Zero modes and low-energy resolvent expansion for three dimensional Schrodinger operators with point interactions

TL;DR

This paper studies the low-energy properties of three-dimensional Schrödinger operators with point interactions, focusing on the resolvent expansion near zero energy and the conditions for zero energy obstructions.

Contribution

It provides a detailed analysis of the low-energy resolvent expansion and characterizes zero energy obstructions for Schrödinger operators with point interactions in three dimensions.

Findings

Derived the low-energy resolvent expansion for the operators.

Identified conditions for the occurrence of zero energy obstructions.

Analyzed the multiplicity of zero energy obstructions.

Abstract

We investigate the low-energy behavior of the resolvent of Schrodinger operators with finitely many point interactions in three dimensions. We also discuss the occurrence and the multiplicity of zero energy obstructions.

Equations

$${}^{\text{“}}-\Delta +\sum _{y\in Y}{\mu }_y{\delta }_y(\cdot {)}^{\text{”}},$$

$$T_Y:=\overline{(-\Delta )\upharpoonright C_0^{\infty }({\mathbb{R}}^3\setminus \{Y\})}$$

$$\{-\Delta_{\alpha,Y}\;\big{|}\;\alpha\equiv(\alpha_{1},\ldots,\alpha_{N})\in(\mathbb{R}\cup\{\infty\})^{N}\},$$

$${\mathcal{G}}_z^y(x):=\frac{e^{\mathrm{i}z|x-y|}}{4\pi |x-y|},{\mathcal{G}}_z^{y{y}^{\prime }}:=\{\begin{array}{ll}\frac{e^{\mathrm{i}z|y-y^{\prime }|}}{4\pi |y-y^{\prime }|}& \text{if }{y}^{\prime }\ne y\\ 0& \text{if }{y}^{\prime }=y,\end{array}$$

$$\Gamma_{\alpha,Y}(z)\;:=\;\Big{(}\Big{(}\alpha_{j}-\frac{\mathrm{i}z}{\,4\pi\,}\Big{)}\delta_{j,k}-\mathcal{G}_{z}^{y_{j}y_{k}}\Big{)}_{\!j,k=1,\dots,N}\,.$$

$$\mathcal{D}(-\Delta_{\alpha,Y})\;=\;\Big{\{}g=F_{z}+\sum_{j,k=1}^{N}(\Gamma_{\alpha,Y}(z)^{-1})_{jk}\,F_{z}(y_{k}){\mathcal{G}}_{z}^{y_{j}}\,,\,F_{z}\in H^{2}(\mathbb{R}^{3})\Big{\}}\,.$$

$$\mathcal{D}(-\Delta_{\alpha,Y})\;=\;\left\{g=F_{z}\\ +\sum_{j=1}^{N}c_{j}\,{\mathcal{G}}_{z}^{y_{j}}\left|\!\!\begin{array}[]{c}F_{z}\in H^{2}(\mathbb{R}^{3})\\ (c_{1},\dots,c_{N})\in\mathbb{C}^{N}\\ \begin{pmatrix}F_{z}(y_{1})\\ \vdots\\ F_{z}(y_{N})\end{pmatrix}=\Gamma_{\alpha,Y}(z)\begin{pmatrix}c_{1}\\ \vdots\\ c_{N}\end{pmatrix}\end{array}\!\!\!\right.\right\}\,.$$

$$(-{\Delta }_{\alpha ,Y}-z^2)g=(-\Delta -z^2)F_z.$$

$$(-{\Delta }_{\alpha ,Y}-z^2{)}^{-1}-(-\Delta -z^2{)}^{-1}=\sum _{j,k=1}^N({\Gamma }_{\alpha ,Y}(z{)}^{-1}{)}_{jk}|{\mathcal{G}}_z^{y_j}⟩⟨\overline{{\mathcal{G}}_z^{y_k}}|.$$

$$\psi =\sum _{j=1}^N{c}_j{\mathcal{G}}_{\mathrm{i}\lambda }^{y_j},$$

$${\mathbf{B}}_{\sigma }:=\mathcal{B}(L^2({\mathbb{R}}^3,⟨x{⟩}^{2+\sigma }dx);L^2({\mathbb{R}}^3,⟨x{⟩}^{-2-\sigma }dx)),$$

$${\mathbb{C}}^{+}\setminus {\mathcal{E}}^{+}\ni z\mapsto (-{\Delta }_{\alpha ,Y}-z^2{)}^{-1}\in {\mathbf{B}}_{\sigma }$$

$$(-{\Delta }_{\alpha ,Y}-z^2{)}^{-1}=\frac{R_{-2}}{z^2}+\frac{R_{-1}}{z}+R_0(z),$$

