Tunneling Estimates for Magnetic Schrödinger Operators

We study the behavior of eigenfunctions in the semiclassical limit for Schr?dinger operators with a simple well potential and a (non-zero) constant magnetic field. We prove an exponential decay estimate on the low-lying eigenfunctions, where the exponent depends explicitly on the magnetic field strength. Received: 30 March 1998 / Accepted: 1 May 1998     

Inverse Spectral Problems for Schrödinger Operators

In this article we find some explicit formulas for the semi-classical wave invariants at the bottom of the well of a Schrödinger operator. As an application of these new formulas for the wave invariants, we improve the inverse spectral results proved by Guillemin and Uribe in [GU]. They proved that under some symmetry assumptions on the potential V(x), the Taylor expansion of V(x) near a non-degenerate global minimum can be recovered from the knowledge of the low-lying eigenvalues of the associated Schrödinger operator in \({\mathbb R^n}\) . We prove similar inverse spectral results using fewer symmetry assumptions. We also show that in dimension 1, no symmetry assumption is needed to recover the Taylor coefficients of V(x).     

A Liouville-type Theorem for Schrödinger Operators

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator P ₁, such that a nonzero subsolution of a symmetric nonnegative operator P ₀ is a ground state. Particularly, if P _j : = ?Δ + V _j , for j = 0,1, are two nonnegative Schrödinger operators defined on \(\Omega\subseteq \mathbb{R}^d\) such that P ₁ is critical in Ω with a ground state φ, the function \(\psi\nleq 0\) is a subsolution of the equation P ₀ u = 0 in Ω and satisfies \(\psi_+\leq C\varphi\) in Ω, then P ₀ is critical in Ω and \(\psi\) is its ground state. In particular, \(\psi\) is (up to a multiplicative constant) the unique positive supersolution of the equation P ₀ u = 0 in Ω. Similar results hold for general symmetric operators, and also on Riemannian manifolds.     

On the Essential Spectrum of Schrödinger Operators on Trees

It is known that the essential spectrum of a Schrödinger operator H on \(\ell ^{2}\left (\mathbb {N}\right )\) is equal to the union of the spectra of right limits of H. The natural generalization of this relation to \(\mathbb {Z}^{n}\) is known to hold as well. In this paper we generalize the notion of right limits to general infinite connected graphs and construct examples of graphs for which the essential spectrum of the Laplacian is strictly bigger than the union of the spectra of its right limits. As these right limits are trees, this result is complemented by the fact that the equality still holds for general bounded operators on regular trees. We prove this and characterize the essential spectrum in the spherically symmetric case.     

Generic Continuous Spectrum for Ergodic Schrödinger Operators

We consider families of discrete Schrödinger operators on the line with potentials generated by a homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon’s Lemma that for a generic continuous sampling function, the set of elements in the associated family of Schrödinger operators that have no eigenvalues is large in a topological or metric sense, respectively. We present a number of applications, particularly to shifts and skew-shifts on the torus.     

On the Essential Spectrum of Two-Dimensional Periodic Magnetic Schrödinger Operators

For two-dimensional Schrödinger operators with a nonzero constant magnetic field perturbed by an infinite number of periodically disposed, long-range magnetic and electric wells, it is proven that when the inter-well distance (R) grows to infinity, the essential spectrum near the eigenvalues of the one well Hamiltonian is located in mini-bands whose widths shrink faster than any exponential with R. This should be compared with our previous result, which stated that, in the case of compactly supported wells, the mini-bands shrink Gaussian-like with R.     

Unique Continuation Principle for Spectral Projections of Schrödinger Operators and Optimal Wegner Estimates for Non-ergodic Random Schrödinger Operators

We prove a unique continuation principle for spectral projections of Schrödinger operators. We consider a Schrödinger operator H = ?Δ + V on ${{\rm L}^2(\mathbb{R}^d)}$ L 2 ( R d ) , and let H _Λ denote its restriction to a finite box Λ with either Dirichlet or periodic boundary condition. We prove unique continuation estimates of the type χ _I (H _Λ ) W χ _I (H _Λ ) ≥ κ χ _I (H _Λ ) with κ > 0 for appropriate potentials W ≥ 0 and intervals I. As an application, we obtain optimal Wegner estimates at all energies for a class of non-ergodic random Schrödinger operators with alloy-type random potentials (‘crooked’ Anderson Hamiltonians). We also prove optimal Wegner estimates at the bottom of the spectrum with the expected dependence on the disorder (the Wegner estimate improves as the disorder increases), a new result even for the usual (ergodic) Anderson Hamiltonian. These estimates are applied to prove localization at high disorder for Anderson Hamiltonians in a fixed interval at the bottom of the spectrum.     

On the Inverse Resonance Problem for Schrödinger Operators

We consider Schrödinger operators on [0, ∞) with compactly supported, possibly complex-valued potentials in L ¹([0, ∞)). It is known (at least in the case of a real-valued potential) that the location of eigenvalues and resonances determines the potential uniquely. From the physical point of view one expects that large resonances are increasingly insignificant for the reconstruction of the potential from the data. In this paper we prove the validity of this statement, i.e., we show conditional stability for finite data. As a by-product we also obtain a uniqueness result for the inverse resonance problem for complex-valued potentials.     

