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Is there a self-adjoint extension of an operator that corresponds to a particle in a box $[a,b] \times [c,d] \subset \mathbb{R}^2$ with a delta potential, i.e., $-\Delta + \lambda \delta_y $ on $L^2([a,b] \times [c,d])$ ?

All the references I've been able to find mention a particle on the line or on $\mathbb{R}^2$.

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  • $\begingroup$ You'd need to specify boundary conditions on a box, to begin... $\endgroup$
    – paul garrett
    Commented Jun 30, 2014 at 18:46
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    $\begingroup$ There is a very (too?) large body of literature on this; you could try Albeverio et al., Solvable models in QM as an entry point. $\endgroup$
    – Christian Remling
    Commented Jun 30, 2014 at 19:29
  • $\begingroup$ I've looked in Solvable Models; unfortunately, they don't have anything on a particle in a box. $\endgroup$
    – daunpunk
    Commented Jun 30, 2014 at 22:42
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    $\begingroup$ I took a look at some papers and apparently what one does is (adapted to the rectangle, as in the OP) consider $-\Delta$ on $C_0^{\infty}(R\setminus\{y\})$ (this of course again will give Dirichlet bc's on $\partial R$). The closure of this operator seems to be symmetric with deficiency $(1,1)$, so the self-adjoint realizations are easily described and one picks one that feels right. $\endgroup$
    – Christian Remling
    Commented Jul 1, 2014 at 1:41
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    $\begingroup$ Some more random musings: Let's denote the closure of $-\Delta$ on $C_0^{\infty}(R\setminus \{ y\})$ by $T$. Then it shouldn't be too hard to show that $D(T)=\{u\in H^2(R): u=0 \textrm{ on }\partial R, u(y)=0 \}$. This is a one-dimensional restriction (by the condition $u(y)=0$) of the self-adjoint Dirichlet Laplacian. This gives the claim on the deficiency indices. $\endgroup$
    – Christian Remling
    Commented Jul 1, 2014 at 17:39

2 Answers 2

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Maybe the following references will be helpful:

http://arxiv.org/abs/quant-ph/0103153 (Self-adjoint extensions of operators and the teaching of quantum mechanics, by G. Bonneau, J. Faraut and G. Valent).

https://www.amherst.edu/media/view/10264/original/gopalakrishnan06.pdf (Self-Adjointness and the Renormalization of Singular Potentials, Bachelor thesis by S. Gopalakrishnan).

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  • $\begingroup$ Thanks; unfortunately, I'd come across both of these, and neither mention the particle in a box with a delta potential. $\endgroup$
    – daunpunk
    Commented Jul 2, 2014 at 9:15
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The paper Ph. Blanchard, R. Figari, A. Mantile "Point Interaction Hamiltonians in Bounded Domains" http://arxiv.org/abs/0704.3249 may contain some answers. In any case, if you have an operator $H$ in $L^2(\Omega)$ and want to study its perturbations by zero-range potentials, you need to know the integral kernel of the resolvent $R(z)=(H-z)^{-1}$, see the discussion in Section 1.4.3 of J. Brüning, V. Geyler, K. Pankrashkin: Spectra of self-adjoint extensions and applications to solvable Schrödinger operators http://arxiv.org/abs/math-ph/0611088 or the older paper V.A. Geyler, V. A. Margulis, I. I. Chuchaev: Potentials of zero radius and Carleman operators, Siberian Math. J. 36 (1995) 714–726.

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