Odinger Operators Research Articles

Optimizing the fundamental eigenvalue gap of quantum graphs

Abstract We study the problem of minimizing or maximizing the fundamental spectral gap of Schr"odinger operators on metric graphs with
either a convex potential or 
a ``single-well'' potential on an appropriate specified subset. (In the case of metric trees, such a subset can be the entire graph.) In the convex case we find
that the minimizing and maximizing potentials are piecewise linear with only a finite number of points of non-smoothness, but give examples showing that the optimal potentials need not be constant. This is a significant departure from the usual scenarios on intervals and domains where the constant potential is typically minimizing. In the single-well case we show that the optimal potentials are piecewise constant with a finite number of jumps, and in both cases give an explicit estimate on the number of points of non-smoothness, respectively jumps, the minimizing potential can have. Furthermore, we show that, unlike on domains, it is not generally possible to find nontrivial bounds on the fundamental gap in terms of the diameter of the graph alone, within the given classes.

Reflectionless Schrodinger operators and Marchenko parametrization

Let $T_q=-d^2/dx^2 +q$ be a Schr\"odinger operator in the space $L_2(\mathbb{R})$. A potential $q$ is called reflectionless if the operator $T_q$ is reflectionless. Let $\mathcal{Q}$ be the set of all reflectionless potentials of the Schr\"odinger operator, and let $\mathcal{M}$ be the set of nonnegative Borel measures on $\mathbb{R}$ with compact support. As shown by Marchenko, each potential $q\in\mathcal{Q}$ can be associated with a unique measure $\mu\in\mathcal{M}$. As a result, we get the bijection $\Theta\colon \mathcal{Q}\to \mathcal{M}$. In this paper, we show that one can define topologies on $\mathcal{Q}$ and $\mathcal{M}$, under which the mapping $\Theta$ is a homeomorphism.

Quantitative Green’s function estimates for lattice quasi-periodic Schrödinger operators

In this paper, we establish quantitative Green's function estimates for some higher dimensional lattice quasi-periodic (QP) Schr\"odinger operators. The resonances in the estimates can be described via a pair of symmetric zeros of certain functions and the estimates apply to the sub-exponential type non-resonant conditions. As the application of quantitative Green's function estimates, we prove both the arithmetic version of Anderson localization and the $(\frac 12-)$-H\"older continuity of the integrated density of states (IDS) for such QP Schr\"odinger operators. This gives an affirmative answer to Bourgain's problem in\cite{Bou00}.

Sharp Weyl laws with singular potentials

We consider the Laplace--Beltrami operator on a three-dimensional Riemannian manifold perturbed by a potential from the Kato class and study whether various forms of Weyl's law remain valid under this perturbation. We show that a pointwise Weyl law holds, modified by an additional term, for any Kato class potential with the standard sharp remainder term. The additional term is always of lower order than the leading term, but it may or may not be of lower order than the sharp remainder term. In particular, we provide examples of singular potentials for which this additional term violates the sharp pointwise Weyl law of the standard Laplace-Beltrami operator. For the proof we extend the method of Avakumovi\'c to the case of Schr\"odinger operators with singular potentials.

The Almost Sure Essential Spectrum of the Doubling Map Model is Connected

We consider discrete Schr\"odinger operators on the half line with potentials generated by the doubling map and continuous sampling functions. We show that the essential spectrum of these operators is always connected. This result is obtained by computing the subgroup of the range of the Schwartzman homomorphism associated with homotopy classes of continuous maps on the suspension of the standard solenoid that factor through the suspension of the doubling map and then showing that this subgroup characterizes the topological structure of the spectrum.

Trace Formulas for Schrödinger Operators on a Lattice

We consider Schr\"odinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We determine trace formulas for the Schr\"odinger operators. The proof is based on the decomposition of the Schr\"odinger operators into a direct integral and a specific representation of fiber operators. The traces of the fiber operators are expressed as finite Fourier series of the quasimomentum. The coefficients of the Fourier series are given in terms of the potentials and cycles in the quotient graph from some specific cycle sets. We also present the trace formulas for the heat kernel and the resolvent of the Schr\"odinger operators and the determinant formulas.