$$\Gamma (z)={\Gamma }_0+z{\Gamma }_1+z^2{\Gamma }_2+O(z^3).$$

$$({\Gamma }_0{)}_{jk}={\alpha }_j{\delta }_{jk}-{\mathcal{G}}_0^{y_j{y}_k},({\Gamma }_1{)}_{jk}=(4\pi i{)}^{-1},({\Gamma }_2{)}_{jk}=(8\pi {)}^{-1}|y_j-y_k|.$$

$$B=P-P(A+P{)}^{-1}P$$

$$A^{-1}=(A+P{)}^{-1}+(A+P{)}^{-1}P(B\upharpoonright P\mathcal{H}{)}^{-1}P(A+P{)}^{-1}.$$

$$\Gamma (z{)}^{-1}=\frac{A_{-2}}{z^2}+\frac{A_{-1}}{z}+O(1),$$

$$\begin{array}{rl}({\Gamma }_{\le 1}(z)+P{)}^{-1}& =[I+z({\Gamma }_0+P{)}^{-1}{\Gamma }_1{]}^{-1}[{\Gamma }_0+P{]}^{-1}\\ & =[I-z({\Gamma }_0+P{)}^{-1}{\Gamma }_1][{\Gamma }_0+P{]}^{-1}+O(z^2).\end{array}$$

$$\begin{split}&\Gamma_{\leq 1}(z)^{-1}=(\Gamma_{\leq 1}(z)+P)^{-1}\\ &\quad+(\Gamma_{\leq 1}(z)+P)^{-1}P\Big{(}\big{(}P-P(\Gamma_{\leq 1}(z)+P)^{-1}P\big{)}\upharpoonright P\,\mathbb{C}^{N}\Big{)}^{-1}P(\Gamma_{\leq 1}(z)+P)^{-1}.\end{split}$$

$$P-P({\Gamma }_{\le 1}(z)+P{)}^{-1}P=zP{\Gamma }_{1}P+O(z^2).$$

$$\begin{split}\Gamma_{\leq 1}^{-1}(z)&=(\Gamma_{\leq 1}(z)+P)^{-1}\\ &+(\Gamma_{\leq 1}(z)+P)^{-1}P\Big{(}\big{(}z\,P\Gamma_{1}P\upharpoonright P\,\mathbb{C}^{N}\big{)}^{-1}+O(1)\Big{)}P(\Gamma_{\leq 1}(z)+P)^{-1}\\ &=z^{-1}P(P\Gamma_{1}P\upharpoonright P\,\mathbb{C}^{N})^{-1}P+O(1)=\Theta(z^{-1})+O(1).\end{split}$$

$$(\,Ker\,\Gamma_{1}\cap\mathbb{R}^{N})\setminus\{0\}=\big{\{}v\in\mathbb{R}^{N}\setminus\{0\}\,|\,v_{1}+\ldots+v_{N}=0\big{\}}.$$

$$⟨{\Gamma }_{2}v,v⟩:=(8\pi {)}^{-1}\sum _{1\le j,k\le N}|y_j-y_k|v_j{v}_k\le 0.$$

$${\int }_{S^2}|⟨w,y⟩|dw=c|y|.$$

$$(8\pi {)}^{-1}\sum _{1\le j,k\le N}|y_j-y_k|v_j{v}_k=(8\pi c{)}^{-1}{\int }_{S^2}\sum _{1\le j,k\le N}|⟨w,y_j-y_k⟩|v_j{v}_{k}dw,$$

$$\sum _{1\le j,k\le N}|{\tilde{y}}_j-{\tilde{y}}_k|v_j{v}_k\le 0,$$

$$\begin{array}{rl}\sum _{1\le j,k\le N}|{\tilde{y}}_j-{\tilde{y}}_k|v_j{v}_k& =2\sum _{1\le j,k\le N}\max \{{\tilde{y}}_j-{\tilde{y}}_k,0\}{v}_j{v}_k\\ & =2{\int }_{t\in \mathbb{R}}\sum _{1\le j,k\le N}[{\tilde{y}}_k<t<{\tilde{y}}_j]v_j{v}_k,\end{array}$$

$$\sum _{{\tilde{y}}_k<t<{\tilde{y}}_j}{v}_j{v}_k\le 0.$$

$$\sum_{\tilde{y}_{k}<t<\tilde{y}_{j}}v_{j}v_{k}=\sum_{j\in J_{t},k\in K_{t}}v_{j}v_{k}=\Big{(}\sum_{j\in J_{t}}v_{j}\Big{)}\Big{(}\sum_{k\in K_{t}}v_{k}\Big{)}=-\Big{(}\sum_{j\in J_{t}}v_{j}\Big{)}^{2}\leq 0,$$

Full text

1 Introduction and main results

A central topic in quantum mechanics is the study of quantum systems subject to very short-range interactions, supported around a submanifold of the ambient space. A relevant situation occurs when the singular interaction is supported on a set of points in the Euclidean space $\mathbb{R}^{d}$. This leads to consider, formally, operators of the form

[TABLE]

where $Y$ is a discrete subset of $\mathbb{R}^{d}$, and $\mu_{y}$, $y\in Y$, are real coupling constants.