Dispersive Estimates for Schrödinger Operators in Dimension Two

We prove L¹(²)L(²) for the two-dimensional Schrödinger operator –+V with the decay rate t^–1. We assume that zero energy is neither an eigenvalue nor a resonance. This condition is formulated as in the recent paper by Jensen and Nenciu on threshold expansions for the two-dimensional resolvent.The author was partially supported by the NSF grant DMS-0300081 and a Sloan Fellowship     

Eigenvalue Estimates for Schrödinger Operators with Complex Potentials

We discuss properties of eigenvalues of non-self-adjoint Schrödinger operators with complex-valued potential V. Among our results are estimates of the sum of powers of imaginary parts of eigenvalues by the L ^p -norm of \({{\Im{V}}}\).     

Perturbations of Magnetic Schrödinger Operators

We study Schrödinger operators H(a, V): = (P – a)² + V acting in L ²(³). We assume that the magnetic field B = rot a may be decomposed as B = B ⁰ + B, where B ⁰ is a very general field having constant direction. The perturbations B and V will be small in a certain sense in the direction of B ⁰, but in the orthogonal plane they may even grow for certain fields B ⁰. Commutator methods are used to derive spectral properties of H(a, V).     

Estimates for Eigenvalues of Schrödinger Operators with Complex-Valued Potentials

New estimates for eigenvalues of non-self-adjoint multi-dimensional Schrödinger operators are obtained in terms of L_p-norms of the potentials. The results cover and improve those known previously, in particular, due to Frank (Bull Lond Math Soc 43(4):745–750, 2011), Safronov (Proc Am Math Soc 138(6):2107–2112, 2010), Laptev and Safronov (Commun Math Phys 292(1):29–54, 2009). We mention the estimations of the eigenvalues situated in the strip around the real axis (in particular, the essential spectrum). The method applied for this case involves the unitary group generated by the Laplacian. The results are extended to the more general case of polyharmonic operators. Schrödinger operators with slowly decaying potentials and belonging to weak Lebesgue’s classes are also considered.     

On Lieb-Thirring Inequalities for Schrödinger Operators with Virtual Level

We consider the operator H=−Δ−V in L₂(ℝ^d), d≥3. For the moments of its negative eigenvalues we prove the estimate Similar estimates hold for the one-dimensional operator with a Dirichlet condition at the origin and for the two-dimensional Aharonov-Bohm operator.     

Asymptotic Number of Scattering Resonances for Generic Schrödinger Operators

Let ?Δ + V be the Schrödinger operator acting on ${L^2(\mathbb{R}^d,\mathbb{C})}$ with ${d\geq 3}$ odd. Here V is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a. Let n _V (r) denote the number of resonances of ?Δ + V with modulus ≤  r. We show that if the potential V is generic in a sense of pluripotential theory then $$n_V(r)=c_d a^dr^d+ O(r^{d-{3\over 16}+\epsilon}) \quad \mbox{as } r \to \infty$$ for any ε > 0, where c _d is a dimensional constant.     

Spectral Theory for Periodic Schrödinger Operators with Reflection Symmetries

Let H=–+V be defined on ^d with smooth potential V, such that In addition we assume that where This is a periodic Schrödinger operator with additional reflection symmetries. We investigate the associated Floquet operators H ^q , q[0,1] ^d . In particular we show that the associated lowest eigenvalues _q are simple if q=(q ₁ ,q ₂ ,,q _d ) satisfies q _j 1/2 for each j=1,2,,d. Supported by Ministerium für Bildung, Wissenschaft und Kunst der Republik ÖsterreichSupported by the European Science Foundation Programme Spectral Theory and Partial Differential Equations (SPECT)     

Schrödinger Operators with Few Bound States

We show that whole-line Schrödinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast. We also prove the following result for one- and two-dimensional Schrödinger operators, H, with bounded positive ground states: Given a potential V, if both H±V are bounded from below by the ground-state energy of H, then V≡0.D. D. was supported in part by NSF grant DMS–0227289.R. K. was supported in part by NSF grant DMS–0401277.B. S. was supported in part by NSF grant DMS–0140592.     

Unique Continuation for Many-Body Schrödinger Operators and the Hohenberg-Kohn Theorem

We prove the strong unique continuation property for many-body Schrödinger operators with an external potential and an interaction potential both in \(L^{p}_{\text {loc}}(\mathbb {R}^{d})\), where p >?2 if d =?3 and \({p = \max (2d/3,2)}\) otherwise, independently of the number of particles. With the same assumptions, we obtain the Hohenberg-Kohn theorem, which is one of the most fundamental results in Density Functional Theory.     

Spectral Gaps for Periodic Schrödinger Operators with Strong Magnetic Fields

We consider Schrödinger operators H^h=(ihd+A)*(ihd+A) with the periodic magnetic field B=dA on covering spaces of compact manifolds. Using methods of a paper by Kordyukov, Mathai and Shubin [14], we prove that, under some assumptions on B, there are in arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of the strong magnetic field h0.Acknowledgement I am very thankful to Bernard Helffer for bringing these problems to my attention and useful discussions and to Mikhail Shubin for his comments.