defined by:
Definition:
An autoadjoint operator is an operator such that
Theorem:
For each hermitic operator , there exists at least one basis constituted by orthonormal eigenvectors. is diagonal in this basis and diagonal elements are eigenvalues.
Proof:
Consider a space of dimension . Let be the eigenvector associated to eigenvalue of . Let us consider a basis the space ( direct sum any basis of ). In this basis:
The first column of is image of . Now, is hermitical thus:
By recurrence, property is prooved..
Theorem:
Eigenvalues of an hermitic operator are real.
Proof:
Consider the spectral equation:
Multiplying it by , one obtains:
Complex conjugated equation of uAu is:
being real and , one has:
Theorem:
Two eigenvectors and associated to two distinct eigenvalues and of an hermitic operator are orthogonal.
Proof:
By definition:
Thus:
The difference between previous two equations implies:
which implies the result.
Let us now presents some methods and tips to solve spectral problems.
chapresospec
Solving spectral problems
[edit  edit source]The fundamental step for solving linear evolution problems by doing the spectral method is the spectral analysis of the linear operator involved. It can be done numerically, but two cases are favourable to do the spectral analysis by hand: case where there are symmetries, and case where a perturbative approach is possible.
Using symmetries
[edit  edit source]Using of symmetries rely on the following fundamental theorem:
Theorem:
If operator commutes with an operator , then eigenvectors of are also eigenvectors of .
Proof is given in appendix chapgroupes. Applications of rotation invariance are presented at section secpotcent. Bloch's theorem deals with translation invariance (see theorem theobloch at section sectheobloch).
Perturbative approximation
[edit  edit source]A perturbative approach can be considered each time operator to diagonalize can be considered as a sum of an operator whose spectral analysis is known and of an operator small with respect to . The problem to be solved is then the following:\index{perturbation method}
bod
Introducing the parameter , it is assumed that can be expanded as:
Let us admit^{[1]} that the eigenvectors can be expanded in : For the i eigenvector:
hyph
Equation ( bod) defines eigenvector, only to a factor. Indeed, if is solution, then is also solution. Let us fix the norm of the eigenvectors to . Phase can also be chosen. We impose that phase of vector is the phase of vector . Approximated vectors and should be exactly orthogonal.
Egalating coefficients of , one gets:
eqortper
Approximated eigenvectors are imposed to be exactly normed and real:
Equalating coefficients in with in product , one gets:
Substituting those expansions into spectral equation bod and equalating coefficients of successive powers of yields to:
oivj
Projecting previous equations onto eigenvectors at zero order, and using conditions eqortper, successive corrections to eigenvectors and eigenvalues are obtained.
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[edit  edit source]Variational approximation
[edit  edit source]In the same way that problem
Problem:
Find such that:

 satisfies boundary conditions on the border of .
can be solved by variational method, spectral problem:
Problem:
Find and such that:

 satisfies boundary conditions on the border of .
can also be solved by variational methods. In case where is self adjoint and is zero (quantum mechanics case), problem can be reduced to a minimization problem. In particular, one can show that:
Theorem:
The eigenvector with lowest energy of self adjoint operator is solution of problem: Find normed such that:
where .
Eigenvalue associated to is .
Demonstration is given in ([ph:mecaq:Cohen73],[ph:mecaq:Pauling60]). Practically, a family of vectors of is chosen and one hopes that eigenvector is well approximated by some linear combination of those vectors:
Solving minimization problem is equivalent to finding coefficients . At chapter chapproncorps, we will see several examples of good choices of families .
Remark: In variational calculations, as well as in perturbative calculations, symmetries should be used each time they occur to simplify solving of spectral problems (see chapter chapproncorps).
 ↑ This is not obvious from a mathematical point of view (see [ma:equad:Kato66])