Abstract In the present paper, we propose and analyse a class of tensor methods for the efficient
numerical computation of the dynamics and spectrum of high-dimensional Hamiltonians. We focus
on the complex-time evolution problems. We apply the quantized-TT (QTT) matrix product states
type tensor approximation that allows to represent N-d tensors generated by the grid
representation of d-dimensional functions and operators with log-volume complexity, O(d log N),
where N is the univariate discretization parameter in space. Making use of the truncated
Cayley transform method allows us to recursively separate the time and space variables and
then introduce the efficient QTT representation of both the temporal and the spatial parts of
the solution to the high-dimensional evolution equation. We prove the exponential convergence
of the m-term time-space separation scheme and describe the efficient tensor-structured
preconditioners for the arising system with multidimensional Hamiltonians. For the class of
"analytic" and low QTT-rank input data, our method allows to compute the solution at a fixed
point in time t=T>0 with an asymptotic complexity of order O(d log N ln^q (1/ε)), where ε>0
is the error bound and q is a fixed small number. The time-and-space separation method via
the QTT-Cayley-transform enables us to construct a global m-term separable (x,t)-representation
of the solution on a very fine time-space grid with complexity of order O(dm^4 log N_t log N),
where N_t is the number of sampling points in time. The latter allows efficient energy
spectrum calculations by FFT (or QTT-FFT) of the autocorrelation function computed on a
sufficiently long time interval [0,T]. Moreover, we show that the spectrum of the Hamiltonian
can also be represented by the poles of the t-Laplace transform of a solution. In particular,
the approach can be an option to compute the dynamics and the spectrum in the time-dependent
molecular Schrödinger equation.
Weighted Estimates of the Cayley Transform Method for Boundary Value Problems in a Banach Space
