1 Introduction. Statement of the main results↩︎

The study of Schrödinger operators with random backgrounds reveals a delicate interplay between order and chaos, where even the slightest noise can fundamentally alter the spectral structure.

Let \(\omega_n\) be independent identically distributed random variables taking values \(0\) and \(1\) with probabilities \(q\) and \(p=1-q\). Let \(\chi\) be the characteristic function of the unit cube \([0,1)^d\). As the coupling constant, \(\varepsilon>0\), in front of a random potential \(\sum_n \omega_n \chi(x-n)\) tends to zero, the operator \[\label{operator} -\Delta + \varepsilon\sum_{n\in {\Bbb Z}^d} \omega_n \chi(x-n)- V,\qquad \text{with}\quad V\geqslant 0,\tag{1}\] undergoes a transition where new discrete eigenvalues are born at the left edge of the essential spectrum. Obtaining precise asymptotic estimates for the number of these eigenvalues, \(N(\varepsilon)\), as \(\varepsilon\to 0\) is not merely an exercise in perturbation theory, but a necessity for understanding the threshold effects, the emergence of bound states, and the system’s overall sensitivity to randomness.

Classical estimates offer a starting point. The standard Cwikel-Lieb-Rozenblum (CLR) inequality [1]–[3] asserts that for any \(\varepsilon \geqslant 0\), the number of discrete eigenvalues satisfies \[\label{cwikel} N(\varepsilon)\leqslant C_d\int_{{\Bbb R}^d}V^{d/2}(x) dx,\qquad d\geqslant 3,\tag{2}\] where the constant \(C_{d}\) depends solely on the dimension \(d\). While this elegant estimate holds even at \(\varepsilon = 0\), it loses its utility when confronting potentials \[\label{notinL} V \notin L^{d/2}({\Bbb R}^d)\tag{3}\] that decay logarithmically at infinity. To address slower-decaying potentials, we establish a sharper boundary for \(\varepsilon > 0\). We prove that the eigenvalue count obeys the refined bound: \[\label{MSa} N(\varepsilon)\leqslant\tilde{C}_d \Bigl( \int_{|x|<R}V^{d/2}(x) dx+\int_{|x|>R}\bigl(V(x)-\tilde{\phi}_{\varepsilon}(x)\bigr)_+^{d/2} dx\Bigr).\tag{4}\] In this formulation, \(R = R(\omega)\) acts as a random truncation radius, and \((\dots)_+\) denotes the positive part, defined as \(f_+ = \frac{\vert{}f\vert{}+f}{2}\). Crucially, \(\tilde{\phi }_{\varepsilon }\) acts as a correction function that asymptotically controls the threshold, behaving as

\[\notag \tilde{\phi}_\varepsilon\sim \frac{C}{ ( \ln (|x|))^{2/d} },\qquad \text{as}\quad |x|\to\infty.\] This construction guarantees that the eigenvalue count remains strictly finite for all potentials that do not exceed \(\tilde{\phi }_{\varepsilon }\): \[\notag V\leqslant\tilde{\phi }_{\varepsilon }\quad \implies \quad N(\varepsilon) < \infty.\] Another crusual property of the family of functions \(\tilde{\phi}_\varepsilon\) is that \[\notag \tilde{\phi}_\varepsilon\to 0,\qquad \text{as}\quad \varepsilon\to0 \quad \text{uniformly on compact sets}.\] Therefore, if \(V\) satisfies the non-integrability condition 3 , the right-hand side of our refined bound 4 tends to infinity as \(\varepsilon\to 0\). This accurately reflects that an infinite number of eigenvalues can emerge.

Determining whether a potential creates infinitely or finitely many bound states is a classical problem. The work of Molchanov and Vainberg [4] established a threshold on how slowly a potential can decay before it yields an infinite number of negative eigenvalues. Our first main result tightens that threshold by presenting a stricter quantitative upper bound on the decay constant \(C\). Here, \(\varkappa _{d}\) denotes the volume of the unit ball in \(\mathbb{R}^{d}\).

To precisely count how many discrete eigenvalues, \(N(\varepsilon)\), emerge from the noise as \(\varepsilon \to 0\), we introduce the following auxiliary functions. For any \(A\) such that \(1 < A^d < \frac{1}{q}\), we define the function \(\eta(x)\) for \(\vert{}x\vert{} > A\) as \[\label{eta} \eta(x)=\frac{ \pi \varkappa_d^{1/d}}{ 4((\log_A|x|)^{1/d}+8+\varkappa_d^{1/d}}.\tag{5}\] Next, we define the barrier function \(\phi_\varepsilon(x)\) for \(\vert{}x\vert{} > A\) as \[\notag \phi_\varepsilon(x)=\min\Bigl\{\sqrt\varepsilon\tanh(\sqrt\varepsilon)\eta(x), \,\, \eta^2(x) \Bigr\}.\] Armed with these tools, we state the following result.

This theorem provides a rigorous quantitative upper bound on the number of discrete eigenvalues \(N(\varepsilon)\), extending the classical Cwikel–Lieb–Rozenbljum (CLR) inequality [1], [2], [3] to a disordered setting. Instead of just proving they exist, it allows one to mathematically count and limit how many bound states can emerge when the system transitions into randomness. The formula incorporates both the base potential \(V\) and the "system parameter" \(\phi _{\varepsilon }\) driven by the coupling constant \(\varepsilon\). It explicitly separates the continuous space effects (the integral over \(|x| < R\)) from the stochastic/random background effects (the integral over \(|x| > R\)).

A key achievement of Theorem 1 is that the spatial boundary \(R(\omega)\) is independent of \(\varepsilon\). This means the geometric boundary separating the localized deterministic region (\(\vert{}x\vert{} < R\)) from the disordered infinity (\(\vert{}x\vert{} > R\)) remains frozen, no matter how weak the coupling becomes. The scaling dynamics of the transition is absorbed entirely into the function \(\phi_\varepsilon\) which tends to zero as \(\varepsilon\to0\).

We can further extend this quantification to the sum of the negative eigenvalues.

Proof. We use the standard layer-cake representation to express the sum of the eigenvalues as an integral of the counting function: \[\notag \sum _{j}|\lambda _{j}|{}^{\gamma }=\gamma \int _{0}^{\infty }\lambda ^{\gamma -1}\tilde{N}(\lambda )\,d\lambda\] Here, \(\tilde{N}(\lambda)\) denotes the number of eigenvalues of the operator ?? lying below \(-\lambda\). By the variational principle, \(\tilde{N}(\lambda)\) does not exceed the number of negative eigenvalues of the operator with the shifted potential \((V - \lambda)_+\). We apply the bound from Theorem 2 to \(\tilde{N}(\lambda)\) by replacing \(V\) with \((V - \lambda)_+\). For almost every \(\omega\), there exists an \(R = R(\omega) > A\) such that: \[\notag \tilde{N}(\lambda )\leqslant 2^{d/2}C_{d}\left(\int _{|x|<R}(V(x)-\lambda )_{+}^{d/2}\,dx+\int _{|x|>R}\bigl((V(x)-\lambda )_{+}-2^{-1}\phi _{2\varepsilon }(x)\bigr)_{+}^{d/2}\,dx\right)\] We substitute this inequality back into the layer-cake integral. This splits the expression into two distinct spatial domains: \[\begin{align} \sum_{j}|\lambda_{j}|^{\gamma }\leqslant\gamma 2^{d/2}C_{d}\Bigl( \int _{0}^{\infty }\lambda ^{\gamma -1}\int _{|x|<R}(V-\lambda )_{+}^{d/2}\,dx\,d\lambda +\\ +\int _{0}^{\infty }\lambda ^{\gamma -1}\int _{|x|>R}\bigl(V-\lambda -2^{-1}\phi _{2\varepsilon }\bigr)_{+}^{d/2}\,dx\,d\lambda \Bigr). \end{align}\] The bound ?? follows by Fubini’s theorem, after we swap the order of integration for both terms. \(\,\,\Box\)

The corollary presents an advancement in Lieb-Thirring-type inequalities ([5], [6]) for Schrödinger-type operators with random potentials. The threshold radius \(R(\omega)\) is random, meaning the boundary where the potential’s effective behavior changes depends on the specific realization of the medium.

