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I am having a hard time making sense of the so-called "delta function potential well" in quantum theory. The Hamiltonian operator is defined as (with $\mathscr D_H\subset \mathscr H=L^2(\mathbb R)$) $$H:\mathscr D_H\rightarrow \mathscr H$$ $$\psi\mapsto H\psi$$ And $$(H\psi)(x):=-\frac{d^2}{dx^2}\psi(x)-\lambda\delta(x)\psi(x).$$ My job is to find the spectrum of this operator given a $\lambda>0$.

My problems are:

  1. How do I construct $\mathscr D_H$?
  2. What definition of the "delta function" is suitable for this kind of job?
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  • $\begingroup$ Q2 is not really relevant here, you normally define $H$ directly, without referring to its ingredients. In one dimension this is fairly straightforward (for example, if you write the "ODE" $Hu=zu$ as a first order system, then it has an obvious interpretation). There is a large literature on these things; see for example the book by Albeverio et al. $\endgroup$
    – Christian Remling
    Commented Aug 24, 2019 at 17:05
  • $\begingroup$ To find the spectrum, you don't really need to worry about any of this. It will be purely absolutely continuous on $[0,\infty)$, of multiplicity $2$, and there will be a negative eigenvalue, which you can just analyze by hand by looking at the ODE $Hu=-Eu$. Again, I'm sure this is discussed in many places, if you search the literature. $\endgroup$
    – Christian Remling
    Commented Aug 24, 2019 at 17:06

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