Tobias Mühlenbruch

Eigenfunctions of transfer operators and cohomology

A. The eigenfunctions with eigenvalues 1 or −1 of the transfer operator of Mayer are in bijective correspondence with the eigenfunc tio s with eigenvalue 1 of a transfer operator connected to the nearest integ er continued fraction algorithm. This is shown by relating these eigenspaces of th ese operators to cohomology groups for the modular group with coe fficients in certain principal series representations.

Hecke operators on period functions for the full modular group

Matrix representations of Hecke operators on classical holomorphical cusp forms and corresponding period polynomials are well known. In this article we define Hecke operators on period functions introduced recently by Lewis and Zagier and show how they are related to the Hecke operators on Maass cusp forms. Moreover we give an explicit general compatibility criterion for formal sums of matrices to represent Hecke operators on period functions. An explicit example of such matrices with only nonnegative entries is constructed. 1991 Mathematics Subject Classification: 11F25, 11F67

Two interacting particles on the half-line

In the case of general compact quantum graphs, many-particle models with singular two-particle interactions were introduced by Bolte and Kerner [J. Phys. A: Math. Theor. 46, 045206 (2013); 46, 045207 (2013)] in order to provide a paradigm for further studies on many-particle quantum chaos. In this note, we discuss various aspects of such singular interactions in a two-particle system restricted to the half-line ℝ+. Among others, we give a description of the spectrum of the two-particle Hamiltonian and obtain upper bounds on the number of eigenstates below the essential spectrum. We also specify conditions under which there is exactly one such eigenstate. As a final result, it is shown that the ground state is unique and decays exponentially as x2+y2→∞.

Generalized Maass wave forms

Borel A., 1997, CAMBRIDGE TRACTS MAT, V130; Bruggeman R.W., 1981, LECT NOTES MATH, V865; BRUGGEMAN RW, 1978, INVENT MATH, V45, P1, DOI 10.1007-BF01406220; Bruggeman R.W., 1994, MONOGRAPHS MATH, V88; Bruinier JH, 2009, MATH ANN, V345, P31, DOI 10.1007-s00208-009-0338-4; Bruinier JH, 2008, MATH ANN, V342, P673, DOI 10.1007-s00208-008-0252-1; Bump Daniel, 1997, CAMBRIDGE STUDIES AD, V55, DOI DOI 10.1017-CBO9780511609572; Dong CY, 2000, COMMUN MATH PHYS, V214, P1, DOI 10.1007-s002200000242; Eichler M., 1957, MATH Z, V67, P267, DOI 10.1007-BF01258863; Eichler M., 1965, ACTA ARITH, V11, P169; Iwaniec H., 2002, GRADUATE STUDIES MAT, V53; Knopp M, 2004, ILLINOIS J MATH, V48, P1345; Knopp M, 2010, INT J NUMBER THEORY, V6, P1083, DOI 10.1142-S179304211000340X; Knopp M, 2003, ACTA ARITH, V110, P117, DOI 10.4064-aa110-2-2; Knopp M, 2003, J NUMBER THEORY, V99, P1, DOI 10.1016-S0022-314X(02)00065-3; Knopp M, 2009, INT J NUMBER THEORY, V5, P1049, DOI 10.1142-S1793042109002547; KNOPP MI, 1974, B AM MATH SOC, V80, P607, DOI 10.1090-S0002-9904-1974-13520-2; Lewis J, 2001, ANN MATH, V153, P191, DOI 10.2307-2661374; Maass H., 1983, LECT MODULAR FUNCTIO; Magnus Wilhelm, 1966, GRUND MATH WISS, V52; Mayer H., 1991, B AM MATH SOC, V25, P55; Muhlenbruch T, 2006, J NUMBER THEORY, V118, P208, DOI 10.1016-j.jnt.2005.09.003; Muhlenbruch T., 2003, THESIS UTRECHT U; Raji W, 2009, FUNCT APPROX COMM MA, V41, P105; Raji W, 2009, INT J NUMBER THEORY, V5, P153; Zhu YC, 1996, J AM MATH SOC, V9, P237, DOI 10.1090-S0894-0347-96-00182-8

On a two-particle bound system on the half-line

In this paper we provide an extension of the model discussed in [arXiv:1504.08283] describing two singularly interacting particles on the half-line. In this model, the particles are interacting only whenever at least one particle is situated at the origin. Stimulated by [arXiv:1503.08814] we then provide a generalisation of this model in order to include additional interactions between the particles leading to a molecular-like state. We give a precise mathematical formulation of the Hamiltonian of the system and perform spectral analysis. In particular, we are interested in the effect of the singular two-particle interactions onto the molecule.