Fermi Isospectrality of Discrete Periodic Schrödinger Operators with Separable Potentials on $$\mathbb {Z}^2$$

Let $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z} $ with $q_1\in \mathbb{Z}_+$ and $q_2\in\mathbb{Z}_+$. Let $\Delta+X$ be the discrete periodic Schr\"odinger operator on $\mathbb{Z}^2$, where $\Delta$ is the discrete Laplacian and $X:\mathbb{Z}^2\to \mathbb{C}$ is $\Gamma$-periodic. In this paper, we develop tools from complex analysis to study the isospectrality of discrete periodic Schr\"odinger operators. We prove that if two $\Gamma$-periodic potentials $X$ and $Y$ are Fermi isospectral and both $X=X_1\oplus X_2$ and $Y= Y_1\oplus Y_2$ are separable functions, then, up to a constant, one dimensional potentials $X_j$ and $Y_j$ are Floquet isospectral, $j=1,2$. This allows us to prove that for any non-constant separable real-valued $\Gamma$-periodic potential, the Fermi variety $F_{\lambda}(V)/\mathbb{Z}^2$ is irreducible for any $\lambda\in \mathbb{C}$, which partially confirms a conjecture of Gieseker, Kn\"{o}rrer and Trubowitz in the early 1990s.

Van der Waals–London interaction of atoms with pseudorelativistic kinetic energy

We consider a multiatomic system where the nuclei are assumed to be point charges at fixed positions. Particles interact via Coulomb potential and electrons have pseudo-relativistic kinetic energy. We prove the van der Waals-London law, which states that the interaction energy between neutral atoms decays as the sixth power of the distance $|D|$ between the atoms. We rigorously compute all the terms in the binding energy up to the order $|D|^{-9}$ with error term of order $\mathcal{O}(|D|^{-10})$ . As intermediate steps we prove exponential decay of eigenfunctions of multiparticle Schr\"odinger operators with permutation symmetry imposed by the Pauli principle and new estimates of the localization error.

Critical perturbations for second-order elliptic operators, I: Square function bounds for layer potentials

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A \nabla$ by first and zero order terms, whose coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the $L^2$ well-posedness of the Dirichlet, Neumann and Regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. For instance, this allows us to claim the first results in the setting of an unbounded domain concerning the solvability of boundary value problems for the magnetic Schr\"odinger operator $-(\nabla-i{\bf a})^2+V$ when the magnetic potential ${\bf a}$ and the electric potential $V$ are accordingly small in the norm of a scale-invariant Lebesgue space. In the present paper, we establish $L^2$ control of the square function via a vector-valued $Tb$ theorem and abstract layer potentials, and use these square function bounds to obtain uniform slice bounds for solutions. The existence and uniqueness of solutions, as well as bounds for the non-tangential maximal operator, are considered in the upcoming paper.

Absence of embedded eigenvalues for non-local Schrödinger operators

We consider non-local Schr\"odinger operators with kinetic terms given by several different types of functions of the Laplacian and potentials decaying to zero at infinity, and derive conditions ruling embedded eigenvalues out. Our goal in this paper is to advance techniques based on virial theorems, Mourre estimates, and an extended version of the Birman-Schwinger principle, previously developed for classical Schr\"odinger operators but thus far not used for non-local operators. We also present a number of specific cases by choosing particular classes of kinetic and potential terms, and discuss existence/non-existence of at-edge eigenvalues in a basic model case in function of the coupling parameter.

The Landscape Law for Tight Binding Hamiltonians

The present paper extends the landscape theory pioneered in [FM, ADFJM2, DFM] to the tight-binding Schr\"odinger operator on $\Z^d$. In particular, we establish upper and lower bounds for the integrated density of states in terms of the counting function based upon the localization landscape.

Continuum limit of the lattice quantum graph Hamiltonian

We consider the quantum graph Hamiltonian on the square lattice in Euclidean space, and we show that the spectrum of the Hamiltonian converges to the corresponding Schr\"odinger operator on the Euclidean space in the continuum limit, and that the corresponding eigenfunctions and eigenprojections also converge in some sense. We employ the discrete Schr\"odinger operator as the intermediate operator, and we use a recent result by the second and third author on the continuum limit of the discrete Schr\"odinger operator.

On the spectrum of the Kronig–Penney model in a constant electric field

We are interested in the nature of the spectrum of the one-dimensional Schr\"odinger operator $$ - \frac{d^2}{dx^2}-Fx + \sum_{n \in \mathbb{Z}}g_n \delta(x-n) \qquad\text{in } L^2(\mathbb{R}) $$ with $F>0$ and two different choices of the coupling constants $\{g_n\}_{n\in \mathbb{Z}}$. In the first model $g_n \equiv \lambda$ and we prove that if $F\in \pi^2 \mathbb{Q}$ then the spectrum is $\mathbb{R}$ and is furthermore absolutely continuous away from an explicit discrete set of points. In the second model the $g_n$ are independent random variables with mean zero and variance $\lambda^2$. Under certain assumptions on the distribution of these random variables we prove that almost surely the spectrum is $\mathbb{R}$ and it is dense pure point if $F < \lambda^2/2$ and purely singular continuous if $F> \lambda^2/2$.

