Nonlocal Schrödinger Problem with Time Dependent Self-Adjoint Operator

Schrödinger equations with time - dependent

We present some general results for the time-dependent mass Hamiltonian problem with H = − 2e∂xx + h(2)(t)e2νx2. This Hamiltonian corresponds to a time-dependent mass (TM) Schrödinger equation with the restriction that there are only P 2 and X2 terms. We give the specific transformations to a different quantum Schrödinger(TQ) equation and to a different time-dependent oscillator (TO) equation. ...

The Laplace operator is essentially self-adjoint

Correspondence of the eigenvalues of a non-self-adjoint operator to those of a self-adjoint operator

We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at ±∞. We use this result to determine the asymptotic distribution of the eigenvalues and to compute some of ...

Localized Modes of the Linear Periodic Schrödinger Operator with a Nonlocal Perturbation

We consider the existence of localized modes corresponding to eigenvalues of the periodic Schrödinger operator −∂ x + V (x) with an interface. The interface is modeled by a jump either in the value or the derivative of V (x) and, in general, does not correspond to a localized perturbation of the perfectly periodic operator. The periodic potentials on each side of the interface can, moreover, be...

The Classical Moment Problem as a Self-Adjoint Finite Difference Operator

This is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators. Among the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix and convergence of Padé approximants appears as the strong resolvent convergence of finite matrix approximations to a Jacobi matrix. As a bonus of this, we o...