Introduction to Mathematical Physics/Some mathematical problems and their solution/Linear evolution problems, spectral method

Spectral point of viewEdit

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 theoremsEdit

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:


  1.  
  2. matrix   is singular
  3.  


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 auto-adjoint 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 problemsEdit

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 symmetriesEdit

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 approximationEdit

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.

Headline textEdit

Variational approximationEdit

In the same way that problem

Problem:

Find   such that:


  1.  

  2.   satisfies boundary conditions on the border   of  .

can be solved by variational method, spectral problem:

Problem:

Find   and   such that:

  1.  

  2.   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).

  1. This is not obvious from a mathematical point of view (see [ma:equad:Kato66])