Theorem 3 highlights a fundamental contrast between one-dimensional and higher-dimensional (\(d \geqslant 3\)) random Schrödinger operators. In dimension \(d=1\), the operator exhibits the property that an arbitrarily weak attractive potential can create at least one bound state. This is reflected in the first term \(1\) of the bound. Furthermore, the dimensionality dictates a different spatial decay and moment requirement on the potential \(V(x)\); for \(d=1\), the first moment \(\int |x|V(x) \, dx\) governs the eigenvalue counting, whereas higher dimensions rely on \(L^{d/2}\) norms.

This theorem establishes an asymptotic upper bound for the number of discrete eigenvalues \(N(\varepsilon)\) as the coupling constant \(\varepsilon\) approaches zero. While general inequalities provide bounds for arbitrary potentials, this result characterizes the specific scaling behavior of bound states near the threshold. Central to this bound is the scaling function \(g(\varepsilon)\), in which the term \(T_{\varepsilon }\) is asymtotically proportional to \(\varepsilon ^{-d}\). Because \(T_{\varepsilon }\) appears in the exponent of the base \(A\), the result indicates that as the perturbation weakens (\(\varepsilon \to 0\)), the number of discrete eigenvalues grows exponentially relative to the inverse power of the coupling constant.

For the one-dimensional case (\(d = 1\)), the statement reduces to the following result. Because the proof of this one-dimensional case differs substantially from the proof of Theorem 4, we state it separately.

While establishing definitive lower bounds is notoriously difficult, we can compare \(N(\varepsilon)\), the number of negative eigenvalues of the random operator \[\label{random} -\Delta+\varepsilon\sum_{n\in {\Bbb Z}^d}\omega_n \chi(x-n)-V(x)\tag{6}\] with \(N_0(\varepsilon)\), the number of eigenvalues of the non-random operator \[\label{notrandom} -\Delta-V\tag{7}\] situated strictly below \(-\varepsilon<0.\) Beyond the obvious inequality \(N_0(\varepsilon)\leqslant N(\varepsilon)\), we establish the following theorem, which quantifies the surplus of negative eigenvalues in the random operator 6 compared to the non-random counterpart 7 .

For the sake of completeness, we provide a proof of the following relatively simple result (see also [4]):

Due to the novelty of our study, direct literature is scarce. Consequently, we limit our bibliography to foundational works analyzing eigenvalue counts in the non-random case and primary contributions in the random case, specifically referencing [7], [4], and [8].

2 Kolmogorov’s Law and Dichotomy↩︎

An operator analogous to 1 can be studied on the lattice \({\Bbb Z}^{d}\): \[\notag \left(H_{\rm disc} u\right)(n)=-\sum_{|m-n|=1} \bigl(u(m)-u(n)\bigr) +\varepsilon\omega_n u(n)-V(n)u(n).\] Here, \(\omega _{n}\) are independent and identically distributed (i.i.d.) random variables taking the values \(0\) and \(1\) with probabilities \(q\) and \(1-q\), respectively. To rigorously establish the spectral properties of \(H_{\rm disc}\), we rely on foundational probabilistic tools.

When the random variables in the sequence are independent, the tail \(\sigma\)-algebra exhibits a fundamental mathematical property known as Kolmogorov’s Zero-One Law. This theorem states that any event \(A\) in the tail \(\sigma\)-algebra of independent random variables is trivial, meaning its probability is always either \(0\) or \(1\):\[\notag P(A) = 0 \text{ or } 1.\] By utilizing the fact that altering finitely many \(\omega _{n}\)’s constitutes a finite-rank perturbation, we translate this probabilistic dichotomy into a deterministic structural result for the operator’s spectrum.

Consequently, the class of nonnegative bounded functions \(V\) on \({\Bbb Z}^d\) splits cleanly into two disjoint sets: \[\notag \{V \text{ with } N(\varepsilon) < \infty \text{ a.s.}\}\qquad \text{and} \qquad\{V \text{ with } N(\varepsilon) = \infty \text{ a.s.}\},\] where \(N(\varepsilon)\) denotes the number of negative eigenvalues of \(H_{\rm disc}\). Although Kolmogorov’s law guarantees this binary alternative, the precise frontier between the finite and infinite eigenvalue regimes is dictated entirely by the spatial decay of the potential \(V\). While this boundary can be completely characterized in simply solvable models (see Section 4), the more realistic, higher-complexity model in this paper allowed us to describe only partial subsets of both regimes.

3 The standard approach fails↩︎

The mathematical study of random media with trapping phenomena traces back to physical frameworks established by Smoluchowski in the context of diffusion-controlled reactions. The exact mathematical results regarding the heat kernel for a random operator in an environment featuring random static traps were pioneered by Alain-Sol Sznitman [9], [10]. These results revolve around the long-time asymptotic behavior of the quenched and annealed heat kernels.In mathematical physics, this model represents Brownian motion in a Poissonian obstacle field. The fundamental random operator under study is the random Schrödinger operator (often called the Dirichlet Laplacian with random traps):\[\notag \tilde{H}_{\omega }=-\frac{1}{2}\Delta +\tilde{V}_{\omega }(x)\] where \(\Delta\) is the Laplacian and \[\notag \tilde{V}_\omega(x) = \sum_{i} W(x - x_i)\] represents a random potential generated by a Poisson point process of static trapping obstacles \(\{x_i\}\) with a constant spatial density \(\nu > 0\). To describe the results, we define \(k_\omega(t,x,y)\) to be the integral kernel of the operator \(\exp(-t \tilde{H}_\omega)\).

The Annealed Heat Kernel Asymptotics. The annealed heat kernel \[\notag\mathbb{E}[k_\omega(t, x, x)]\] averages the return probability over all possible random trap configurations. Sznitman rigorously formalized the exact leading-order long-time decay, known as the Lifshitz tail effect.