Heuristically, (1) can be interpreted as the Hamiltonian for a non-relativistic quantum particle interacting with “point obstacles” of strengths $\mu_{y}$, located at $y\in Y$.

From a mathematical point of view, Schrödinger operators with point (delta-like) interactions have been intensively studied, since the first rigorous realization by Berezin and Faddeev Berezin-Faddeev-1961 , and subsequent characterizations by many other authors Albeverio-Fenstad-HoeghKrohn-1979_singPert_NonstAnal ; Zorbas-1980 ; Grossmann-HK-Mebkhout-1980 ; Grossmann-HK-Mebkhout-1980_CMPperiodic ; Dabrowski-Grosse-1985 ; MO-2016 (see the surveys DFT-brief_review_2008 ; Albeverio-Figari , the monograph of Albeverio, Gesztesy, Høegh-Krohn, and Holden albeverio-solvable , and references therein for a thorough discussion).

In this work we focus on the case of finitely many point interactions in three dimensions. Our aim is to provide a detailed spectral analysis at the bottom of the continuous spectrum, i.e. at zero energy. A similar analysis has been done in Cornean-MY for the two dimensional case, with application to the $L^{p}$-boundedness of the wave operators.

We start by recalling some well-known facts on the construction and the main properties of Schrödinger operators with point interactions.

We fix a natural number $N\geqslant 1$ and the set $Y=\{y_{1},\dots,y_{N}\}\subseteq\mathbb{R}^{3}$ of distinct centers of the singular interactions. Consider

[TABLE]

as an operator closure with respect to the Hilbert space $L^{2}(\mathbb{R}^{3})$. It is a closed, densely defined, non-negative, symmetric operator on $L^{2}(\mathbb{R}^{3})$, with deficiency index $N$. Hence, it admits an $N^{2}$-real parameter family of self-adjoint extensions. Among these, there is an $N$-parameter family of local extension, denoted by

[TABLE]

whose domain of self-adjointness is qualified by certain local boundary conditions at the singularity centers.

The self-adjoint operators $-\Delta_{\alpha,Y}$ provide rigorous realizations of the formal Hamiltonian (1), the coupling parameters $\alpha_{j}$, $j=1,\ldots,N$, being now proportional to the inverse scattering length of the interaction at the center $y_{j}$. In particular, if for some $j\in\{1,\ldots,N\}$ one has $\alpha_{j}=\infty$, then no actual interaction is present at the point $y_{j}$, and in practice things are as if one discards the point $y_{j}$. When all $\alpha_{j}=\infty$, one recovers the Friedrichs extension of $T_{Y}$, namely the self-adjoint realization of $-\Delta$ on $L^{2}(\mathbb{R}^{3})$. Owing to the discussion above, we may henceforth assume, without loss of generality, that $\alpha$ runs over $\mathbb{R}^{N}$.

We review the basic properties of $-\Delta_{\alpha,Y}$, from (albeverio-solvable, , Section II.1.1) and Posilicano2000_Krein-like_formula (see also DFT-brief_review_2008 ; DAncona-Pierfelice-Teta-2006 ; Iandoli-Scandone-2017 ; DMSY-2017 ). We introduce first some notations.

For $z\in\mathbb{C}$ and $x,y,y^{\prime}\in\mathbb{R}^{3}$, set

[TABLE]

and

[TABLE]

The function $z\mapsto\Gamma_{\alpha,Y}(z)$ has values in the space of $N\times N$ symmetric, complex valued matrices and is clearly entire, whence $z\mapsto\Gamma_{\alpha,Y}(z)^{-1}$ is meromorphic in $\mathbb{C}$. It is known that $\Gamma_{\alpha,Y}(z)^{-1}$ has at most $N$ poles in the open upper half-plane $\mathbb{C}^{+}$, which are all located along the positive imaginary semi-axis. We denote by $\mathcal{E}^{+}$ the set of such poles. Moreover, we denote by $\mathcal{E}^{0}$ the set of poles of $\Gamma_{\alpha,Y}(z)^{-1}$ on the real line. Observe that $\mathcal{E}^{0}$ is finite and symmetric with respect to $z=0$. Actually, either $\mathcal{E}^{0}=\emptyset$ or $\mathcal{E}^{0}=\{0\}$. This follows by a generalization of the Rellich Uniqueness Theorem (sjo_lecture, , Theorem 2.4), valid for a large class of compactly supported perturbations of the Laplacian, introduced by Sjöstrand and Zworski in Sjo-Zwo . For an introduction to the classical theory of the Rellich Uniqueness Theorem, we refer to the monograph of Lax and Phillips Lax-Phi . More recently, the absence of non-zero real poles for $\Gamma_{\alpha,Y}^{-1}$ has been proved through different techniques by Galtbayar-Yajima Galtbayar-Yajima-2019 , and by the author in collaboration with Michelangeli MS-2020 .