On Lieb–Robinson Bounds for the Bose–Hubbard Model

We consider the dynamics of the Bose-Hubbard model on general lattices and prove a Lieb-Robinson bound for observables whose supports are separated by an initially almost particle-free region. We further obtain a maximal velocity bound for particle transport through an initially empty region which also applies to long-range hopping. Our techniques originate in the proofs of maximal velocity bounds for Schr\"odinger operators and scattering theory in non-relativistic QED.

An Optimal Multiplier Theorem for Grushin Operators in the Plane, II

In a previous work we proved a spectral multiplier theorem of Mihlin--H\"ormander type for two-dimensional Grushin operators $-\partial_x^2 - V(x) \partial_y^2$, where $V$ is a doubling single-well potential, yielding the surprising result that the optimal smoothness requirement on the multiplier is independent of $V$. Here we refine this result, by replacing the $L^\infty$ Sobolev condition on the multiplier with a sharper $L^2$ condition. As a consequence, we obtain the sharp range of $L^1$ boundedness for the associated Bochner--Riesz means. The key new ingredient of the proof is a precise pointwise estimate in the transition region for eigenfunctions of one-dimensional Schr\"odinger operators with doubling single-well potentials.

Weyl’s Laws and Connes’ Integration Formulas for Matrix-Valued $$L\!\log \!L$$-Orlicz Potentials

Thanks to the Birman-Schwinger principle, Weyl's laws for Birman-Schwinger operators yields semiclassical Weyl's laws for the corresponding Schr\"odinger operators. In a recent preprint Rozenblum established quite general Weyl's laws for Birman-Schwinger operators associated with pseudodifferential operators of critical order and potentials that are product of $L\log L$-Orlicz functions and Alfhors-regular measures supported on a submanifold. In this paper, for matrix-valued $L\log L$-Orlicz potentials supported on the whole manifold, Rozenblum's results are direct consequences of the Cwikel-type estimates on tori recently established by Sukochev-Zanin. As applications we obtain CLR-type inequalities and semiclassical Weyl's laws for critical Schr\"odinger operators associated with matrix-valued$L\log L$-Orlicz potentials. Finally, we explain how the Weyl's laws of this paper imply a strong version of Connes' integration formula for matrix-valued $L\log L$-Orlicz potentials.

Analyticity of the Lyapunov exponent of meromorphic monotonic cocycles

We consider meromorphic monotonic cocycles, and show that the Lyapunov exponent is analytic. We then apply the result to the quasiperiodic Schrödinger operators that have meromorphic monotone potentials.

Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues

Let $H_0$ be a discrete periodic Schr\"odinger operator on $\ell^2(\mathbb{Z}^d)$: $$H_0=-\Delta+V,$$ where $\Delta$ is the discrete Laplacian and $V: \mathbb{Z}^d\to \mathbb{C}$ is periodic. We prove that for any $d\geq3$, the Fermi variety at every energy level is irreducible (modulo periodicity). For $d=2$, we prove that the Fermi variety at every energy level except for the average of the potential is irreducible (modulo periodicity) and the Fermi variety at the average of the potential has at most two irreducible components (modulo periodicity). This is sharp since for $d=2$ and a constant potential $V$, the Fermi variety at $V$-level has exactly two irreducible components (modulo periodicity). We also prove that the Bloch variety is irreducible (modulo periodicity) for any $d\geq 2$. As applications, we prove that when $V$ is a real-valued periodic function, the level set of any extrema of any spectral band functions, spectral band edges in particular, has dimension at most $d-2$ for any $d\geq 3$, and finite cardinality for $d=2$. We also show that $H=-\Delta +V+v$ does not have any embedded eigenvalues provided that $v$ decays super-exponentially.

Minimal families of limit operators

We study two abstract scenarios, where an operator family has a certain minimality property. In both scenarios, it is shown that norm, spectrum and resolvent are the same for all family members. Both abstract settings are illustrated by practically relevant examples, including discrete Schr\odinger operators with periodic, quasiperiodic, almost-periodic, Sturmian and pseudo-ergodic potential. The main tool is the method of limit operators, known from studies of Fredholm operators and convergence of projection methods. We close by connecting this tool to the study of subwords of the operator potential.

The absence of the Efimov effect in systems of one- and two-dimensional particles

We study virtual levels of $N$-particle Schr\"odinger operators and prove that if the particles are one-dimensional and $N\ge 3$, then virtual levels at the bottom of the essential spectrum correspond to eigenvalues. The same is true for two-dimensional particles if $N\ge 4$. These results are applied to prove the non-existence of the Efimov effect in systems of $N\ge 4$ one-dimensional or $N\ge 5$ two-dimensional particles.