For a \(d\)-dimensional space \(\mathbb{R}^{d}\), as time \(t \to \infty\):\[\label{SZ1}\lim _{t\rightarrow \infty }t^{-\frac{d}{d+2}}\log \mathbb{E}[k_{\omega }(t,x,x)]=-c(d,\nu )\tag{8}\] The precise value of the constant is defined explicitly by:\[\notag c(d,\nu )=\left(\frac{d+2}{d}\right)\left(\frac{2\nu \varkappa _{d}}{d}\right)^{\frac{2}{d+2}}\lambda _{1}(B_{1})^{\frac{d}{d+2}}\] where \(\varkappa _{d}\) is the volume of the unit ball in \(\mathbb{R}^{d}\), and \(\lambda_1(B_1)\) is the principal (lowest) eigenvalue of the Dirichlet Laplacian on a unit ball \(B_{1}\).

This exact result reflects that the survival of a Brownian particle over long periods relies entirely on finding a vast, naturally occurring "empty clearing" of radius \(R \sim t^{1/(d+2)}\) that is entirely free of traps.

The Quenched Heat Kernel Asymptotics. The quenched heat kernel \(k_\omega(t, x, x)\) tracks the return probability within a single, fixed almost every realization of the random environment. Sznitman established that for almost every realization \(\omega\), the individual sample behaves entirely differently from the ensemble average because large clearings are exponentially rare but inevitably exist somewhere far away. For a fixed realization \(\omega\), as \(t \to \infty\):\[\label{SZ2} \lim _{t\rightarrow \infty }\frac{\log k_{\omega }(t,x,x)}{t/(\log t)^{2/d}}=-c_{\text{quenched}}(d)\tag{9}\] Notice that the decay rate changes from \(t^{\frac{d}{d+2}}\) in the annealed case to \(\frac{t}{(\log t)^{2/d}}\) in the quenched case.

A physical interpretation of this fact is the following. Because the environment is fixed, the particle cannot rely on a clearing being natively present exactly where it starts. Instead, it must pay an entropic transport cost to travel across space to reach the nearest massive trap-free clearing.

To achieve these exact results, Sznitman developed a specialized mathematical technique called the “Enlargement of Obstacles”. Instead of dealing with irregular and granular random potential formulas, this method uses coarse-graining to replace thousands of random scattered traps with a smooth, macroscopically larger effective boundary condition. This breakthrough allowed for sharp, mathematically exact multi-scale estimates of the principal Dirichlet eigenvalues on random domains, ultimately yielding the precise geometric constants listed above.

Limitations of the Standard Approach. The rigorous mapping of heat kernel expectations to variational problems and Dirichlet eigenvalues was established in classic papers by Donsker and Varadhan [11], [12] on the large deviation principle (LDP) for Brownian motion. Specifically, 8 was established for the unperturbed Anderson model. While a precise understanding of heat kernel behavior theoretically justifies applying Lieb’s approach [2] (which is based on the Feynman–Kac formula) for obtaining eigenvalue bounds, this method ultimately breaks down. As demonstrated by Molchanov and Vainberg [4], using the estimate \[\notag {\Bbb E}[N(\varepsilon)]\leqslant C(\sigma)\int_{{\Bbb R}^d} V(x)\int_{\frac{\sigma}{V(x)} }^\infty {\Bbb E}[k_\omega(t,x,x)]dt dx,\qquad \sigma>0,\] reveals that the annealed estimate 8 is too coarse; it only guarantees a finite number of negative eigenvalues if the potential decays faster than \(C/(\ln\vert{}x\vert{})^{1+2/d}\) at infinity.

While employing the quenched asymptotics 9 would provide the necessary sharpness, this approach fails in practice and its formal application leads to an incorrect result. The rate at which the quenched kernel \(k_{\omega }\) converges to its asymptotic limit depends wildly and non-uniformly on the spatial variable \(x\). This lack of uniformity renders the standard methodology inapplicable.

4 One exactly solvable model↩︎

This section establishes a simplified, mathematically rigorous baseline for studying random operators, providing a solvable benchmark to determine conditions for when a random system has finitely or infinitely many bound states.

Let \(\{x_n\}_{n \in \mathbb{N}}\) be a sequence of independent and identically distributed (i.i.d.) random variables with a probability density function \(f(x) = e^{-x}\) on \([0, \infty).\) We define interval endpoints \(\{a_n\}_{n \geqslant 0}\) via the partial sums:\[\notag a_n = \sum_{j=1}^n x_j, \quad a_0 = 0.\] Consider the differential operator \(\mathcal{H}\) acting as the orthogonal sum of operators on the disjoint intervals \([a_n, a_{n+1}]\): \[\label{H1} \mathcal{H} = \bigoplus_{n=0}^{\infty} \left( -\frac{d^2}{dx^2} - V(x) \right).\tag{10}\] This operator is subject to Dirichlet boundary conditions at the endpoints \(a_{n}\). Assume that the potential \(V(x)\) is piecewise constant, defined as \[\notag V(x) = v_n\geqslant 0 \quad \text{for} \quad x \in [a_n, a_{n+1}).\]

The negative spectrum of \(\mathcal{H}\) is a discrete, random set formed by the eigenvalues of each isolated interval \([a_n, a_{n+1}]\). For a given interval index \(n\) and mode index \(m\), the negative eigenvalues are given by: \[\notag E_{n,m}=\left(\frac{m\pi }{x_{n+1}}\right)^{2}-v_{n}<0\] where \(n \in \{0, 1, 2, \dots\}\) and \(m \in \mathbb{N} = \{1, 2, 3, \dots\}\), subject to the condition that \(v_n > \left(\frac{m\pi}{x_{n+1}}\right)^2\). The condition for the \(n\)-th interval to possess at least one negative eigenvalue is \[\label{x62v} x_{n+1} > \frac{\pi}{\sqrt{v_n}}.\tag{11}\] We know that \(\{x_n\}_{n \in \mathbb{N}}\) are i.i.d. variables that follow an exponential distribution whose density is \(e^{-x}\). Thus the probability that 11 holds is \[\notag P_n = \int_{\frac{\pi}{\sqrt{v_n}}}^{\infty} e^{- x} \,dx=e^{- \frac{\pi}{\sqrt{v_n}}}.\] Applying the Borel-Cantelli lemmas, we obtain the following result.

Remark. This model exhibits unique behavior because the potential \(V\) introduces an additional layer of randomness, contrasting with our previous model where the potential was deterministic.

5 Painting the space in two colors↩︎

For a fixed scaling factor \(A > 1\), consider a sequence of expanding "cubic" layers in \(\mathbb{R}^{d}\). Each layer is covered by a collection of \(d\)-dimensional half-open cubes having a side length \(n \in \mathbb{N}\). These cubes are defined by: \[\notag Q_{\tilde{n}, n}= n\cdot(\tilde{n}+ [0,1)^d),\qquad \text{with}\quad \tilde{n}\in {\Bbb Z}^d.\] The outer boundary of the \(n\)-th layer is located at a distance of \(R_n = A^{n^d}\) from the origin.

The number of the cubes \(Q_{\tilde{n}, n}\) covering the \(n\)-th layer, denoted by \(N_n\), does not exceed the following quantity: \[N_n\leqslant\frac{2^d}{n^d} ((A^{ n^d}+n)^d-(A^{(n-1)^d}-n)^d)\]

Now, consider a single cube \(Q_{\tilde{n}, n}\) of side length \(n\) consisting of \(n^d\) unit cells. Suppose each unit cell is painted black with probability \(p\) and white with probability \(q=1-p\) independently.