The following facts are known.

Proposition 1

1. (i)

The domain of $-\Delta_{\alpha,Y}$ has the following representation, for any $z\in\mathbb{C}^{+}\!\setminus\!\mathcal{E}^{+}$:

[TABLE]

Equivalently, for any $z\in\mathbb{C}^{+}\!\setminus\!\mathcal{E}^{+}$,

[TABLE]

At fixed $z$, the decompositions above are unique. 2. (ii)

With respect to the decompositions (6)-(7), one has

[TABLE] 3. (iii)

For $z\in\mathbb{C}^{+}\!\setminus\!\mathcal{E}^{+}$, we have the resolvent identity

[TABLE] 4. (iv)

The spectrum $\sigma(-\Delta_{\alpha,Y})$ of $-\Delta_{\alpha,Y}$ consists of at most $N$ non-positive eigenvalues and the absolutely continuous part $\sigma_{\mathrm{ac}}(-\Delta_{\alpha,Y})=[0,\infty)$, the singular continuous spectrum is absent.

Parts (i) and (ii) of Proposition 1 above originate from Grossmann-HK-Mebkhout-1980_CMPperiodic and are discussed in (albeverio-solvable, , Theorem II.1.1.3), in particular (7) is highlighted in DFT-brief_review_2008 . Part (iii) was first proved in Grossmann-HK-Mebkhout-1980 ; Grossmann-HK-Mebkhout-1980_CMPperiodic (see also (albeverio-solvable, , equation (II.1.1.33))). Part (iv) is discussed in (albeverio-solvable, , Theorem II.1.1.4), where it is stated that $\sigma_{p}(-\Delta_{\alpha,Y})\subset(-\infty,0)$. An errata at the end of the monograph (see also dellapana ; gri_nov ) specifies that a zero eigenvalue embedded in the continuous spectrum can actually occur: in fact for every $N\geq 2$ one can find a configuration $Y$ of the $N$ centers, and coupling parameters $\alpha_{1},\ldots\alpha_{N}$ such that $0\in\sigma_{p}(-\Delta_{\alpha,Y})$ – see the discussion in Section 4.

Next, let us discuss in detail the spectral properties of $-\Delta_{\alpha,Y}$, whose resolvent is characterized by (9) as an explicit rank-$N$ perturbation of the free resolvent. For negative eigenvalues, the situation is well-understood (albeverio-solvable, , Theorem II.1.1.4).

Proposition 2

There is a one to one correspondence between the poles $\mathrm{i}\lambda\in\mathcal{E}^{+}$ of $\Gamma_{\alpha,Y}(z)^{-1}$ and the negative eigenvalues $-\lambda^{2}$ of $-\Delta_{\alpha,Y}$, counting the multiplicity. The eigenfunctions associated to the eigenvalue $-\lambda^{2}<0$ have the form

[TABLE]

where $(c_{1},\dots,c_{N})\in\,Ker\,\Gamma_{\alpha,Y}(\mathrm{i}\lambda)\setminus\{0\}$.

Our main purpose is to analyze the spectral behavior of $-\Delta_{\alpha,Y}$ at $z=0$, and more generally when $z$ approaches the real line. The starting point is a classical version of the Limiting Absorption Principle for the free Laplacian. Given $\sigma>0$, we consider the Banach space

[TABLE]

where $\langle x\rangle:=\sqrt{1+|x|^{2}}$, and $\mathcal{B}(X;Y)$ denotes the space of linear bounded operators from $X$ to $Y$. We have the following result Agmon ; Kuroda ; Jensen-Kato .

Proposition 3 (Limiting Absorption Principle for $-\Delta$)

Let $\sigma>0$. For any $z\in\mathbb{C}^{+}$, we have $(-\Delta-z^{2})^{-1}\in\mathbf{B}_{\sigma}$. Moreover, the map $\mathbb{C}^{+}\ni z\mapsto(-\Delta-z^{2})^{-1}\in\mathbf{B}_{\sigma}$ can be continuously extended to the real line.

Owing to the resolvent formula (9), and observing that for any $z\in\mathbb{C}^{+}\cup\mathbb{R}$ the projectors $|\mathcal{G}_{z}^{y_{j}}\rangle\langle\overline{\mathcal{G}_{z}^{y_{k}}}|$ belong to $\mathbf{B}_{\sigma}$, it is easy to deduce that also $-\Delta_{\alpha,Y}$ satisfies a Limiting Absorption Principle.

Proposition 4 (Limiting Absorption Principle for $-\Delta_{\alpha,Y}$)

Let $\sigma>0$. For every $z\in\mathbb{C}^{+}\setminus\mathcal{E}^{+}$, we have $(-\Delta_{\alpha,Y}-z^{2})^{-1}\in\mathbf{B}_{\sigma}$. Moreover, the map

[TABLE]

can be continuously extended to $\mathbb{R}\setminus\mathcal{E}^{0}$.

