What Conditions Make This Differential Operator Self-Adjoint?

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In summary, a self-adjoint operator is a linear operator on a vector space that is equal to its own adjoint and has many useful properties. They are important in areas such as differential equations, spectral theory, and quantum mechanics. A self-adjoint operator is related to a Hermitian matrix and can only exist on a complex inner product space. In quantum mechanics, self-adjoint operators represent physical observables and are used to describe the measurement process.
  • #1
MarkovMarakov
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Homework Statement



When is the following operator self-adjoint? I am looking for constraints on [itex]f_n[/itex]'s s.t. The operator below becomes self-adjoint.

[itex]O:= \sum_{n=0}^4 f_n(x){d^n\over dx^n}[/itex] subjected to boundary conditions [itex]y(0)=y'(0)=y(1)=y'(1)=0[/itex] and where [itex]f_n[/itex]'s are real functions.

Thanks.

Homework Equations


See above.

The Attempt at a Solution


Totally clueless. Please help.
 
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  • #2
Well, what is the definition of "self adjoint"? That would be a good place to start.
 

FAQ: What Conditions Make This Differential Operator Self-Adjoint?

What is a self-adjoint operator?

A self-adjoint operator is a linear operator on a vector space that is equal to its own adjoint. This means that the operator and its adjoint have the same matrix representation, making it a symmetric matrix.

What is the significance of self-adjoint operators in mathematics?

Self-adjoint operators are important in mathematics because they have many useful properties. They are used in the study of differential equations, spectral theory, and quantum mechanics, to name a few areas. They also have real eigenvalues and orthogonal eigenvectors, making them easier to solve and analyze.

How are self-adjoint operators related to Hermitian matrices?

A self-adjoint operator is essentially the linear operator version of a Hermitian matrix. Both have the property of being equal to their own conjugate transpose. In other words, a self-adjoint operator is the abstract representation of a Hermitian matrix in linear algebra.

Can any linear operator be self-adjoint?

No, not all linear operators are self-adjoint. Only operators on a complex inner product space can be self-adjoint. This means that the vector space must have a defined inner product and the operator must preserve this inner product in order to be self-adjoint.

How are self-adjoint operators used in quantum mechanics?

In quantum mechanics, self-adjoint operators represent physical observables such as energy, momentum, and spin. They have real eigenvalues which correspond to the possible outcomes of a measurement, and their eigenvectors represent the states of the system. The measurement process in quantum mechanics can be described using self-adjoint operators and their eigenvalues.

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