We define a "boundary condition" where the number of black cells does not exceed the number of cells on the cube’s boundary, \(n^d-(n-1)^d\). The probability that the number of black cells in the cube does not exceed \(n^d-(n-1)^d\) is not larger than \[\sum_{k=(n-1)^d}^{n^d}\frac{n^d!}{k!(n^d-k)!} q^{k}(1-q)^{n^d-k}.\] Therefore, applying a union bound, the probability \(P_n\) that at least one such cube exists in the \(n\)-th layer is bounded by: \[P_n\leqslant\frac{2^d}{n^d} ((A^{ n^d}+n)^d-(A^{(n-1)^d}-n)^d)\sum_{k=(n-1)^d}^{n^d}\frac{n^d!}{k!(n^d-k)!} q^{k}(1-q)^{n^d-k}.\] By the Borel-Cantelli Lemma, the property occurs finitely often almost surely if \(\sum_n P_n<\infty.\) Given that \(N_n\) grows as \(A^{dn^d}\) and the probability decays as \(q^{n^d}\), the series converges if the growth of the volume is offset by the decay of the probability, specifically when \[A^d<q^{-1}.\]

Thus, we obtain the following result.

Finally, note that the Euclidean distance \(|x|\) from the origin to the cube \(Q_{\tilde{n}, n}\) satisfies \[\notag A^{(n-1)^d}-n\sqrt d \leqslant|x|\leqslant A^{ n^d}+n\sqrt d.\] Consequently, as the distance approaches infinity, the layer index \(n\) scales as: \[\notag n=\Bigl(\frac{\ln |x|}{\ln A}\Bigr)^{1/d} (1+o(1)),\qquad \text{as}\quad |x|\to\infty.\] However, the nature of our results requires a more accurate estimate of \(n\) in terms of \(|x|\). \[\label{n60x} n\leqslant\left(\log_A|x|\right)^{1/d} +2,\qquad \text{for}\quad n\geqslant n_1(A).\tag{12}\]

6 Lowest eigenvalue for the Neumann problem↩︎

For a bounded domain \(\Omega \subset \mathbb{R}^n\) and a subset \(X \subset \Omega\) with \(|X| = m\), let \(\lambda_1(\Omega, X)\) be the first eigenvalue of: \[-\Delta u+\epsilon \chi _{X}u=\lambda u\text{\;in\;}\Omega ,\quad \frac{\partial u}{\partial \nu }=0\text{\;on\;}\partial \Omega\] where \(\chi _{X}\) is the indicator function of \(X\). If \(B\) is a ball with \(|B| = |\Omega|\) and \(S\) is a spherical layer at the boundary of \(B\) with \(|S| = |X|\), then \(\lambda_1(\Omega, X) \geqslant\lambda_1(B, S)\). Put differently, we intend to prove that the lowest eigenvalue of the Neumann problem with a localized potential is minimized when the domain is a ball and the potential is concentrated in a spherical layer at the boundary.

The first eigenvalue \(\lambda_1(\Omega, X)\) is characterized by the Rayleigh quotient: \[\lambda _{1}(\Omega ,X)=\inf _{u\in H^{1}(\Omega ),u\ne 0}\frac{\int _{\Omega }|\nabla u|{}^{2}dx+\epsilon \int _{X}|u|^{2}dx}{\int _{\Omega }|u|^{2}dx}\] For \(\epsilon > 0\), the first eigenfunction \(u\) is strictly positive and can be chosen such that \(\int_{\Omega} u^2 dx = 1\). Thus: \[\lambda _{1}(\Omega ,X)=\int _{\Omega }|\nabla u|{}^{2}dx+\epsilon \int _{X}u^{2}dx\]

Let \(B\) be the ball centered at the origin such that \(|B| = |\Omega|\). By the Faber-Krahn inequality principle [13], [14] and the Polya-Szego theorem, for any \(u \in H^1(\Omega)\), its spherically symmetric decreasing rearrangement \(u^* \in H^1(B)\) satisfies: \[\int_B (u^*)^2 dx = \int_{\Omega} u^2 dx\] \[\int_B \vert\nabla u^*\vert{}^2 dx \leqslant\int_{\Omega} \vert\nabla u\vert^2 dx\]

To minimize the term \(\epsilon \int_X u^2 dx\), we use the Hardy-Littlewood inequality (see [15]) which states that for two non-negative functions \(f\) and \(g\), the integral of their product is maximized when both functions are symmetrically rearranged: \[\int_\Omega f g dx\leqslant\int_B f^* g^* dx.\] Consequently, \[\int_\Omega (1-\chi_X)|u|^2 dx\leqslant\int_B (1-\chi_S) |u^*|^2 dx,\] which implies that \[\epsilon \int_\Omega \chi_X|u|^2 dx\geqslant\epsilon \int_B \chi_S |u^*|^2 dx.\] The interpretation of this inequality is the following. To minimize the term \(\epsilon \int_X u^2 dx\), must place the potential \(\epsilon\) where the eigenfunction \(u\) takes its smallest values. In the Neumann case, the lowest eigenvalue for a fixed domain is minimized when the potential \(\epsilon \chi_X\) is "as far as possible" from the "center" of the domain.

Combining these steps, let \(u\) be the first eigenfunction for \((\Omega, X)\). Then: \[\lambda _{1}(\Omega ,X)=\int _{\Omega }|\nabla u|{}^{2}dx+\epsilon \int _{X}u^{2}dx\geqslant\int _{B}|\nabla u^{*}|{}^{2}dx+\epsilon \int _{S}(u^{*})^{2}dx\] where \(S\) is the spherical layer of volume \(|X|\) at the boundary of \(B\). Since \(u^{*}\) is a valid test function for the Neumann problem on \(B\) with potential \(\epsilon \chi_S\), the right-hand side is greater than or equal to the true first eigenvalue \(\lambda_1(B, S)\) by the variational principle.

Consider a particle confined within a one-dimensional, symmetric potential well. Our goal is to estimate its lowest energy eigenvalue,\(\lambda\). We define the Schrödinger operator \[\label{HL} -\frac{d^2}{d x^2}+ \varepsilon\chi_L\tag{13}\] with the Neumann conditions on the boundary of the interval \([-L,L]\), where \(\chi_L\) denotes the characteriostic function of the set \[L-1<|x|<L, \quad \text{for}\quad L>1.\] By introdicing the potential, we subject the particle to a potential barrier \(\varepsilon>0\).

