A Self-adjointness Criterion for the Schrödinger Operator with Infinitely Many Point Interactions and Its Application to Random Operators

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.

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

$$\begin{aligned} \mathop {\mathrm{sgn}}\nolimits V |V|^{1/2}G_0|V|^{1/2} \phi =-\phi ,\quad \psi = G_0 |V|^{1/2}\phi , \end{aligned}$$

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

$$\begin{aligned} \alpha _0 = - \mu '(0)C^{-1},\quad C=\left| \int _{\mathbb {R}^3}|V|^{1/2}\phi \mathrm{d}x\right| ^2. \end{aligned}$$

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

$$\begin{aligned} |(u,\Delta \phi )_{L^2(U)}|\le M \Vert \phi \Vert _{L^2(U)} \end{aligned}$$

(31)

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

$$\begin{aligned} \Vert u\Vert _{H^2(V)}\le C(M + \Vert u\Vert _{L^2(U)}), \end{aligned}$$

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

$$\begin{aligned} \mathop {\mathrm{dist}}\nolimits (\partial U, V)\ge \delta \end{aligned}$$

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

$$\begin{aligned} \Vert u\Vert _{H^2(V)}^2 \le C \left( \Vert \Delta u \Vert _{L^2(U)}^2+ \Vert u \Vert _{L^2(U)}^2 \right) . \end{aligned}$$

(32)

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

$$\begin{aligned} |(u, \Delta \phi )_{L^2(Q)}| = |(\Delta u, \phi )_{L^2(Q)}| \le \Vert \Delta u\Vert _{L^2(Q)} \Vert \phi \Vert _{L^2(Q)} \end{aligned}$$

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

$$\begin{aligned} \Vert u\Vert _{H^2(Q')}^2\le C(\Vert \Delta u\Vert _{L^2(Q)}^2 + \Vert u\Vert _{L^2(Q)}^2) \end{aligned}$$

(33)

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)

$$\begin{aligned} \Vert u\Vert _{H^2(V)}^2&\le \sum _{Q'} \Vert u\Vert _{H^2(Q')}^2\\&\le C \sum _{Q'} \left( \Vert \Delta u\Vert _{L^2(Q)}^2 + \Vert u\Vert _{L^2(Q)}^2 \right) \\&\le 2^{d}C \left( \Vert \Delta u\Vert _{L^2(U)}^2 + \Vert u\Vert _{L^2(U)}^2 \right) , \end{aligned}$$

where we use the fact Q can overlap at most $2^d$ times. $\square$