The role of the weight function for adjoint DO

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SUMMARY

The discussion focuses on the significance of the weight function \( w(t) \) in defining adjoint and s-adjoint operators within the context of inner products on spaces of square integrable functions. It is established that the weight function is crucial for determining the admissibility of states in a physical system, as it influences the finiteness of the weighted \( L^2 \)-norm and the orthogonality of functions. These properties are essential for understanding the physical implications of the mathematical framework being discussed.

PREREQUISITES
  • Understanding of adjoint and s-adjoint operators in functional analysis
  • Familiarity with inner product spaces and square integrable functions
  • Knowledge of weighted \( L^2 \)-norms
  • Basic concepts in quantum mechanics related to admissible states
NEXT STEPS
  • Research the mathematical foundations of adjoint operators in Hilbert spaces
  • Explore the implications of weight functions in quantum mechanics
  • Study the properties of \( L^2 \) spaces and their applications in physics
  • Investigate examples of physical systems where weight functions play a critical role
USEFUL FOR

Mathematicians, physicists, and students in advanced mathematics or quantum mechanics who are interested in the applications of weight functions in operator theory and their relevance to physical systems.

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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