In the region where \(|x|<L-1\), the eigenfunction \(u\) corresponding to the eigenvalue \(\lambda>0\) equals \(u=A \cos (\sqrt \lambda x)\). On the other hand, in the region \(L-1<|x|<L\), it equals \(u=B \cosh(\sqrt{\varepsilon-\lambda }(L-x))\). Matching \(u\) and \(u'\) at the popint \(x=L-1\) leads to \[\begin{align} A \cos (\sqrt \lambda (L-1))=B \cosh(\sqrt{\varepsilon-\lambda })\\ A \sqrt \lambda \sin (\sqrt \lambda (L-1))=B \sqrt{\varepsilon-\lambda }\sinh(\sqrt{\varepsilon-\lambda }) \end{align}\] Dividing one equation by the other, we obtain that \[\label{eqn1} \sqrt \lambda \tan(\sqrt \lambda (L-1))= \sqrt{\varepsilon-\lambda }\tanh(\sqrt{\varepsilon-\lambda })\tag{14}\] It is very well known that \(\tanh x<x\) for \(x>0\). Consequently, \[\label{eqn2} \sqrt \lambda \tan(\sqrt \lambda (L-1))<\varepsilon-\lambda\tag{15}\] By analyzing the monotonicity of the left (increasing) and right (decreasing) sides of 14 , we guarantee a unique solution for the eigenvalue \(\lambda\). Namely, the right hand side of 14 is a monotonically decreasing function of \(\lambda\) on the interval \([0,\varepsilon]\), while the left hand side is monotonically increasing from \(0\) to \(\infty\) on the interval \([0, (\pi/(2(L-1)))^2]\). Consequently, the equation 14 has a unique solution on the intersection of these two intervals, and this solution satisfies \[\notag \sqrt \lambda(L-1)<\pi/2.\] Therefore, 15 implies the inequality \[\notag \lambda<\varepsilon/L.\]

For a given \(\varepsilon>0\), we define the system parameter \(\delta\in(0,1)\) to be \[\notag \delta=1-\frac{\tanh \sqrt \varepsilon}{\sqrt \varepsilon}.\] If we know in advance that \(0<\sqrt \lambda (L-1)<\pi/4\), then \(\tan (\sqrt \lambda (L-1) )<\frac{4}{\pi} \sqrt \lambda (L-1)\). Furthermore, we have \[\notag \sqrt{\varepsilon-\lambda} \tanh \sqrt{\varepsilon-\lambda}\geqslant(1-\delta)(\varepsilon-\lambda).\] Combining the two inequalities yields the lower bound \[\notag \lambda>\frac{ \varepsilon\pi (1-\delta)}{ 4L+1}.\] This leads to the following result.

Separating the variables we obtain the following result.

Combining Theorems  10 and  11 with Corollary  2 yields the following statement.

We now define the function \(\phi_\varepsilon\) by setting \[\notag \phi_\varepsilon(x)=\min\Bigl\{\frac{ \varepsilon\pi (1-\delta)\varkappa_d^{1/d}}{ 4((\log_A|x|)^{1/d}+8+\varkappa_d^{1/d}}, \,\, \Bigl( \frac{\pi \varkappa_d^{1/d}}{4 ( \log_A |x|)^{1/d} +8 } \Bigr)^2 \Bigr\},\qquad |x|>A.\] Let \(\tilde{\chi}_R\) denote the characteristic function of the region \(\{x\in {\Bbb R}^d:\quad |x|>R\}\). We claim that for any \(\omega\), there exists \(R_0(\omega)>0\) such that the operator \[\notag -\Delta+\sum_{n\in {\Bbb Z}^d}\omega_n \chi(x-n)- \phi_\varepsilon(|x|)\tilde{\chi}_R\] does not have negative eigenvalues (i.e. it is positive) for all \(R>R_0(\omega)\). Moreover,

7 Key Estimates and Inequalities↩︎

The number of negative eigenvalues of a Schrödinger operator with a sum of potentials \(V_1 + V_2\) is estimated using the variational principle. This estimate relates the total number of bound states (negative energy levels) of the combined system to the bound states related to the potentials \(V_1\) and \(V_2\) separately.

The relationship between the number of negative eigenvalues \(N(V)\) of an operator \(-\Delta - V\) for the combined and separate potentials is generally described by the subadditivity of the counting function. Namely, for any two potentials \(V_{1}\) and \(V_{2}\), the number of negative eigenvalues of the operator with the sum potential satisfies: \[\notag N(V_{1}+V_{2})\leqslant N(V_{1}/(1-\tau ))+N(V_{2}/\tau )\]for any \(0 < \tau< 1.\) This arises because the quadratic form of the combined operator is the sum of the individual forms, and the dimension of the subspace where the sum is negative is constrained by the dimensions of the subspaces where the components are negative. Applying this to a perturbed system with \(\chi_n(x) = \chi(x-n)\), we obtain: \[\notag N(V- \varepsilon\sum_{n\in {\Bbb Z}^d}\omega_n \chi_n)\leqslant N(\tilde{\chi}_R \phi_{2\varepsilon}- 2\varepsilon\sum_{n\in {\Bbb Z}^d}\omega_n \chi_n)+N(2V-\tilde{\chi}_R\phi_{2\varepsilon})\] Choosing \(R>R_0(\omega)\), we obtain that \[\label{N60N402V41} N(V- \varepsilon\sum_{n\in {\Bbb Z}^d}\omega_n \chi_n)\leqslant N(2V-\tilde{\chi}_R\phi_{2\varepsilon})\tag{16}\] For a Schrödinger operator \(-\Delta - V\) acting on the Hilbert space \(L^2(\mathbb{R}^d)\), with \(d\geqslant 3\) :\[\label{CRL} N(V)\leqslant C_{d}\int _{\mathbb{R}^{d}}V_+(x)^{d/2}\,dx\tag{17}\] Here, \(N(V)\) is the total number of negative eigenvalues, counting multiplicities (physically, the number of bound states an electron can occupy), \(C_{d}\) is a universal constant that depends only on the spatial dimension \(d\), and \(V_+(x)\) is the positive part of the potential function \(V\): \[\notag V_+(x)=\max(V(x), 0).\] Consequently, if follows from 16 and 17 that \[\notag N(V- \varepsilon\sum_{n\in {\Bbb Z}^d}\omega_n \chi_n)\leqslant 2^{d/2} C_d \Bigl( \int_{|x|<R}V^{d/2}(x) dx+\int_{|x|>R}\bigl(V(x)-2^{-1} \phi_{2\varepsilon}\bigr)_+^{d/2} dx\Bigr)\] This proves Theorem 2.

For the case \(d=1\), we replace the Cwikel-Lieb-Rozenblum (CLR) inequality with the Bargmann inequality. In one dimension, the CLR bound fails because a negative potential, no matter how small or fast-decaying, always produces at least one bound state. In the one-dimensional case, we utilize specific bounds for the number of negative eigenvalues \(N(V)\) of the operator \[\notag -\frac{d^2}{dx^2} - V.\] According to the Bargmann inequality, the number of bound states is controlled by the first moment of the potential:\[\label{Bargmann}N(V) \leqslant 1 + \int_{-\infty}^{\infty} |x| V_+(x) , dx.\tag{18}\] Applying the subadditivity principle and the reduction in 16 to the one-dimensional case, we can use the Bargmann bound 18 to obtain: \[\notag N\left(V- \varepsilon\sum_{n\in {\Bbb Z}_+}\omega_n \chi_n\right) \leqslant 1+\int _{-\infty}^{\infty }|x|\left(2V(x)- \tilde{\chi }_{R}\phi _{2\epsilon }(x)\right)_{+}dx.\] Splitting the integration domain at \(R\), we arrive at the following estimate for \(d=1\):\[\notag N\left(V- \varepsilon\sum_{n\in {\Bbb Z}_+}\omega_n \chi_n\right) \leqslant 1+2\left(\int _{|x|<R}|x|V_{+}(x)dx+\int _{|x|>R}|x| \left(V(x)-\frac{1}{2} \phi _{2\epsilon }(x)\right)_{+}dx\right).\] This provides the necessary bound on the number of negative eigenvalues for the perturbed one-dimensional system, analogous to the result obtained via the CLR inequality in higher dimensions.

