Abstract. This paper deals with general structural properties of one-dimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and Denisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper [19]. The treatment of the continuous case in the present paper depends on the same basic ideas. 1.

In principle, a wide variety of sequential decision problems -- ranging from dynamic resource allocation in telecommunication networks to financial risk management -- can be formulated in terms of stochastic control and solved by the algorithms of dynamic programming. Such algorithms compute and store a value function, which evaluates expected future reward as a function of current state. Unfortunately, exact computation of the value function typically requires time and storage that grow proportionately with the number of states, and consequently, the enormous state spaces that arise in practical applications render the algorithms intractable. In this thesis, we study tractable methods that approximate the value function. Our work builds on research in an area of artificial intelligence known as reinforcement learning. A point of focus of this thesis is temporal-difference learning -- a stochastic algorithm inspired to some extent by phenomena observed in animal behavior. Given a selection of...

We present the construction of an exponentially accurate time--dependent Born-- Oppenheimer approximation for molecular quantum mechanics. We study molecular systems whose electron masses are held fixed and whose nuclear masses are proportional to # -4 , where # is a small expansion parameter. By optimal truncation of an asymptotic expansion, we construct approximate solutions to the time--dependent Schrodinger equation that agree with exact normalized solutions up to errors whose norms are bounded by C exp # -#/# 2 # , for some C and # > 0. # Partially Supported by National Science Foundation Grant DMS--9703751. 1 1 Introduction In this paper we construct exponentially accurate approximate solutions to the time--dependent Schrodinger equation for a molecular system. The small parameter that governs the approximation is the usual Born--Oppenheimer expansion parameter #, where # 4 is the ratio of the electron mass divided by the mean nuclear mass. The approximate solutions we c...

We discuss the existence and completeness of scattering for onedimensional systems with different spatial asymptotics at ± oo, for example 2 4- V(x) where V(x) = 0 (resp. sin x) if x < 0 (resp. x> 0). We then extend our results to higher dimensional systems periodic, except for a localised impurity, in all but one space dimension. A new method, "the twisting trick", is presented for proving the absence of singular continuous spectrum, and some independent applications of this trick are given in an appendix.

We consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity. More precisely, let X be an n × p matrix, and let its rows be i.i.d. complex normal vectors with mean 0 and covariance �p. We show that for a large class of covariance matrices �p, the largest eigenvalue of X ∗ X is asymptotically distributed (after recentering and rescaling) as the Tracy–Widom distribution that appears in the study of the Gaussian unitary ensemble. We give explicit formulas for the centering and scaling sequences that are easy to implement and involve only the spectral distribution of the population covariance, n and p. The main theorem applies to a number of covariance models found in applications. For example, well-behaved Toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms (under some conditions on the mass of the atoms) are among the models the theorem can handle. Generalizations of the theorem to certain spiked versions of our models and a.s. results about the largest eigenvalue are given. We also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics.

We present a novel approach to obtaining the basic facts (including Lidskii's theorem on the equality of the matrix and spectral traces) about determinants and traces of trace class operators on a separable Hilbert space. We also discuss Fredholm theory, "regularized " determinants and Fredholm theory on the trace ideals, c#~(p < oo). 1.

Bohmian mechanics is the most naively obvious embedding imaginable of Schrödinger’s equation into a completely coherent physical theory. It describes a world in which particles move in a highly non-Newtonian sort of way, one which may at first appear to have little to do with the spectrum of predictions of quantum mechanics. It turns out, however, that as a consequence of the defining dynamical equations of Bohmian mechanics, when a system has wave function ψ its configuration is typically random, with probability density ρ given by |ψ | 2, the quantum equilibrium distribution. It also turns out that the entire quantum formalism, operators as observables and all the rest, naturally emerges in Bohmian mechanics from the analysis of “measurements. ” This analysis reveals the status of operators as observables in the description of quantum phenomena, and facilitates a clear view of the range of applicability of the usual quantum mechanical formulas. Dedicated to Elliott Lieb on the occasion of his 70th birthday. Elliott will be (we fear unpleasantly) surprised to learn that he bears a greater responsibility for this paper than he could possibly imagine. We would of course like to think that our work addresses in some way the concern suggested by the title of his recent talks, The

We calculate the firing rate of the quadratic integrate-and-fire neuron in response to a colored noise input current. Such an input current is a good approximation to the noise due to the random bombardment of spikes, with the correlation time of the noise corresponding to the decay time of the synapses. The key parameter that determines the firing rate is the ratio of the correlation time of the colored noise, ¿s, to the neuronal time constant, ¿m. We calculate the firing rate exactly in two limits: when the ratio, ¿s=¿m, goes to zero (white noise) and when it goes to infinity. The correction to the short correlation time limit is O.¿s=¿m/, which is qualitatively different from that of the leaky integrate-and-fire neuron, where the correction is O. p ¿s=¿m/. The difference is due to the different boundary conditions of the probability density function of the membrane potential of the neuron at firing threshold. The correction to the long correlation time limit is O.¿m=¿s/. By combining the short and long correlation time limits, we derive an expression that provides a good approximation to the firing rate over the whole range of ¿s=¿m in the suprathreshold regime— that is, in a regime in which the average current is sufficient to make the cell fire. In the subthreshold regime, the expression breaks down somewhat when ¿s becomes large compared to ¿m.

Abstract. Let K be a connected Lie group of compact type and let T ∗ (K) be its cotangent bundle. This paper considers geometric quantization of T ∗ (K), first using the vertical polarization and then using a natural Kähler polarization obtained by identifying T ∗ (K) with the complexified group KC. The first main result is that the Hilbert space obtained by using the Kähler polarization is naturally identifiable with the generalized Segal–Bargmann space introduced by the author from a different point of view, namely that of heat kernels. The second main result is that the pairing map of geometric quantization coincides with the generalized Segal–Bargmann transform introduced by the author. This means that in this case the pairing map is a constant multiple of a unitary map. For both results it is essential that the half-form correction be included when using the Kähler polarization. These results should be understood in the context of results of K. Wren and of the author with B. Driver concerning the quantization of 1+1-dimensional Yang–Mills theory. Together with those results the present paper may be seen as an instance of “quantization commuting with reduction.” Contents

Abstract. We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is the asymptotic behavior of the integrated density of states at the boundary of the spectrum. In particular, we are interested in Lifshitz tails and the occurrence of a classical and a quantum regime. In the last section we discuss regularity properties of the IDS. Our emphasis is on the discussion of fundamental problems and central ideas to handle them. Finally, we discuss further developments and problems of current