Our main result is a resolvent expansion in a neighborhood of $z=0$, which in view of the previous discussion is the only possible singular point on the real line for the map $z\mapsto(-\Delta_{\alpha,Y}-z^{2})^{-1}\in\mathbf{B}_{\sigma}$.

Theorem 1.1

In a (real) neighborhood of $z=0$, we have the expansion

[TABLE]

where $R_{-2},\,R_{-1}\in\mathbf{B}_{\sigma}$ and $z\mapsto R_{0}(z)$ is a continuous $\mathbf{B}_{\sigma}$-valued map. Moreover, $R_{-2}\neq 0$ if and only if zero is an eigenvalue for $-\Delta_{\alpha,Y}$.

Remark 1

For Schrödinger operators of the form $-\Delta+V$, the Limiting Absorption Principle and the analogous of Theorem 1.1 can be proved under suitable short-range assumptions on the scalar potential $V$ (see e.g. the classical papers Agmon ; Jensen-Kato ). In this case, moreover, it is well-known that $R_{-1}\neq 0$ if and only if there exists a generalized eigenfunction at $z=0$ (a zero-energy resonance for $-\Delta+V$), namely a function $\psi\in L^{2}(\mathbb{R}^{3},\langle x\rangle^{-1-\sigma}dx)\setminus L^{2}(\mathbb{R}^{3})$, for any $\sigma>0$, which satisfies $(-\Delta+V)\psi=0$ as a distributional identity on $\mathbb{R}^{3}$. As it will be clear from the proof of Theorem 1.1, a similar characterization holds true also for $-\Delta_{\alpha,Y}$ (see Remark 2).

2 Asymptotic for $\Gamma_{\alpha,Y}(z)^{-1}$ as $z\to 0$

We fix $N\geq 1$, $\alpha\in\mathbb{R}^{N}$ and $Y\subseteq\mathbb{R}^{3}$, and we set $\Gamma(z):=\Gamma_{\alpha,Y}(z)$.

We shall use the notation $O(z^{k})$, $k\in\mathbb{Z}$, to denote a meromorphic $M^{N}(\mathbb{C})$-valued function whose Laurent expansion in a neighborhood of $z=0$ contains only terms of degree $\geq k$. In particular, $O(1)$ denotes an analytic map in a neighborhood of $z=0$. We also write $\Theta(z^{k})$ to denote a function of the form $Az^{k}$, with $A\in\,M^{N}(\mathbb{C})\setminus\{0\}$.

In a neighborhood of $z=0$, we can expand

[TABLE]

Explicitly, we have

[TABLE]

In particular, $\Gamma_{0}$, $\Gamma_{2}$ are real symmetric matrices, while $\Gamma_{1}$ is skew-Hermitian, i.e. $\Gamma_{1}^{*}=-\Gamma_{1}$. Our aim is to characterize the small $z$ behavior of $\Gamma(z)^{-1}$. Preliminary, we recall the following useful result due to Jensen and Nenciu JN .

Lemma 1 (Jensen-Nenciu)

Let $A$ be a closed operator in a Hilbert space $\mathcal{H}$ and $P$ a projection, such that $A+P$ has a bounded inverse. Then $A$ has a bounded inverse if and only if

[TABLE]

has a bounded inverse in $P\mathcal{H}$, and in this case

[TABLE]

We can state now the main result of this Section.

Proposition 5

In a neighborhood of $z=0$ we have the Laurent expansion

[TABLE]

where $A_{-2},\,A_{-1}\in M^{N}(\mathbb{C})$. Moreover,

1. (i)

$A_{-2}\neq 0$* if and only if $\,Ker\,\Gamma_{0}\,\cap\,Ker\,\Gamma_{1}\neq\{0\}$,* 2. (ii)

$A_{-1}\neq 0$* if and only if $\,Ker\,\Gamma_{0}\not\subseteq\,Ker\,\Gamma_{1}$.*

Proof

If $\Gamma_{0}=\Gamma(0)$ is non-singular, then $\Gamma(z)^{-1}$ is analytic in a sufficiently small neighborhood of $z=0$. Assume now that $\Gamma_{0}$ is singular. Let us distinguish two cases:

Case 1: $Ker\,\Gamma_{0}\,\cap\,Ker\,\Gamma_{1}=\{0\}$. Let us set $\Gamma_{\leq 1}(z):=\Gamma_{0}+z\Gamma_{1}$, and observe that for $z$ small enough, $z\neq 0$, the matrix $\Gamma_{\leq 1}(z)$ is non-singular. Suppose indeed that $\Gamma_{\leq 1}(z)v=0$ for some $v\in\mathbb{C}^{N}$. If $\Gamma_{0}v\neq 0$, then for small $z$ we also have $\Gamma_{\leq 1}(z)v\neq 0$, a contradiction. Hence $\Gamma_{0}v=0$, which for $z\neq 0$ implies $\Gamma_{1}v=0$, and using the hypothesis $Ker\,\Gamma_{0}\,\cap\,Ker\,\Gamma_{1}=\{0\}$ we deduce that $v=0$. Observe also that for $z$ small enough, $z\neq 0$, the matrix $\Gamma(z)$ in non-singular, with $\Gamma(z)^{-1}=\Gamma_{\leq 1}(z)^{-1}+O(1)$.

