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Homework Help: Eigenfunction of all shift operators

  1. Oct 8, 2013 #1
    Prove that if a continuous function [itex] e\left( x \right) [/itex] on [itex]\mathbb{R}[/itex] is eigenfunction of all shift operators, i.e. [itex] e\left( x+t \right) = \lambda_t e\left( x \right) [/itex] for all x and t and some constants [itex] \lambda_t [/itex], then it is an exponential function, i.e. [itex] e\left( x \right)= Ce^{ax}[/itex] for some constants C and a.

    Thanks in Advance.
  2. jcsd
  3. Oct 8, 2013 #2

    Ray Vickson

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