Introduction Yogi Berra is reputed to have said, "Prediction is difficult, especially about the future." Lists of open problems are typically lists of problems on which you expect progress in a reasonable time scale and so they involve an element of prediction. We have seen remarkable progress in the past fifty years in our understanding of Schrodinger operators, as I discussed in Simon [1]. In this companion piece, I present fifteen open problems. In 1984, I presented a list of open problem in Mathematical Physics, including thirteen in Schrodinger operators. Depending on how you count (since some are multiple), five have been solved. We will focus on two main areas: anomalous transport (Section 2) where I expect progress in my lifetime, and Coulomb energies where some of the problems are so vast and so far from current technology that I do not expect them to be solved in my lifetime. (There is a story behind the use of this phrase. I have heard that when Jeans lectured in

Abstract: We analyze the time evolution of a one-dimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation η(t). We show that for generic η(t), which includes the sum of any finite number of harmonics, the system, started in a bound state will get fully ionized as t →∞. This is irrespective of the magnitude or frequency (resonant or not) of η(t). There are however exceptional, very non-generic η(t), that do not lead to full ionization, which include rather simple explicit periodic functions. For these η(t) the system evolves to a nontrivial localized stationary state which is related to eigenfunctions of the Floquet 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 Szegö type condition.

The paper concerns the magnetic Schrödinger operator H(a, V) =

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Abstract. The existence and uniqueness of the initial value problem for Schrödinger and wave equations in the presence of a (large) time dependent potential is studied. The usual Strichartz estimates for such linear evolutions are shown to hold true with optimal assumptions on the potentials. As a byproduct, one obtains a counterexample to the two dimensional double endpoint inhomogeneous Strichartz estimate. 1.

We obtain several essential self-adjointness conditions for the Schrödinger type operator HV = D ∗ D + V, where D is a first order elliptic differential operator acting on the space of sections of a hermitian vector bundle E over a manifold M with positive smooth measure dµ, and V is a Hermitian bundle endomorphism. These conditions are expressed in terms of completeness of certain metrics on M naturally associated with HV. These results generalize the

Abstract. We present a direct discretization of the electronic Schrödinger equation. It is based on one-dimensional Meyer wavelets from which we build an anisotropic multiresolution analysis for general particle spaces by a tensor product construction. We restrict these spaces to the case of antisymmetric functions. To obtain finite-dimensional subspaces we first discuss semi-discretization with respect to the scale parameter by means of sparse grids which relies on mixed regularity and decay properties of the electronic wave functions. We then propose different techniques for a discretization with respect to the position parameter. Furthermore we present the results of our numerical experiments using this new generalized sparse grid methods for Schrödinger’s equation.

Since it became clear that the band structure of the spectrum of periodic Sturm-Liouville operators t =-(d )+q(r) does not survive a spherically symmetric extension to Schrödinger operators T =-#+VwithV(x)=q(|x|) for x N\{1}, a wealth of detailed information about the spectrum of such operators has been acquired. The observation of eigenvalues embedded in the essential spectrum of T with exponentially decaying eigenfunctions provided evidence for the existence of intervals of dense point spectrum, eventually proved by spherical separation into perturbed Sturm-Liouville operators t c ). Subsequently, a numerical approach was employed to investigate the distribution of eigenvalues of T more closely. An eigenvalue was discovered below the essential spectrum in the case d 2, and it turned out that there are in fact infinitely many, accumulating at 0 . Moreover, a method based on oscillation theory made it possible to count eigenvalues of t c contributing to an interval of dense point spectrum of T . We gained evidence that an asymptotic formula, valid for c does in fact produce correct numbers even for small values of the coupling constant, such that a rather precise picture of the spectrum of radially periodic Schrödinger operators has now been obtained.

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