Point Spectrum in Schrödinger Operators

The paper demonstrates that for every $d ransition{title}:ge 2$, there exist smooth, nonnegative potentials with arbitrarily small ${V}$ ransition{math}$_\infty$, $]I ransition{math}_d ranisiton{Moving}$ ransition{subtitle}0(.)}, and $]I ransition{math}_d ransition{Power}0 υ$ with pure point spectrum in Schrödinger operators. The construction uses annular lenses and barriers to create barriers supporting compactly supported approximate solutions on a prescribed bounded interval $({2 transit}3 ransition{empty}Pets”) and scales exponential decay. A global spectral statement is derived from local block estimates using spectral-averaging criterion, ensuring the existence of a bounded operator $F_\lambda$ with pure point spectrum. Problem and principal result The paper addresses Barry Simon’s multidimensional L2L^2 conjecture for Schrödinger operators HV=−Δ+Von L2(Rd),H_V=-\Delta+V\quad\text{on }L^2(\mathbb R^d), which asserts, in particular, that the weighted condition Id,0(V)=∫Rd∣x∣1−d∣V(x)∣2 dx<∞\mathcal I{d,0}(V) =\int_{\mathbb R^d}|x|^{1-d}|V(x)|^2,dx<\infty forces an absolutely continuous spectral component of infinite multiplicity at almost every positive energy. The conjecture was motivated by the one-dimensional Deift–Killip theorem and by classical scattering results under pointwise decay assumptions. The main theorem gives a substantially stronger counterexample. For every d≥2d\ge 2, every 0≤σ<1/30\le \sigma<1/3, and every δ>0\delta>0, the authors construct a smooth, bounded, nonnegative potential VV satisfying ∥V∥∞+Id,0(V)+Id,σ(V)<δ|V|\infty+\mathcal I{d,0}(V)+\mathcal I_{d,\sigma}(V)<\delta such that σ(HV)=σess(HV)=[0,∞),Pc(HV)=0.\sigma(H_V)=\sigma_{\mathrm{ess}}(H_V)=[0,\infty), \qquad P_c(H_V)=0. Thus the operator has no absolutely continuous or singular continuous spectrum on the positive half-line. Its spectrum there is pure point, with eigenfunctions forming a complete orthonormal basis; in dimensions two and three, the positive eigenvalues have finite multiplicity and are dense in [0,∞)[0,\infty) (2610.31637). The contradiction with Simon’s conjecture is therefore not caused merely by the appearance of singular continuous spectrum. The construction eliminates the entire continuous spectral subspace while preserving the essential spectrum of the free Laplacian and while making the potential arbitrarily small in both the supremum norm and a stronger weighted HV=−Δ+Von L2(Rd),H_V=-\Delta+V\quad\text{on }L^2(\mathbb R^d),0 …