SUSY transformations with complex factorization constants. Application to spectral singularities

Supersymmetric (SUSY) transformation operators corresponding to complex factorization constants are analyzed as operators acting in the Hilbert space of functions square integrable on the positive semiaxis. Obtained results are applied to Hamiltonians possessing spectral singularities which are non-Hermitian SUSY partners of selfadjoint operators. A new regularization procedure for the resolution of the identity operator in terms of continuous biorthonormal set of the non-Hermitian Hamiltonian eigenfunctions is proposed. It is also shown that the continuous spectrum eigenfunction has zero binorm (in the sense of distributions) at the singular point.Comment: Thanks to A. Sokolov a number of inaccuracies are correcte

SUSY transformation of the Green function and a trace formula

An integral relation is established between the Green functions corresponding to two Hamiltonians which are supersymmetric (SUSY) partners and in general may possess both discrete and continuous spectra. It is shown that when the continuous spectrum is present the trace of the difference of the Green functions for SUSY partners is a finite quantity which may or may not be equal to zero despite the divergence of the traces of each Green function. Our findings are illustrated by using the free particle example considered both on the whole real line and on a half line

Supersymmetry of the Nonstationary Schr\"odinger equation and Time-Dependent Exactly Solvable Quantum Models

New exactly solvable quantum models are obtained with the help of the supersymmetric extencion of the nonstationary Schr/"odinger equation.Comment: Talk at the 8th International Conference "Symmetry Methods in Physics". Dubna, Russia, 28 July - 2 August, 199

Fluctuations of atomic energy levels due to axion dark matter

The amplitude of the pseudoscalar (axion) or scalar field fluctuates on a time scale of order of million field oscillation periods which is a typical coherence time in the virialized axion galactic dark matter halo model. This causes fluctuations of frequencies of atomic clocks on the same time scale. We show that this effect may be employed to search for the axion and scalar field dark matter with atomic and nuclear clocks. We re-purpose the results of the atomic clocks experiments comparing the variations of frequencies of hyperfine transitions in Rb and Cs atoms as well as in hydrogen atom vs cavity frequency fluctuations, and extract new limits on the axion coupling constant $f_a$ for masses in the range $2\times 10^{-17}\text{ eV}\lesssim m \lesssim 10^{-13}\text{ eV}$. We also show that similar energy shifts arise in the second-order perturbation theory with linear in the pseudoscalar field interaction. These shifts may be potentially measured with nuclear clocks based on the low-energy transition in $^{229}$Th nucleus. We propose a procedure which could, in principle, help determine the axion mass if the axion dark matter signal is present in experimental data sets.Comment: New section about difference between axion signal and white nois