Defect-resonance conjecture for truncated Schrödinger operators

In $\mathbb{R}^d$, where $d$ is an arbitrary positive integer, let $H=-\Delta+V$ be a Schrödinger operator with $V(\boldsymbol{x})$ of unbounded support. Suppose that $H$ has a bound state $\Phi$ with eigenvalue $E$ isolated from the rest of the spectrum, so that

$$\sigma(H)\cap B_{\rho}(E)=\{E\},$$

for some $\rho>0$. Let

$$\boldsymbol{\mu}=\int_{\mathbb{R}^d}\boldsymbol{x}|\Phi(\boldsymbol{x})|^2\,\mathrm{d}\boldsymbol{x}$$

be the bound state's center of mass. For $M>0$, define

$$V_{\mathrm{trunc}}(\boldsymbol{x})=\begin{cases}V(\boldsymbol{x})&\boldsymbol{x}\in B_M(\boldsymbol{\mu}),\\0&\boldsymbol{x}\notin B_M(\boldsymbol{\mu}),\end{cases}$$

and set $H_{\mathrm{trunc}}=-\Delta+V_{\mathrm{trunc}}$. Defect-resonance conjecture. For sufficiently large $M$, provided that $\Phi$ is nonzero at some $\boldsymbol{x}\notin B_M(\boldsymbol{\mu})$, the operator $H_{\mathrm{trunc}}$ has a resonance $z$ in the lower half of the complex plane such that $|z-E|$ is exponentially small in $M$. If $\Phi$ vanishes outside $B_M(\boldsymbol{\mu})$, it is instead a bound state of $H_{\mathrm{trunc}}$. This conjecture proposes that truncating a structure away from a localized bound state converts that state into an exponentially close resonance; establishing it beyond the one-dimensional setting remains open because the methods in the source rely on ordinary differential equation and Floquet theory, while higher-dimensional defect states are known in several models.