8 Proof of Theorem 4↩︎

According to assumptions of Theorem 4, \[\notag V(x) = \frac{C}{\left(4 ( \log_A |x|)^{1/d} +8 +\varkappa_d^{1/d}\right)^2},\] where \[\notag C< \pi^2 \varkappa_d^{2/d} .\] The first integral on the right hand side of ?? does not contain \(\varepsilon\). Since \(V\), it is bounded, this term contributes a constant, independent of \(\varepsilon\),.

We can replace \(\phi_{2\varepsilon}(x)\) by the function defined as the minimum of two terms: \[\notag\phi_\varepsilon(x)=\min\Bigl\{\frac{ \varepsilon\pi (1-\delta)\varkappa_d^{1/d}}{ 4((\log_A|x|)^{1/d}+8+\varkappa_d^{1/d}}, \,\, \Bigl( \frac{\pi \varkappa_d^{1/d}}{4 ( \log_A |x|)^{1/d} +8 +\varkappa_d^{1/d}} \Bigr)^2 \Bigr\},\qquad |x|>A.\] For large \(|x|\) (where \(\log_A|x|\) is large), the first term is strictly larger than the second term which is larger than \(V\). Therefore, \[\begin{align} \int_{|x|>R}&\Bigl(V(x)-\frac{1}{2}\phi _{2\varepsilon }(x)\Bigr)_+^{d/2} dx = \\ \int_{|x|>R} &\left(\frac{C}{\left(4 ( \log_A |x|)^{1/d} +8 +\varkappa_d^{1/d}\right)^2} - \frac{ \varepsilon\pi (1-\delta_{2\varepsilon})\varkappa_d^{1/d}}{ 4(\log_A|x|)^{1/d}+8+\varkappa_d^{1/d}} \right)_+^{d/2}dx. \end{align}\] The positive part \((\dots)_+\) implies the integrand is non-zero when \[\notag \log _{A}|x|<\left(\frac{C}{4\varepsilon \pi (1-\delta _{2\varepsilon })\varkappa _{d}^{1/d}}-2-\varkappa _{d}^{1/d}/4\right)^d=T_\varepsilon.\]

Using spherical coordinates in the domain \(|x| > R\) and substituting \(t = \log_A|x|\), we conclude that the second integral equals \[I(\varepsilon)=|{\Bbb S}_d|\int _{\log_A R}^{\infty }\left(\frac{C}{\left(4t^{1/d}+8+\varkappa_d^{1/d}\right)^2}-\frac{\varepsilon \pi \varkappa _{d}^{1/d}}{4 t^{1/d}+8+\varkappa_d^{1/d}}\right)_{+}^{d/2}A^{t\cdot d}\ln A\,dt\] where \(|{\Bbb S}_d|=d \varkappa_d\) is the area of the unit sphere. We factor out \(\frac{C}{\left(4t^{1/d}+8+\varkappa_d^{1/d}\right)^2}\) from the parenthesis: \[\begin{align} \left(\frac{C}{\left(4t^{1/d}+8+\varkappa_d^{1/d}\right)^2}-\frac{\varepsilon \pi \varkappa _{d}^{1/d}}{4 t^{1/d}+8+\varkappa_d^{1/d}}\right)_{+}^{d/2}=\\ \frac{C^{d/2}}{\left(4t^{1/d}+8+\varkappa _{d}^{1/d}\right)^{d}} \left(1-\frac{\varepsilon(1-\delta_{2\varepsilon}) \pi \varkappa _{d}^{1/d} \left(4t^{1/d}+8+\varkappa _{d}^{1/d}\right)}{C}\right)_{+}^{d/2}\\= \frac{(4\varepsilon(1-\delta_{2\varepsilon}) \pi)^{d/2} \varkappa _{d}^{1/2} T_\varepsilon^{1/2}}{\left(4t^{1/d}+8+\varkappa _{d}^{1/d}\right)^{d}} \left(1-\left(\frac{t}{T_{\varepsilon }}\right)^{1/d}\right)_+^{d/2}, \end{align}\] where \(T_\varepsilon = \left(\frac{C}{4\varepsilon \pi (1-\delta _{2\varepsilon })\varkappa _{d}^{1/d}}-2-\varkappa _{d}^{1/d}/4\right)^d\).

Then the asymptotics of the integral becomes:\[\notag I(\varepsilon )\sim |{\Bbb S}_{d}|4^{-d}C^{d/2}\ln A\int _{\log _{A}R}^{T_{\varepsilon }}\frac{1}{t}\left(1-\left(\frac{t}{T_{\varepsilon }}\right)^{1/d}\right)^{d/2}A^{dt}\,dt.\] Obviously, the integral \(\int _{\log _{A}R}^{T_{\varepsilon }}\frac{1}{t}\left(1-\left(\frac{t}{T_{\varepsilon }}\right)^{1/d}\right)^{d/2}A^{dt}\,dt\) can be replaced by \(\int _{T_{\varepsilon }/2}^{T_{\varepsilon }}\frac{1}{t}\left(1-\left(\frac{t}{T_{\varepsilon }}\right)^{1/d}\right)^{d/2}A^{dt}\,dt\), because \[\notag \int _{\log _{A}R}^{T_{\varepsilon }/2}\frac{1}{t}\left(1-\left(\frac{t}{T_{\varepsilon }}\right)^{1/d}\right)^{d/2}A^{dt}\,dt\leqslant\frac{1}{\log _{A}R} A^{dT_{\varepsilon }/2}.\] Changing the variables, we obtain \[\notag \begin{align} I(\varepsilon )\sim |{\Bbb S}_{d}| 4^{-d}C^{d/2}\ln A\int _{1/2}^{1}\frac{1}{t}\left(1-t^{1/d}\right)^{d/2}A^{dT_{\varepsilon }t}\,dt\sim \\ |{\Bbb S}_{d}|4^{-d}C^{d/2}\ln A\int _{0}^{\infty}\left(\frac{s}{d}\right)^{d/2}A^{dT_{\varepsilon }(1-s)}\,ds. \end{align}\] Finally, computing the remaining exponential integral, the asymptotics as \(\varepsilon \to 0\) are given by:\[\notag I(\varepsilon )\sim |{\Bbb S}_{d}| 4^{-d}C^{d/2}\ln A \frac{A^{dT_{\varepsilon}}}{d^{d+1} (T_{\varepsilon} \ln A)^{d/2 + 1}} \Gamma\left(\frac{d}{2} + 1\right).\] Using the identities for the surface area of a \(d\)-dimensional sphere \(\vert{}{\Bbb S}_{d}\vert{} = \frac{2\pi^{d/2}}{\Gamma(d/2)}\) and \(\Gamma(\frac{d}{2}+1) = \frac{d}{2}\Gamma(\frac{d}{2})\), we have \(\vert{}{\Bbb S}_{d}\vert{} \Gamma(\frac{d}{2}+1) = d\pi^{d/2}\). Thus, we obtain \[\notag I(\varepsilon )\sim \left(\frac{C\pi }{16d^{2}}\right)^{d/2}\frac{A^{dT_{\varepsilon }}}{T_{\varepsilon }^{d/2+1}(\ln A)^{d/2}}\quad \text{as\;}\varepsilon \rightarrow 0.\]

