Abstract
We prove the Schrödinger operator with infinitely many point interactions in \(\mathbb {R}^d\)\((d=1,2,3)\) is self-adjoint if the support \(\Gamma \) of the interactions is decomposed into infinitely many bounded subsets \(\{\Gamma _j\}_j\) such that \(\inf _{j\not =k}\mathop {\mathrm{dist}}\nolimits (\Gamma _j,\Gamma _k)>0\). Using this fact, we prove the self-adjointness of the Schrödinger operator with point interactions on a random perturbation of a lattice or on the Poisson configuration. We also determine the spectrum of the Schrödinger operators with random point interactions of Poisson–Anderson type.
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Acknowledgements
The work of T. M. is partially supported by JSPS KAKENHI Grant No. JP18K03329. The work of F. N. is partially supported by JSPS KAKENHI Grant No. JP26400145.
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Appendix
Appendix
1.1 Renormalization Procedure for Point Interactions
For readers’ convenience, we quote a result about the renormalization procedure (1)
in the case \(d=3\) and \(\Gamma =O:=\{0\}\) ([2, Theorem I-1.2.5]).
Theorem 23
([2]). Let \(d=3\). Let V be a real-valued function on \(\mathbb {R}^3\) belonging to Rollnik class (i.e., \(\iint _{\mathbb {R}^3\times \mathbb {R}^3}|V(x)||V(x')||x-x'|^{-2}\mathrm{d}x\mathrm{d}x'<\infty \)), and \((1+|\cdot |)V\in L^1(\mathbb {R}^3)\). Let \(\lambda (\epsilon )=\epsilon \mu (\epsilon )\), where \(\mu (\epsilon )\) is real analytic near \(\epsilon =0\), \(\mu (0)=1\) and \(\mu '(0)\not =0\). Let \(G_0=(-\Delta )^{-1}\), defined as an integral operator with integral kernel \(1/(4\pi |x-x'|)\). If two functions \(\phi \in L^2(\mathbb {R}^3)\) and \(\psi \in L^2_\mathrm{loc}(\mathbb {R}^3){\setminus } L^2(\mathbb {R}^3)\) satisfy
where \(\mathop {\mathrm{sgn}}\nolimits z= z/|z|\) (\(z\not =0\)) and \(\mathop {\mathrm{sgn}}\nolimits 0 =1\), then \(\psi \) is called a zero-energy resonance for \(-\Delta +V\).
- (i)
If a zero-energy resonance for \(-\Delta +V\) exists, then the norm resolvent limit of \(H_{\epsilon }\) in (1) coincides with \(H_{O,\alpha }\), \(\alpha =(\alpha _0)\), for some \(\alpha _0\in \mathbb {R}\).
- (ii)
If no zero-energy resonance for \(-\Delta +V\) exists, then the norm resolvent limit of \(H_{\epsilon }\) in (1) coincides with the free Laplacian \(H_0\).
It is easy to see that \(\psi \) satisfies the equation \((-\Delta +V)\psi =0\). In the case (i), an explicit formula for \(\alpha _0\) is given in [2, Theorem I-1.2.5]. If \(-\Delta +V\) has simple zero-energy resonance \(\psi \) and no zero-energy eigenfunction, it is given as
Thus, \(\alpha _0\) appears in the coefficient of \(\lambda (\epsilon )\) as \(\lambda (\epsilon )=\epsilon - C\alpha _0 \epsilon ^2 + O(\epsilon ^3)\). It is known that if V is nonnegative, then there is no zero-energy resonance for \(-\Delta +V\). So, the occurrence of the case (i) requires V to have negative part.
A similar result is obtained also in the case \(d=2\), but \(\alpha _0\) appears in the coefficient of \(\lambda (\epsilon )\) in more complicated way (see [2, Theorem I-5.5]).
1.2 Elliptic Inner Regularity Estimate
The following is a special case of the elliptic inner regularity theorem ([4, Theorem 6.3]).
Theorem 24
Let U be an open set in \(\mathbb {R}^d\) and \(u\in L^2(U)\). Assume that there exists a positive constant M such that
holds for every \(\phi \in C_0^\infty (U)\). Then, \(u\in H^2_\mathrm{loc}(U)\). Moreover, for any open set V such that \(\overline{V}\) is a compact subset of U, there exists a positive constant C dependent only on U and V such that
where M is the constant in (31).
From Theorem 24, we have the following corollary useful for our purpose.
Corollary 25
Let U, V be open sets in \(\mathbb {R}^d\) such that \(\overline{V}\subset U\) and
for some positive constant \(\delta \). Let \(u\in L^2(U)\) such that \(\Delta u\in L^2(U)\) in the distributional sense. Then, \(u\in H^2_\mathrm{loc}(U)\), and there exists a constant C dependent only on \(\delta \) and the dimension d such that
Proof
Put \(\epsilon =\delta /(2d)\). For \(x_0\in \mathbb {R}^d\), consider open cubes \(Q=x_0 + (-\epsilon ,\epsilon )^d\) and \(Q'=x_0 + (-\epsilon /2,\epsilon /2)^d\). When \(Q\subset U\), we have
for every \(\phi \in C_0^\infty (Q)\). Then, the assumption of Theorem 24 is satisfied with \(U=Q\), \(V=Q'\), and \(M=\Vert \Delta u\Vert _{L^2(Q)}\), and we have
for some positive constant C dependent only on \(\delta \) and dimension d. We collect all the cubes \(Q'\) such that the center \(x_0\in \epsilon \mathbb {Z}^d\) and \(Q'\cap V \not =\emptyset \). Notice that \(Q\subset U\) for such \(Q'\). Thus, we have by (33)
where we use the fact Q can overlap at most \(2^d\) times. \(\square \)
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Kaminaga, M., Mine, T. & Nakano, F. A Self-adjointness Criterion for the Schrödinger Operator with Infinitely Many Point Interactions and Its Application to Random Operators. Ann. Henri Poincaré 21, 405–435 (2020). https://doi.org/10.1007/s00023-019-00869-1
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DOI: https://doi.org/10.1007/s00023-019-00869-1