Sum over Eigenvalues in Spectral Theory

[lokofer]

"sum" over Eigenvalues...

Is there any mathematical meaning or it's used in Calculus or other 2branch" of mathematics de expression:

$$\sum_{n} e^{-u\lambda (n) }$$

where every "lambda" is just an Eigenvalue of a linear operator:

$$L[y]=-\lambda _{n} y$$

We Physicist know it as the "partition function" and is used in Statistical Mechanics and Quantum physics...in first approximmation:

$$\sum_{n} e^{-u\lambda (n) } = \iint dxdpe^{-uH}$$

Where all the "eigenvalues" are positive...here i would like to hear if this "approach" using integrals would be rigorous from a math point of view...

 

[Clausius2]

Is there any mathematical meaning or it's used in Calculus or other 2branch" of mathematics de expression:

$$\sum_{n} e^{-u\lambda (n) }$$

where every "lambda" is just an Eigenvalue of a linear operator:

$$L[y]=-\lambda _{n} y$$

:

I think it is rigorous as far as $$u$$ is small enough. That's what happens with the constant that multiplies the square of the principal quantum number when summing for obtaining the partition function for the translational motion, isn't it?.

We Physicist know it as the "partition function" and is used in Statistical Mechanics and Quantum physics...in first approximmation:

We Engineers call it the "partition function" too. :)

 

[lokofer]

Uh..sorry then "Clausius"... perhaps you as an engineer have heard about "Semiclassical approach" in Physics so you approximate the series by means of an integral, to calculate "Thermodinamic" entities...

 

[matt grime]

It's spectral theory as done by analysts, say, or measure theorists. (Lots of things are called spectral and are unrelated, so be careful).