Spectral triangles of Schrödinger operators with complex potentials

Consider the Schrödinger operator with a complex-valued potential v of period Let and be the eigenvalues of L that are close to respectively, with periodic (for n even), antiperiodic (for n odd), and Dirichelet boundary conditions on [0,1], and let be the diameter of the spectral triangle with vertices We prove the following statement: If then v(x) is a Gevrey function, and moreover      

Pseudomodes for Schrödinger operators with complex potentials

For one-dimensional Schrödinger operators with complex-valued potentials, we construct pseudomodes corresponding to large pseudoeigenvalues. We develop a first systematic non-semi-classical approach, which results in a substantial progress in achieving optimal conditions and conclusions as well as in covering a wide class of previously inaccessible potentials, including discontinuous ones. Applications of the present results to higher-dimensional Schrödinger operators are also discussed.     

Schrödinger operators with slowly decaying potentials

Several recent papers have obtained bounds on the distribution of eigenvalues of non-self-adjoint Schrödinger operators and resonances of self-adjoint operators. In this paper we describe two new methods of obtaining such bounds when the potential decays more slowly than previously permitted.     

Schrödinger operators with highly singular oscillating potentials

We investigate the Feynman-Kac semigroupP _t ^V and its densityp ^V(t,.,.),t>0, associated with the Schrödinger operator ?1/2Δ+V on ?^d\{0}.V will be a highly singular, oscillating potential like $V\left( x \right) = k \cdot \left\| x \right\|^{ - 1} \cdot \sin \left( {\left\| x \right\|^{ - m} } \right)$ with arbitraryk, l, m>0. We derive conditions (onk,l,m) which are sufficientand necessary for the existence of constants α, β, γ, ∈ ? such that for allt, x, y p ^V(t, x, y)≤γ·p(βt, x, y)·e^at. On the other hand, also conditions are derived which imply thatp ^V (t, x, y)≡∞ for allt, x, y. The aim is to see to which extent quick oscillations can lead to annihilations of the singularities ofV. For this purpose, we analyse the above example in great detail. Note that forl≥2 the potential is so singular that none of the usual perturbation techniques applies.     

Monodromy-free Schrödinger operators with quadratically increasing potentials

We consider one-dimensional monodromy-free Schrödinger operators with quadratically increasing rational potentials. It is shown that all these operators can be obtained from the operator -?² + x² by finitely many rational Darboux transformations. An explicit expression is found for the corresponding potentials in terms of Hermite polynomials.     

Spectral estimates for Schrödinger operators with sparse potentials on graphs

A construction of “sparse potentials,” suggested by the authors for the lattice \mathbbZ^d {\mathbb{Z}^d} , d > 2, is extended to a large class of combinatorial and metric graphs whose global dimension is a number D > 2. For the Schr?dinger operator − Δ − αV on such graphs, with a sparse potential V, we study the behavior (as α → ∞) of the number N_(−Δ − αV) of negative eigenvalues of − Δ − αV. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of N_(−Δ − αV) under very mild regularity assumptions. A similar construction works also for the lattice \mathbbZ² {\mathbb{Z}^2} , where D = 2. Bibliography: 13 titles.     

Spectral representation for Schrödinger operators with long-range potentials

A spectral representation for the self-adjoint Schrödinger operator H = ?Δ + V(x), x? R³, is obtained, where V(x) is a long-range potential: $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(1</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, grad $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(3</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, $\text{ΛV(x) = O(¦ x}\text{s}\text{?}{}^{\mathrm{?\delta }}\text{) (δ > 0), Λ}$ being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator $\text{F}$ from PL₂(R³) onto $\text{L}{}_2\text{((0, ∞); L}{}_2\text{(Ω)), P}$ being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ),

$\text{(α(H)(Pf,g)=}\text{∫}\text{0}\text{∞}\text{(α(λ)(Ff)(λ),(Fg)(λ))L}{}_2\text{(ω) dλ}$

.     

Schrödinger operators with singular potentials from the space of multiplicators

We study Schrödinger operatorsT+Q, whereT=?Δ is the Laplace operator andQ is the multiplication operator by a generalized function (distribution). We also consider generalizations for the case of the polyharmonic operatorT = (-δ) ⁿ      

Persistence of Anderson localization in Schrödinger operators with decaying random potentials

We show persistence of both Anderson and dynamical localization in Schrödinger operators with non-positive (attractive) random decaying potential. We consider an Anderson-type Schrödinger operator with a non-positive ergodic random potential, and multiply the random potential by a decaying envelope function. If the envelope function decays slower than |x|^-2 at infinity, we prove that the operator has infinitely many eigenvalues below zero. For envelopes decaying as |x|^-α at infinity, we determine the number of bound states below a given energy E<0, asymptotically as α↓0. To show that bound states located at the bottom of the spectrum are related to the phenomenon of Anderson localization in the corresponding ergodic model, we prove: (a) these states are exponentially localized with a localization length that is uniform in the decay exponent α; (b) dynamical localization holds uniformly in α.     