9 Proof of Theorem  5↩︎

For the reader’s convenience, we recall that \[\notag V(x) = \frac{C}{\left(4 \log_A \vert{}x\vert{} + 10 \right)^2}, \quad\text{ and } \quad T_{\varepsilon} = \frac{C}{8\varepsilon \pi (1-\delta_{2\varepsilon})} - \frac{5}{2}.\] According to Calogero’s bound in one dimension, the number of bound states \(N\) of the operator \(-d^2/dx^2 - W(x)\) is bounded from above. For a potential \(W(x)\geqslant 0\) that monotonically decreases with distance from the origin, the inequality is: \[\notag N\leqslant 1+\frac{2}{\pi }\int _{-\infty }^{\infty }|W(x)|{}^{1/2}\,dx.\] Applying the inequality 16 , we obtain that \[\notag N(\varepsilon)\leqslant N(2V-\alpha\tilde{\chi}_R\phi_{2\varepsilon})\leqslant 1+\frac{2}{\pi }\int _{-\infty}^\infty (2V(x)-\tilde{\chi}_R\phi_{2\varepsilon})_+^{1/2}\,dx,\] We evaluate this integral in two parts: for \(|x|\leqslant R\) (which contributes a bounded constant), and for \(|x| > R\). To find the leading-order asymptotics of the tail integral as \(\varepsilon \to 0\), we expand the parameter \(\delta\) and the function \(\phi_{2\varepsilon}(x)\) with \(d=1\). The one-dimensional function \(\phi_{2\varepsilon}(x)\) becomes:\[\notag\phi_\varepsilon(x)=\min\Bigl\{\frac{ 2\varepsilon\pi (1-\delta_\varepsilon)}{ 4\log_A|x|+10}, \,\, \Bigl( \frac{2\pi }{4 \log_A |x|+10} \Bigr)^2 \Bigr\},\qquad |x|>A.\] For large \(|x|\), the first term is strictly larger than the second, which is larger than \(V\). Thus, we can replace \(\phi_{2\varepsilon}(x)\) by \(\frac{ \varepsilon\pi (1-\delta_{2\varepsilon})}{ \log_A|x|+5/2}\). Substituting this into the positive part of the function yields: \[\notag \int _{|x|>R}\left(V(x)-\frac{1}{2}\phi _{2\varepsilon }(x)\right)_{+}^{1/2}dx=2 \int _{R}^{\infty }\left( \frac{C}{\left(4 \log_A \vert{}x\vert{} + 10 \right)^2}- \frac{ \varepsilon\pi (1-\delta_{2\varepsilon})}{ 2\log_A|x|+5}\right)_{+}^{1/2}dx\] The integrand is non-zero when:\[\notag \frac{C}{\left(4 \log_A \vert{}x\vert{} + 10 \right)^2} > \frac{ 2\varepsilon\pi (1-\delta_{2\varepsilon})}{ 4\log_A|x|+10} \quad \iff \quad \log _{A}|x| < \frac{C}{8\varepsilon \pi (1-\delta_{2\varepsilon})}-\frac{5}{2}.\] Letting \(T_{\varepsilon} = \frac{C}{8\varepsilon \pi (1-\delta_{2\varepsilon})}-\frac{5}{2}\) and substituting \(t = \log_A|x|\), we conclude that the integral equals \[\notag I(\varepsilon) = 2\int _{\log_A R}^{\infty }\left( \frac{C}{\left(4 t + 10 \right)^2}- \frac{ \varepsilon\pi (1-\delta_{2\varepsilon})}{ 2t+5}\right)_{+}^{1/2}A^t\ln A \, dt.\] Factoring out \(\frac{C}{\left(4 t + 10 \right)^2}\) from the parenthesis gives the following result: \[\notag I(\varepsilon )=2\sqrt{C}\ln A \sqrt{\frac{T_{\varepsilon }}{T_{\varepsilon }+\frac{5}{2}}}\int _{\log _{A}R}^{\infty }\frac{1}{4t+10}\left(1-\frac{t}{T_{\varepsilon }}\right)_{+}^{1/2}A^{t}\,dt.\] The asymptotics of the integral is therefore \[\notag I(\varepsilon) \sim \frac{ \sqrt{C}\ln A }{2}\int _{\log _{A}R}^{T_{\varepsilon }}\frac{1}{t}\left(1-\frac{t}{T_{\varepsilon}}\right)^{1/2}A^{t}\,dt.\] Using Laplace’s method for asymptotic evaluation, the dominant contribution arises near the point \(t = T_{\varepsilon}\). Replacing the lower limit \(\log _{A}R\) with \(T_{\varepsilon}/2\) and changing variables, we get: \[\notag I(\varepsilon) \sim \frac{ \sqrt{C}\ln A }{2}\int _{0}^{\infty} s^{1/2}A^{T_{\varepsilon}(1-s)}\,ds\] Consequently, evaluating this asymptotic limit as \(\varepsilon \to 0\) gives: \[\notag I(\varepsilon )\sim \frac{\sqrt{\pi C}\cdot A^{T_{\varepsilon }}}{4 T_{\varepsilon }^{3/2}\sqrt{\ln A}}.\] This proves Theorem 5.

10 Existence of white cubes of length \(n\)↩︎

Proof. Indeed, the \(n\)-th layer occupies the region between the boundaries \(A^{(n-1)^{d}}\) and \(A^{n^{d}}\). The total volume of this layer (approximately) grows proportionally to \(A^{d\cdot n^{d}}\). Since each individual cube \(Q_{\tilde{n},n}\) has a volume of \(n^{d}\), we can place a large number \(M_{n}\) of disjoint (independent) cubes entirely within the \(n\)-th layer. As \(n \to \infty\), this number scales as: \[\notag M_{n}\approx \frac{A^{d\cdot n^{d}}}{n^{d}}.\]

For a single cube of side length \(n\) to be completely white, all of its \(n^{d}\) unit cells must be white. Since each cell is white independently with probability \(q\), the probability that a specific cube is entirely white is:\[\notag P(\text{Cube\;is\;all\;white})=q^{n^{d}}.\] Conversely, the probability that a specific cube is not entirely white is \(1 - q^{n^d}\). Because the \(M_{n}\) cubes are disjoint, their cell configurations are mutually independent. The probability \(P(E_n)\) that none of the \(M_{n}\) cubes in the \(n\)-th layer are entirely white is: \[\notag P(E_{n})=\left(1-q^{n^{d}}\right)^{M_{n}}.\] Using the standard inequality \(1 - x \leqslant e^{-x}\), we bound this probability by:\[\notag P(E_{n})\leqslant\exp \left(-M_{n}\cdot q^{n^{d}}\right)=\exp \left(-\frac{A^{d\cdot n^{d}}q^{n^{d}}}{n^{d}}\right)=\exp \left(-\frac{(A^{d}q)^{n^{d}}}{n^{d}}\right)\]

We want the event \(E_{n}\) (the \(n\)-th layer contains no all-white cubes) to occur only finitely many times. According to the first Borel-Cantelli Lemma, this happens with probability 1 if the sum of the probabilities converges:\(\sum _{n=1}^{\infty }P(E_{n})<\infty.\) If \(A^d q > 1\): The term \((A^d q)^{n^d}\) grows double-exponentially fast. As a result, \(P(E_n)\) decays to \(0\) extremely quickly (faster than any exponential), guaranteeing that the series \(\sum_n P(E_n)\) converges.

