The ability of the Rigged Hilbert Space formalism to deal with continuous spectrum is demonstrated within the example of the square barrier potential. The non-square integrable solutions of the time-independent Schrödinger equation are used to define Dirac kets, which are (generalized) eigenvectors of the Hamiltonian. These Dirac kets are antilinear functionals over the space of physical wave f...

Following [Hub93, §3], we introduce the spectrum of continuous valuations Cont(A) for a Huber ring A and the adic spectrum Spa(A,A) for a Huber pair (A,A). We also draw heavily from [Con14; Wed12]. These notes are from the arithmetic geometry learning seminar on adic spaces held at the University of Michigan during the Winter 2017 semester, organized by Bhargav Bhatt. See [Dat17; Ste17] for oth...

We recall the notion of cuspform for Γ = SL2(Z) acting on the upper half-plane, and prove that the orthogonal complement in L(Γ\H) to cuspforms is spanned by pseudo-Eisenstein series, which in turn are expressible as wave packets of Eisenstein series Es. Further we have a Plancherel theorem for this space. The non-trivial harmonic analysis resides in the application of the Fourier transform on ...

Abstract. We prove several new results on the absolutely continuous spectrum of perturbed one-dimensional Stark operators. First, we find new classes of potentials, characterized mainly by smoothness condition, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may o...

Some new conditions are obtained for the absolutely continuous spectrum of a Stark operator to fill the entire real line. §

We construct examples of potentials V (x) satisfying |V (x)| ≤ h(x) 1+x , where the function h(x) is growing arbitrarily slowly, such that the corresponding Schrödinger operator has imbedded singular continuous spectrum. This solves one of the fifteen “twenty-first century” problems for Schrödinger operators posed by Barry Simon in [22]. The construction also provides the first example of a Sch...

The ability of the Rigged Hilbert Space formalism to deal with continuous spectrum is demonstrated within the example of the square barrier potential. The non-square integrable solutions of the time-independent Schrödinger equation are used to define Dirac kets, which are (generalized) eigenvectors of the Hamiltonian. These Dirac kets are antilinear functionals over the space of physical wave f...

In this paper, we describe the Pascal automorphism and present a sketch of the proof that its spectrum is continuous in the orthogonal complement of the constants.

We give new examples of discrete Schrödinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hull X of the potential is strictly ergodic, then the existence of just one potential x in X for which the operator has no eigenvalues implies that there is a generic set in X for which the operator has purely singular continuous spectrum. A su...

We show that the map from minimal-parabolic pseudo-Eisenstein series to their decomposition coefficients against minimal-parabolic Eisenstein series is an isometry to its image. The argument characterizing the image is sketched, but is incomplete without characterizing the other, less-continuous parts of the spectral decomposition. Thus, we do not quite prove Plancherel for this part of the spe...