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A lot of books find the bound state energies of the single delta function potential centred at $x=0$ by integrating the Schrodinger equation around $x=0$, using symmetric limits and making the width zero.

By heuristic argument it should not be equal to zero as the function is even.

Check

https://www.wolframalpha.com/input/?i=lim+x-%3E0+integrate+exp(-%7Cx%7C)+dx+from+-infinity+to+infinity

Please, let me know what is wrong or if there is some technicality with respect to theory of distributions playing around here.

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  • $\begingroup$ You are integrating over the whole space whereas you need to integrate from $-\epsilon$ to $\epsilon$ about the discontinuity so that, with $\epsilon\to 0$, you capture the change in $\psi'$ across that discontinuity. See en.wikipedia.org/wiki/Delta_potential $\endgroup$ – ZeroTheHero Sep 27 '17 at 7:25
  • $\begingroup$ What exactly is your question? $\endgroup$ – By Symmetry Sep 27 '17 at 12:23

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