The Schrödinger operator

In this paper we study the maximum dissipative extension of the Schrödinger operator, introduce the generalized indefinite metric space, obtain the representation of the maximum dissipative extension of the Schrödinger operator in the natural boundary space and make preparation for the further study of the longtime chaotic behavior of the infinite-dimensional dynamics system in the Schrödinger equation.

     

“Localized” self-adjointness of Schrödinger type operators on Riemannian manifolds

We prove self-adjointness of the Schrödinger type operator , where ∇ is a Hermitian connection on a Hermitian vector bundle E over a complete Riemannian manifold M with positive smooth measure dμ which is fixed independently of the metric, and V∈L_loc¹(EndE) is a Hermitian bundle endomorphism. Self-adjointness of H_V is deduced from the self-adjointness of the corresponding “localized” operator. This is an extension of a result by Cycon. The proof uses the scheme of Cycon, but requires a refined integration by parts technique as well as the use of a family of cut-off functions which are constructed by a non-trivial smoothing procedure due to Karcher.     

Sharp essential self-adjointness of relativistic Schrödinger operators with a singular potential

This paper is devoted to the study of essential self-adjointness of a relativistic Schrödinger operator with a singular homogeneous potential. From an explicit condition on the coefficient of the singular term, we provide a sufficient and necessary condition for essential self-adjointness.     

Inverse scattering for the magnetic Schrödinger operator

We show that fixed energy scattering measurements for the magnetic Schrödinger operator uniquely determine the magnetic field and electric potential in dimensions n?3. The magnetic potential, its first derivatives, and the electric potential are assumed to be exponentially decaying. This improves an earlier result of Eskin and Ralston (1995) [5] which considered potentials with many derivatives. The proof is close to arguments in inverse boundary problems, and is based on constructing complex geometrical optics solutions to the Schrödinger equation via a pseudodifferential conjugation argument.     

A Szeg? condition for a multidimensional Schrödinger operator

We consider spectral properties of a Schrödinger operator perturbed by a potential vanishing at infinity and prove that the corresponding spectral measure satisfies a Szeg?-type condition.     

Asymptotics of resonances for a Schrödinger operator with vector values

We show that the resonance counting function for a Schrödinger operator in dimension one has an asymptotic expansion and calculate an explicit expression for the leading term in some situations.     

The spectral bounds of the discrete Schrödinger operator

Let H(λ)=−Δ+λb be a discrete Schrödinger operator on ?²(Zd) with a potential b and a non-negative coupling constant λ. When b≡0, it is well known that σ(−Δ)=[0,4d]. When b?0, let and be the bounds of the spectrum of the Schrödinger operator. One of the aims of this paper is to study the influence of the potential b on the bounds 0 and 4d of the spectrum of −Δ. More precisely, we give a necessary and sufficient condition on the potential b such that s(−Δ+λb) is strictly positive for λ small enough. We obtain a similar necessary and sufficient condition on the potential b such that M(−Δ+λb) is lower than 4d for λ small enough. In dimensions d=1 and d=2, the situation is more precise. The following result was proved by Killip and Simon (2003) (for d=1) in [5], then by Damanik et al. (2003) (for d=1 and d=2) in [3]:

     

A Liouville type theorem for the Schrödinger operator

In this paper we prove that the equation on a complete Riemannian manifold of dimension without boundary and with nonnegative Ricci curvature admits no positive solution provided that is a function satisfying and where , and are constants depending only on the dimension, thus generalizing similar results in P. Li and S. T. Yau (Acta Math. 156 (1986), 153-201), J. Li (J. Funct. Anal. 100 (1991), 233-256) and E. R. Negrin (J. Funct. Anal. 127 (1995), 198-203) in all of which is assumed to be subharmonic. We also give a generalization in case the Ricci curvature of is not necessarily positive but its negative part has quadratic decay under the additional assumption that is unbounded from above.

     

Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space

We study a semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space. Key results are semiboundedness theorem of the Schrödinger operator, Laplace-type asymptotic formula and IMS localization formula. We also make a remark on the semiclassical problem of a Schrödinger operator on a path space over a Riemannian manifold.     

Measures on double or resonant eigenvalues for linear Schrödinger operator

In this paper we consider linear Schrödinger operator with double or resonant eigenvalues. The main result is the bound of the measure (in a suitable space of functions) of the potentials leading to such double or resonant eigenvalues. Namely we present measure type estimates evaluating neighborhoods of the so-called double or resonant set.     

The self-adjointness conditions for a higher order differential operator with an operator coefficient

Certain sufficient conditions are found for self-adjointness of the differential operator generated by the expressionl (y)=(–1) ⁿ y ²ⁿ +Q (x)y, – <x <, where Q(x) is for each fixed value of x a bounded self-adjoint operator acting from the Hilbert space H into H, and y(x) is a vector function of H₁ for which .Translated from Matematicheskie Zametki, Vol. 5, No. 6, pp. 697–707, June, 1969.     

The transformation operator for Schrödinger operators on almost periodic infinite-gap backgrounds

We investigate the kernels of the transformation operators for one-dimensional Schrödinger operators with potentials, which are asymptotically close to Bohr almost periodic infinite-gap potentials.     

On perturbations of quasiperiodic Schrödinger operators

Using relative oscillation theory and the reducibility result of Eliasson, we study perturbations of quasiperiodic Schrödinger operators. In particular, we derive relative oscillation criteria and eigenvalue asymptotics for critical potentials.     

Schrödinger equation and the oscillatory semigroup for the Hermite operator

We discuss the regularity of the oscillatory semigroup e^itH, where H=-Δ+|x|² is the n-dimensional Hermite operator. The main result is a Strichartz-type estimate for the oscillatory semigroup e^itH in terms of the mixed L^p spaces. The result can be interpreted as the regularity of solution to the Schrödinger equation with potential V(x)=|x|².     

Quadratic forms for the 1-D semilinear Schrödinger equation

This paper is concerned with 1-D quadratic semilinear
Schrödinger equations. We study local well posedness in classical Sobolev space of the associated initial value problem and periodic boundary value problem. Our main interest is to obtain the lowest value of which guarantees the desired local well posedness result. We prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered.

     

On Schrödinger semigroups and related topics

This paper deals with two related subjects. In the first part, we give generation theorems, relying on (weak) compactness arguments, for perturbed positive semigroups in general ordered Banach spaces with additive norm on the positive cone. The second part provides new functional analytic developments on semigroup theory for Schrödinger operators in Lp spaces with (L¹) Δ-bounded potentials without restriction on the (L¹) Δ-bound. In particular, our formalism enlarges a priori the classical Kato class and its subsequent refinements. The connection with form-perturbation theory is also dealt with.     

Essential self-adjointness of the Schrödinger operator with magnetic vector potential

We prove essential self-adjointness of the operator under weak conditions on the coefficients.     

On a class of Schrödinger systems with discontinuous nonlinearities

Using a non-smooth critical point theory for locally Lipschitz functionals, we investigate a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow up at infinity. The existence of nontrivial solution is obtained.     

On symplectic self-adjointness of Hamiltonian operator matrices

Symplectic self-adjointness of Hamiltonian operator matrices is studied, which is important to symplectic elasticity and optimal control. For the cases of diagonal domain and off-diagonal domain, necessary and sufficient conditions are shown. The proofs use Frobenius-Schur factorizations of unbounded operator matrices.Under additional assumptions, sufficient conditions based on perturbation method are obtained. The theory is applied to a problem in symplectic elasticity.