Is this licitthen why?(functional analysis question)

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In summary, the conversation discusses the possibility of using a sum-integral approximation in the context of a Hamiltonian operator and its energies (eigenvalues). The speaker also mentions using functional analysis to justify this approach. Other participants in the conversation recommend looking up spectral analysis and using a textbook or online resources to learn more about the topic. The conversation also addresses the issue of convergence and recommends using a library for further resources.
  • #1
lokofer
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I think i put this question before...:confused: :confused: but i can't be very sure.. let's suppose we have a Hamiltonian operator:

[tex] H= - \frac{d^2 }{dx^ 2}+V(x) [/tex] so its "energies" (eigenvalues)

satisfy that [tex] E(n)=-E(-n) [/tex] then here comes the question..is licit legal (at least as an approximation) to take:

[tex] Z(u)=\sum_{n=-\infty}^{n=\infty}e^{iuE(n)}\sim \iint dxdpe^{iup^2 +iuV(x)} [/tex] ??

The explanation is clear..you substitute a "discrete" sum over energies by a continuous sum over all the energies..classically the energy of the system is [tex] E=p^2 +V(x) [/tex] (time independent potential) , so it would be similar to take the "sum-integral" approximation (valid at least at first order ??) i know that perhaps using functional analysis you could justify my approach, as a physicist when dealing with Statistical mechanics we use it all the time since "sums" are very hard to evaluate, except when E(n)=log(n) or E(n)=n :grumpy: :grumpy: usually the exponential is "real" :rolleyes: :rolleyes: but i think that for this case this wouldn't be a problem..also if we kenw Z(u) we could obtain the inverse of the potential taking:

[tex] A\int_{-\infty}^{\infty}du \frac{Z(u)e^{-iut}}{\sqrt (u) } = V^{-1}(t) [/tex] :tongue2: :tongue2:
 
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  • #2
Really, there is only a small number of times I can say "look up spectral analysis" before it starts becoming really really annoying, Jose.
 
  • #3
Sorry..i remember you that I'm not physicist..if you could recommend me a good "arxiv2 or other online paper about preliminaries of spectral analysis where my approximation is used and justified i would be very grateful...:rolleyes: :rolleyes: the only thing i know is that the "exponential sum" will be the trace of a certain operator [tex] Tr[e^{iu\hat H } [/tex]
 
  • #4
Buy any textbook on functional analysis, use google (spectral theory functional analysis), find someone's lecture notes. (and arxiv is not the p lace to learn about classical mathematics.) And this has nothing to do with you being, nor not being a physicist, but with you not listening to what people (shmoe, halls, hurkyl and me at anyrate) tell you.
 
  • #5
- I can't buy any book I'm actally unemployed :frown: :frown:

- perhaps a "trivial" solution to my problem is just put:

[tex] \sum_{n=-\infty}^{\infty}e^{iuE(n)} \sim \int_{E}dEe^{iuE} [/tex]

if the "eigenvalue" are energies of a certain Hamiltonian operator [tex] H=E=p^2 + V(x) [/tex] then dE=dxdp (momentum and position) .

Using this is the only "justification", my teacher gave us when speaking about partition functions :tongue2: the problem here is that perhaps you can't use the Euler-Mc Laurin sum formula to improve the integral-series convergence.
 
  • #6
Your school probably has a library...
 

1. What is functional analysis?

Functional analysis is a branch of mathematics that deals with the study of vector spaces and linear transformations between those spaces. It is also used in other fields such as engineering, physics, and economics to analyze and describe complex systems.

2. How is functional analysis used in scientific research?

Functional analysis is used in scientific research to understand and model complex systems. It allows scientists to break down a system into smaller, more manageable parts and study how these parts interact with each other. This can provide insights into the behavior and functioning of the system as a whole.

3. What are the main components of functional analysis?

The main components of functional analysis include vector spaces, linear transformations, and operators. Vector spaces are sets of objects that can be added together and multiplied by scalars. Linear transformations are functions that preserve vector space structure, while operators are functions that map vectors to other vectors.

4. How is functional analysis different from other branches of mathematics?

Functional analysis is different from other branches of mathematics because it focuses on the study of infinite-dimensional spaces and functions. This allows for the analysis of complex systems that cannot be described using finite-dimensional spaces. It also has applications in a wide range of fields, including physics, engineering, and economics.

5. What are some real-world applications of functional analysis?

Functional analysis has many real-world applications, including image and signal processing, data analysis, optimization, and control theory. It is also used in the study of quantum mechanics, electromagnetism, and fluid dynamics. Additionally, functional analysis has applications in economics, finance, and game theory.

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