In order to invert $\Gamma_{\leq 1}(z)$, we use the Jensen-Nenciu Lemma. Let $P:\mathbb{C}^{N}\to\mathbb{C}^{N}$ be the orthogonal projection onto $Ker\,\Gamma_{0}$. Since $\Gamma_{0}^{*}=\Gamma_{0}$, we have that $\Gamma_{0}+P$ is non-singular, whence the same is $\Gamma_{\leq 1}(z)+P$ for small $z$, with $(\Gamma_{\leq 1}(z)+P)^{-1}=O(1)$. More precisely,

[TABLE]

By Lemma 1 we get

[TABLE]

Observe that $(\Gamma_{0}+P)^{-1}P=P$, and since $\Gamma_{0}^{*}=\Gamma_{0}$ we also have $P(\Gamma_{0}+P)^{-1}=P$. Using these relations and (13), we compute

[TABLE]

Substituting into (14) we obtain

[TABLE]

Case 2: $Ker\,\Gamma_{0}\,\cap\,Ker\,\Gamma_{1}\neq\{0\}$. We start by proving that $\,Ker\,\Gamma_{1}\,\cap\,Ker\,\Gamma_{2}\,=\{0\}$. Since $\Gamma_{2}$ is real symmetric, and $\Gamma_{1}$ is purely imaginary and skew-symmetric, it is sufficient to show that the quadratic form associated to $\Gamma_{2}$ is strictly negative on

[TABLE]

To this aim, we prove preliminary that for any $v\in\mathbb{R}^{N}$, with $v_{1}+\ldots+v_{N}=0$,

[TABLE]

The key point is to use the so called averaging trick. By rotational and scaling invariance, we can see that there exists a positive constant $c$ such that, for any $y\in\mathbb{R}^{3}$,

[TABLE]

It follows that

[TABLE]

and then it is sufficient to prove that, for a fixed $w\in S^{2}$,

[TABLE]

where we set $\tilde{y}_{j}:=\langle w,y_{j}\rangle$ for $j=1,\ldots,N$. We have

[TABLE]

where we use the Iverson bracket notation $[P]$, which equals $1$ if the statement $P$ is true and 0 if it is false. So it is enough to prove that, for almost every $t\in\mathbb{R}$,

[TABLE]

For every $t\in\mathbb{R}\setminus\{\tilde{y}_{1},\ldots\tilde{y}_{N}\}$, define J_{t}:=\{j\,\big{|}\,\tilde{y}_{j}>t\}, K_{t}:=\{k\,\big{|}\,\tilde{y}_{k}<t\}. We have

[TABLE]

where we use, in the last equality, the hypothesis $v_{1}+\ldots+v_{N}=0$.

Assume now that we have the equality in (16), for a suitable vector $v\in\mathbb{R}^{N}$ with $v_{1}+\ldots v_{N}=0$. It follows from the identity (17) that for almost every $w\in S^{2}$

[TABLE]

In particular, we can choose $w\in S^{2}$ satisfying (20), and such that the quantities $\tilde{y}_{j}=\langle w,y_{j}\rangle$ are pairwise distinct, say $\tilde{y}_{1}>\tilde{y}_{2}>\ldots>\tilde{y}_{N}$. Owing to (18)-(19), we deduce that

[TABLE]

for every $t$ in a full-measure set $\mathcal{T}\subset\mathbb{R}$. In particular, choosing $t_{1},\ldots t_{N-1}\in\mathcal{T}$, with $t_{n}\in(\tilde{y}_{n+1},\tilde{y}_{n})$ for $n=1,\ldots,N-1$, we obtain from (21) and (19) that

[TABLE]

This implies $v=0$, concluding the proof that $\,Ker\,\Gamma_{1}\,\cap\,Ker\,\Gamma_{2}\,=\{0\}$.

Now, let us set $\Gamma_{\leq 2}(z):=\Gamma_{\leq 1}(z)+z^{2}\Gamma_{2}$. Arguing as in Case 1, and using the property $\,Ker\,\Gamma_{1}\,\cap\,Ker\,\Gamma_{2}\,=\{0\}$, we deduce that for $z$ small enough, $z\neq 0$, the matrix $\Gamma_{\leq 2}(z)$ is non-singular. In particular, for $z\neq 0$ small enough, also $\Gamma(z)$ is non-singular, with $\Gamma(z)^{-1}=\Gamma_{\leq 2}(z)^{-1}+O(1)$.