\(\Box\)

Combining this theorem with the formula for the first Dirichlet eigenvalue of the negative Laplacian on a \(d\)-dimensional cube of side length \(n\), \[\notag \lambda_1 = \frac{d\pi^2}{n^2},\] we obtain the following:

Proof. The result follows immediately from the Dirichlet-Neumann bracketing. \(\,\,\Box\)

This corollary is equivalent to Theorem 7.

11 Lower estimates↩︎

Here we study the one-dimensional operator. Let \(\tilde{\chi}_L(x)\) be the characteristic function of the set \({\Bbb R}\setminus \left[-\frac{L}{2}, \frac{L}{2}\right]\), which equals \(1\) outside the interval and \(0\) inside. To find the lowest eigenvalue \(\lambda\) of the differential operator \(\hat{L} =- \frac{d^2}{dx^2} + \epsilon \tilde{\chi}_L(x)\) acting on the Hilbert space \(L^2(\mathbb{R})\), we solve the eigenvalue equation \(\hat{L}\psi(x) = \lambda \psi(x)\). The operator \(\hat{L} = -\frac{d^2}{dx^2} + \epsilon \tilde{\chi}_L(x)\) represents a particle in a symmetric finite potential well of width \(L\) and height \(\epsilon\). For a bound state in \(L^2(\mathbb{R})\), the eigenvalue must lie in the range \(0 < \lambda < \epsilon\). Because the potential is symmetric under reflection \(x \to -x\), the ground state (lowest eigenvalue) wavefunction must be symmetric (even parity). We can write the general solution as:\[\notag \psi (x)=\begin{cases}A\cos (\sqrt\lambda x)&\text{for\;}|x|\leqslant\frac{L}{2}\\ Be^{-\sqrt{\epsilon-\lambda} |x|}&\text{for\;}|x|>\frac{L}{2}.\end{cases}\]

To ensure \(\psi(x)\) belongs to the domain of the differential operator, both \(\psi(x)\) and its first derivative \(\psi'(x)\) must be continuous at the boundaries \(x = \pm \frac{L}{2}\). Matching them at \(x = \frac{L}{2}\) gives

\[A\cos \left(\frac{\sqrt\lambda L}{2}\right)=Be^{-\sqrt{\epsilon-\lambda} L/2},\] and \[-\sqrt\lambda A\sin \left(\frac{\sqrt\lambda L}{2}\right)=-\sqrt{\epsilon-\lambda} Be^{-\sqrt{\epsilon-\lambda} L/2}.\] Thus, the lowest eigenvalue \(\lambda\) is the smallest positive solution to the following equation: \[\notag \sqrt{\lambda }\tan \left(\frac{\sqrt{\lambda }L}{2}\right)=\sqrt{\epsilon -\lambda }.\] This equaion is equivalent to \[\notag \cos \left(\frac{ \sqrt{\lambda}L}{2} \right)=\sqrt{\frac{\lambda}{\epsilon}}, \qquad \text{or}\quad \frac{\sqrt{\lambda}}{\cos \left(\frac{ \sqrt{\lambda}L}{2} \right)}=\sqrt{\epsilon}.\] Since the function on the left hand side of the latter equation is strictly increasing, we obtain that \(\lambda<C/L^2\) if and only if \[\notag \frac{\sqrt{C}}{L\cos \left(\frac{ \sqrt{C}}{2} \right)}>\sqrt{\epsilon}.\]

Let \(d=1\) and define the potential \(V(x)\) as: \[\notag V(x)=\frac{C}{(\log_A |x|)^2}, \qquad \text{where }\quad A>1/q,\quad \text{and}\quad 0<C<\pi^2.\] Let also \(\theta_\varepsilon\) be the characteristic function of the set \[\notag \left\{ x\in{\Bbb R}:\qquad \sqrt{\frac{C}{\varepsilon}}<\log_A|x|<\sqrt{ \frac{C}{\varepsilon} } \frac{1}{ \cos\left( \frac{\sqrt{C}}{2} \right) } \right\}.\] Define now \(N_1(\varepsilon)\) as the number of negative eigenvalues of the operator \[\label{opH} -\frac{d^2}{dx^2} +\varepsilon\sum_{n\in {\Bbb Z}}\omega_n\chi(x-n)- \theta_\varepsilon V(x).\tag{19}\] Using the bounds derived in the unperturbed problem, we obtain the following asymptotic lower bound for the number of negative eigenvalues: \[\label{liminf} \liminf_{\varepsilon\to0} \sqrt{\varepsilon} N_1(\varepsilon)\geqslant\frac{\sqrt{C}}{2}\left(\frac{1}{ \cos \left( \sqrt{C} / 2 \right) }- 1\right).\tag{20}\]

Proof of 20 For almost every \(\omega\), there exists an \(n_0(\omega) \in \mathbb{N}\) such that for all \(n > n_0(\omega)\), the spherical layer \(A^{n-1} < \vert{}x\vert{} < A^n\) contains an interval of length \(n\) where the random potential \(\sum_j \omega_j \chi(x-j)\) vanishes. Let us now introduce the layers \[\notag \mathcal{L}_n=\left \{ x\in {\Bbb R}:\quad \frac{A^{n-1}+A^{n-2}}{2} <|x|< \frac{A^{n}+A^{n+1}}{2}\right \}.\] For sufficiently large \(n\), each layer \(\mathcal{L}_{n}\) contains an interval of length \(n\) where the random potential \(\sum_j \omega_j \chi(x-j)\) vanishes identically. Moreover, the minimal distance from this zero-potential interval to the boundary of \(\mathcal{L}_{n}\) is bounded below by \(\frac{A^{n-1}-A^{n-2}}{2}\).

By the domain monotonicity of eigenvalues, we can decouple the operator 19 into independent operators on each layer \(\mathcal{L}_{2n}\) with Dirichlet boundary conditions.

If \(\mathcal{L}_n\subset {\rm supp}(\theta_\varepsilon)\), and \(\varepsilon>0\) is sufficiently large, then the operator \[\notag -\frac{d^2}{dx^2} +\varepsilon\sum_{j\in {\Bbb Z}}\omega_j\chi(x-j)- \theta_\varepsilon V(x).\] on \(\mathcal{L}_n\) possesses at least one negative eigenvalue. Consequently, for large \(\varepsilon\), \[\notag N_1(\varepsilon)\geqslant\#\{n \in {\Bbb N}: \quad n/2\in {\Bbb N},\quad \mathcal{L}_n\subset {\rm supp}(\theta_\varepsilon) \}.\] This implies the statement. \(\,\,\Box\)

Proof. By the splitting principle, for any \(\varepsilon>0\), \[\notag N(\varepsilon)\geqslant N_0(\varepsilon)+ N_1(\varepsilon)-3.\qquad \Box\]

This corollary is equivalent to Theorem 6.

References↩︎