The role of the weight function for adjoint DO

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Discussion Overview

The discussion revolves around the role of the weight function \( w(t) \) in the definition of adjoint and s-adjoint operators, particularly in relation to its relevance in physical applications.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant expresses curiosity about the significance of the weight function \( w(t) \) in defining adjoint and s-adjoint operators and its relevance in physical applications.
  • Another participant seeks clarification, suggesting that the discussion pertains to weight functions in the context of inner products on spaces of square integrable functions.
  • A participant notes that the weight factor influences which functions have a finite weighted \( L^2 \)-norm, thereby determining admissible states for the system and orthogonality among these states, indicating potential physical relevance.
  • There is a suggestion that specifying the physical system or class of systems being discussed would provide more concrete context for the relevance of the weight function.

Areas of Agreement / Disagreement

Participants appear to agree on the importance of the weight function in determining admissible states and orthogonality, but the discussion remains open regarding its specific applications and the types of physical systems involved.

Contextual Notes

The discussion lacks specific examples of physical systems and does not resolve the implications of the weight function in various contexts.

Jianphys17
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Hi at all, I've a curiosity about the role that the weight function w(t) she has, into the define of adjoint & s-adjoint op.
It is relevant in physical applications or not ?
 
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Can you give an example of what you are talking about? Am I correct that you are talking about weight functions appearing in the definition of an inner product on a space of (equivalence classes of) square integrable functions?
 
Yes..
 
Well, one could say that the weight factor determined for which functions the (weighted) ##L^2##-norm is finite, hence which functions are admissible states for the system under consideration. It also determines which functions are orthogonal to each other. To me it seems that these properties (being an admissible state and being orthogonal to other admissible states) are quite relevant from a physical point of view.

To become more concrete, one would have to specify which physical system (or: class of systems) is under consideration.
 
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