Sobolev spaces related to Schrödinger operators with polynomial potentials

The aim of this note is to prove the following theorem. Let

On the absence of positive eigenvalues of Schrödinger operators with rough potentials

We prove the absence of positive eigenvalues of Schrödinger operators $ H=-\Delta+V $ on Euclidean spaces $ \mathbb{R}^n $ for a certain class of rough potentials $V$. To describe our class of potentials fix an exponent $q\in[n/2,\infty]$ (or $q\in(1,\infty]$, if $n=2$) and let $\beta(q)=(2q-n)/(2q)$. For the potential $V$ we assume that $V\in L^{n/2}_{{\rm{loc}}}(\mathbb{R}^n)$ (or $V\in L^{r}_{{\rm{loc}}}(\mathbb{R}^n)$, $r>1$, if $n=2$) and$\begin{equation*}$$\lim_{R\to\infty}R^{\beta(q)}||V||_{L^q(R\leq |x|\leq 2R)}=0\,.$$\end{equation*}$Under these assumptions we prove that the operator $H$ does not admit positive eigenvalues. The case $q=\infty$ was considered by Kato [K]. The absence of positive eigenvalues follows from a uniform Carleman inequality of the form$\begin{equation*}$$||W_m u||_{l^a(L^{p(q)})(\mathbb R^n)}\leq C_q||W_m|x|^{\beta(q)}(\Delta+1)u||_{l^a(L^{p(q)})(\mathbb{R}^n)}$$\end{equation*}$for all smooth compactly supported functions $u$ and a suitable sequence of weights $W_m$, where $p(q)$ and $p(q)$ are dual exponents with the property that $1/p(q)-1/p(q)=1/q$.     

Local behavior of solutions of the stationary Schrödinger equation with singular potentials and bounds on the density of states of Schrödinger operators

We study the local behavior of solutions of the stationary Schrödinger equation with singular potentials, establishing a local decomposition into a homogeneous harmonic polynomial and a lower order term. Combining a corollary to this result with a quantitative unique continuation principle for singular potentials, we obtain log-Hölder continuity for the density of states outer-measure in one, two, and three dimensions for Schrödinger operators with singular potentials, results that hold for the density of states measure when it exists.     

Schrödinger operators with negative potentials and Lane–Emden densities

We consider the Schrödinger operator $?\mathrm{\Delta }+V$ for negative potentials V, on open sets with positive first eigenvalue of the Dirichlet–Laplacian. We show that the spectrum of $?\mathrm{\Delta }+V$ is positive, provided that V is greater than a negative multiple of the logarithmic gradient of the solution to the Lane–Emden equation $?\mathrm{\Delta }u=u^{q?1}$ (for some $1\le q<2$). In this case, the ground state energy of $?\mathrm{\Delta }+V$ is greater than the first eigenvalue of the Dirichlet–Laplacian, up to an explicit multiplicative factor. This is achieved by means of suitable Hardy-type inequalities, that we prove in this paper.     

On essential self-adjointness for Schrödinger operators with wildly oscillating potentials

A sufficient condition is given for the operator T₀: C^∞₀(R^m) → L²(R^m) given by

$\text{T}{}_0\text{∑}\text{K?1}\text{M}\text{(}\text{i?}\text{?x}{}_1\text{+b}{}_1\text{)a}{}_{\mathrm{1K}}\text{(}\text{i?}\text{?N}{}_{\mathrm{k}}\text{+b}{}_{\mathrm{k}}\text{)+q}$

to be essentially self-adjoint. This condition is sufficiently general to admit certain potentials q having unbounded oscillations in a neighborhood of ∞.     

Some estimates for the Schrödinger type operators with nonnegative potentials

In this paper we consider the Schrödinger operator ?Δ + V on \({\mathbb R^d}\), where the nonnegative potential V belongs to the reverse Hölder class \({B_{q_{_1}}}\) for some \({q_{_1}\geq \frac{d}{2}}\) with d ≥ 3. Let \({H^1_L(\mathbb R^d)}\) denote the Hardy space related to the Schrödinger operator L = ?Δ + V and \({BMO_L(\mathbb R^d)}\) be the dual space of \({H^1_L(\mathbb R^d)}\). We show that the Schrödinger type operator \({\nabla(-\Delta +V)^{-\beta}}\) is bounded from \({H^1_L(\mathbb R^d)}\) into \({L^p(\mathbb R^d)}\) for \({p=\frac{d}{d-(2\beta-1)}}\) with \({ \frac{1}{2}<\beta<\frac{3}{2} }\) and that it is also bounded from \({L^p(\mathbb R^d)}\) into \({BMO_L(\mathbb R^d)}\) for \({p=\frac{d}{2\beta-1}}\) with \({ \frac{1}{2}<\beta< 2}\).     

Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials

Let L=?Δ+V be a Schrödinger operator on ? ^d , d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to L ^p (? ^d ), for some p>d /2. Let K _t be the semigroup generated by ?L. We say that an L ¹(? ^d )-function f belongs to the Hardy space \(H^{1}_{L}\) associated with L if sup?_t>0|K _t f| belongs to L ¹(? ^d ). We prove that \(f\in H^{1}_{L}\) if and only if R _j f∈L ¹(? ^d ) for j=1,…,d, where R _j =(?/? x _j )L ^?1/2 are the Riesz transforms associated with L.