Applied Spectral Theory: Deriving Math Techniques in EE/Physics

[thegreenlaser]

I'd like to see if/how the spectral theorem(s) can be used to derive a range of mathematical techniques used in electrical engineering/physics:

• (Generalized) Fourier series
• Fourier transforms
• Laplace transforms
• Green's functions
• Sturm-Liouville problem solution method
• Dirac notation QM
• etc.

Are there any books that go through this sort of thing? I've read Kreyszig's functional analysis book, which gave me a decent introduction to the subject, but I felt like it wasn't quite deep enough. I still can't fully see how to get from the spectral theorem to all the things I listed above. Any help is appreciated...

 

[Greg Bernhardt]

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

 

[thegreenlaser]

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

Not really, unfortunately. I'm still looking for books which show me how to use the spectral theorem to solve problems. Some do it for compact and compact resolvent operators, but I'm more interested in operators which have a non-empty continuous spectrum.

 

[Daverz]

This book has very good coverage of most of these topics:

https://www.amazon.com/dp/0486664449/?tag=pfamazon01-20
http://store.doverpublications.com/0486664449.html

 

[jasonRF]

This book has very good coverage of most of these topics:

https://www.amazon.com/dp/0486664449/?tag=pfamazon01-20
http://store.doverpublications.com/0486664449.html

I have read most of this book and worked ~1/3 of the exercises, and agree that it is pretty good. It does derive Fourier and Laplace transforms from sturm-liouville problems with continuous spectra. The book is worth a look but I think it is somewhat dated, and uses some less common conventions. A more modern approach is by Stakgold,
https://www.amazon.com/dp/0471610224/?tag=pfamazon01-20
which is also worth a look. I'm guessing personal preference would dictate which one a given person would prefer. I went with Friedman because it was shorter so would require less investment of time to work through linearly (edit: Stakgold is also at a slightly higher level, so Friedman was a little easier to dip into on my free time).

Hopefully thegreenlaser has access to a library to check these out, although used copies of both can be found for little money. Good luck!

jason

 

[thegreenlaser]

Thanks guys! My university library has both of those books, so I'll check them out in the next little while.

 

[Daverz]

I have read most of this book and worked ~1/3 of the exercises, and agree that it is pretty good. It does derive Fourier and Laplace transforms from sturm-liouville problems with continuous spectra. The book is worth a look but I think it is somewhat dated, and uses some less common conventions. A more modern approach is by Stakgold,

Thanks for the comment. I chose the Friedman book based mostly on price and easy availability.