In order to invert $\Gamma_{\leq 2}(z)$, we use the Jensen-Nenciu Lemma. Let $P:\mathbb{C}^{N}\to\mathbb{C}^{N}$ be the orthogonal projection onto $Ker\,\Gamma_{0}\,\cap\,Ker\,\Gamma_{1}$. Owing to the relations $\Gamma_{0}^{*}=\Gamma_{0}$, $\Gamma_{1}^{*}=-\Gamma_{1}$, we deduce that for $z$ small enough $\Gamma_{\leq 1}(z)+P$ is non-singular, with

[TABLE]

For small $z$, also $\Gamma_{\leq 2}(z)+P$ is non-singular, with

[TABLE]

With similar computations as in Case 1, we get

[TABLE]

Expansion (12) is thus proved in any case. Moreover, statements (i) and (ii) easily follow from the discussion above.

3 Proof of the main Theorem

This Section is devoted to the proof of Theorem 1.1. Let us fix $N\geq 1$, $\alpha\in\mathbb{R}^{N}$ and $Y\subseteq\mathbb{R}^{3}$, and set $\Gamma(z):=\Gamma_{\alpha,Y}(z)$. Preliminary, observe that the low-energy expansion (11) follows by combining the resolvent formula (9) with the small $z$ expansion (12) for $\Gamma(z)^{-1}$. We prove now that $R_{-2}\neq 0$ if and only if $0\in\sigma(-\Delta_{\alpha,Y})$, which in view of Proposition 5, part (i), is equivalent to prove that $Ker\,\Gamma_{0}\,\cap\,Ker\,\Gamma_{1}\neq\{0\}$ if and only if $0\in\sigma(-\Delta_{\alpha,Y})$.

Suppose first that there exists $c=(c_{1},\ldots,c_{N})\neq 0\in\,Ker\,\Gamma_{0}\,\cap\,Ker\,\Gamma_{1}$. We are going to show that the non-zero function

[TABLE]

belongs to $Ker\,(-\Delta_{\alpha,Y})$. First of all, observe that the condition $\Gamma_{1}\,c=0$ is equivalent to $c_{1}+\ldots+c_{N}=0$, which implies $\psi\in L^{2}(\mathbb{R}^{3})$.

Let us fix $z\in\mathbb{C}^{+}\setminus\mathcal{E}^{+}$, and write

[TABLE]

where

[TABLE]

Observe that $F_{z}\in H^{2}(\mathbb{R}^{3})$. Moreover, for every $k\in\{1,\ldots,N\}$,

[TABLE]

where in the second equality we use that $\Gamma_{0}\,c=\Gamma_{1}\,c=0$. By virtue of representation (7), we conclude that $\psi\in\mathcal{D}(-\Delta_{\alpha,Y})$. Moreover, formula (8) yields

[TABLE]

which shows that $\psi\in\,Ker(-\Delta_{\alpha,Y})$.

Let us discuss now the opposite implication. To this aim, consider a function $\psi\in\,Ker\,(-\Delta_{\alpha,Y})\setminus\{0\}$. For a fixed $z=i\lambda\in\mathbb{C}^{+}\setminus\mathcal{E}^{+}$, we can write

[TABLE]

for some non-zero $F_{z}\in H^{2}(\mathbb{R}^{3})$, where

[TABLE]

Observe that the $c_{j}$’s are necessarily independent of $z$, since $\mathcal{G}_{i\lambda}^{y_{j}}\not\in H^{2}(\mathbb{R}^{3})$ for any $j$. Moreover, the condition $\psi\in L^{2}(\mathbb{R}^{3})$ implies $c_{1}+\ldots+c_{n}=0$, namely $\Gamma_{1}c=0$. Owing to (8) and the representation (25), the relation $-\Delta_{\alpha,Y}\psi=0$ is equivalent to

[TABLE]

We show now that $\|F_{i\lambda}\|_{H^{2}}\to 0$ as $\lambda\downarrow 0$, whence also $F_{i\lambda}\to 0$ as $\lambda\downarrow 0$, uniformly on compact subsets of $\mathbb{R}^{3}$. This implies

[TABLE]

and the identity

[TABLE]

which conclude the proof.

In order to show that $\|F_{i\lambda}\|_{H^{2}}\to 0$ as $\lambda\downarrow 0$, we start with the estimate

[TABLE]

Observe moreover that $\widehat{F_{i\lambda}}(p)=\lambda^{2}(p^{2}+\lambda^{2})^{-1}\widehat{\psi}(p)$. By dominate convergence we get $\|F_{i\lambda}\|_{L^{2}}=o(1)$, which combined with (27) yields $\|F_{i\lambda}\|_{H^{2}}=o(1)$, as desired.

