Quantized-TT-cayley transform for computing the dynamics and the spectrum of high-dimensional hamiltonians

In the present paper, we propose and analyse a class of tensor methods for the eifcient 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 logN), 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 eifcient 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 eifcient 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 logN lnq 1ε ), 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(dm4 log Nt logN), where Nt is the number of sampling points in time. The latter allows eifcient energy spectrum calculations by FFT (or QTT-FFT) of the autocorrelation function computed on a suifciently 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 timedependent molecular Schrödinger equation. © 2011 Institute of Mathematics, National Academy of Sciences.