Remark 2

By Proposition 5(ii), there is a $\Theta(z^{-1})$ term in the expansion of $\Gamma(z)^{-1}$ at $z=0$ if and only if there exists $c\in\mathbb{R}^{n}$ such that $\Gamma_{0}c=0$, $\Gamma_{1}c\neq 0$. In this case, the function defined by (24) belongs to $L^{2}(\mathbb{R}^{3},\langle x\rangle^{-1-\sigma}dx)\setminus L^{2}(\mathbb{R}^{3})$, for any $\sigma>0$, and formally satisfies $-\Delta_{\alpha,Y}\psi=0$, whence $\psi$ can be interpreted as a zero energy resonance for $-\Delta_{\alpha,Y}$. Hence, as anticipated in Remark 1, we have that $R_{-1}\neq 0$ in expansion (11) if and only if there exists a zero energy resonance, analogously to the case of classical Schrödinger operators.

4 Occurrence and multiplicity of zero energy obstructions

In this Section we discuss the occurrence and the multiplicity of obstructions at zero energy for the resolvent of $-\Delta_{\alpha,Y}$, depending on the choice of the set $Y$ of centers of interactions and the coupling parameters $\alpha_{1},\ldots\alpha_{N}$.

In the single center case, it is easy to check that the only possible obstruction at $z=0$ is a resonance, attained if and only if $\alpha=0$. In general, for any $N$ and for any given configuration of the centers, there exists a measure zero set of choices of the parameters $\alpha_{1},\ldots,\alpha_{N}$ which leads to a zero-energy resonance. By means of the discussion in Section 2 and Section 3, we can define the multiplicity of the zero-energy resonance as

[TABLE]

We conjecture that, as $N$ increases, one can find $Y$ and $\alpha$ such that $r_{\alpha,Y}$ becomes arbitrarily large.

As anticipated in Section 1, when $N=2$ we can find a simple zero eigenvalue by choosing $\alpha_{1}=\alpha_{2}=-(4\pi d)^{-1}$, where $d$ is the distance between the two centers. For $N\geq 3$, a zero eigenvalue occurs for specific geometric configurations of the centers of interactions and for a measure zero set of choices of $\alpha_{1},\ldots,\alpha_{N}$. Owing to the discussion in Section 2 and Section 3, the multiplicity of the zero eigenvalue is given by

[TABLE]

Let us discuss now the maximal possible value for $e_{\alpha,Y}$ as the number of centers of interactions increases.

• •

$N=3$. We can take $Y$ as the vertices of an equilateral triangle of side-length one, and $\alpha_{1}=\alpha_{2}=\alpha_{3}=-(4\pi)^{-1}$. With this choice we get $e_{\alpha,Y}=2$.

• •

$N=4$. We can take $Y$ as the vertices of a regular tetrahedron of side-length one, and $\alpha_{1}=\alpha_{2}=\alpha_{3}=\alpha_{4}=-(4\pi)^{-1}$. With this choice we get $e_{\alpha,Y}=3$.

• •

$N=5$. Observe that we can not find five points in $\mathbb{R}^{3}$ with constant pairwise distances. It easily follows that the maximal value for $e_{\alpha,Y}$ is still three.

One could conjecture that for $N\geq 4$ the maximal value of $e_{\alpha,Y}$ is three. Nevertheless, it is also conceivable that for large $N$ there exist complicated geometrical configurations which lead to a higher multiplicity. Such kind of mechanism is well-known in similar contexts. Consider, for example, the problem in combinatorics to determine the chromatic number of the unit distance graph on $\mathbb{R}^{3}$, that is the graph with vertices set $V=\mathbb{R}^{3}$ and edges set $E=\{(x,y)\in\mathbb{R}^{3}\times\mathbb{R}^{3}\,|\,|x-y|=1\}$. Owing to the compactness principle by De Bruijn and Erdős Erdos this is equivalent, under the axiom of choice, to determine the highest chromatic number of a finite graph embedded in $\mathbb{R}^{3}$ in such a way all its edges have length one. For a graph with $N$ vertices, we have the following situation:

• •

$N=3$. We can consider an equilateral triangle of side-length one, which has chromatic number three.

• •

$N=4$. We can consider a regular tetrahedron of side-length one, which has chromatic number four.

• •

$N=5$. The highest possible chromatic number is still four.

• •

$N=14$. There is a configuration of $14$ points in $\mathbb{R}^{3}$, the Moser-Raiskii spindle, with chromatic number five raiskii ; szewor .

• •

For large $N$, the highest possible chromatic number is known to be between $6$ and $12$ Nechushtan ; radototh ; coulson .

It is evident that there are similarities between the two problems, and it would be interesting to understand if they are actually related. In particular, one may take $Y$ as the vertices of the Moser-Raiskii spindle and wondering whether there exists $\alpha=(\alpha_{1},\ldots,\alpha_{14})$ such that $e_{\alpha,Y}=4$.

Acknowledgments

The author acknowledges an anonymous referee for the useful suggestions, and for pointing out relevant references.