Let L = d/dx +u(x) be the one-dimensional Schrödinger operator and H(x, y, t) be the corresponding heat kernel. We prove that the nth Hadamard’s coefficient Hn(x, y) is equal to 0 if and only if there exists a differential operator M of order 2n− 1 such that L = M. Thus, the heat expansion is finite if and only if the potential u(x) is a rational solution of the KdV hierarchy decaying at infini...

Starting from Katsuno and Mendelzon postulates, a new set of postulates which is minimal and complete has been deened, these postulates characterize an update operator. The update operator class which is obtained is larger than Katsuno and Mendelzon's one, since it includes updating operators which are not inert, and which allow for the existence of unreachable states. The main property is that...

Consider the Floquet operator of a time independent quantum system, periodically perturbed by a rank one kick, acting on a separable Hilbert space: e0 e |φ〉〈φ| where T and κ are the period and the coupling constant respectively. Assume the spectrum of the self adjoint operator H0 is pure point, simple, bounded from below and the gaps between the eigenvalues (λn) grow like: λn+1 − λn ∼ Cn with d...

The variational quantum eigensolver is one of the most promising algorithms for near-term computers. It has potential to solve chemistry problems involving strongly correlated electrons with relatively low-depth circuits, which are otherwise difficult on classical eigenstate constructed from a number factorized unitary coupled-cluster terms applied onto an initial (single-reference) state. Curr...

Abstract The eigenvectors of the particle number operator in second quantization are characterized by block sparsity their matrix product state representations. This is shown to generalize other classes operators. Imposing yields a scheme for conserving that commonly used applications physics. Operations on such structures, rank truncation, and implications numerical algorithms discussed. Expli...

Let E be an optimal elliptic curve of conductor N , such that the L-function LE(s) of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split. The Gross-Zagier theorem gives a formula that expresses the Birch and Swinnerton-Dyer conjectural order the Shafarevich-Tate group of E over K as a rational number. We extract an integer factor from...

indeed has $D=b¥cdot c^{*}$ for $b^{*}=(0,¥ldots,0,1)$ and $c^{*}$ with entries $¥frac{1}{2}(¥beta_{k}-a_{k})$ . (As Jan Willems once pointed out, all the entries are constants; however, the zeros and ones are stiff structure constants, while only the last row has “soft” parameters, to be encompassed by a rank-one control matrix.) Section 1 presents a canonic decomposition of the state space of...

Low-rank tensor approximation techniques attempt to mitigate the overwhelming complexity of linear algebra tasks arising from high-dimensional applications. In this work, we study the low-rank approximability of solutions to linear systems and eigenvalue problems on Hilbert spaces. Although this question is central to the success of all existing solvers based on low-rank tensor techniques, very...

We consider the problem of designing an optimal quantum detector to minimize the probability of a detection error when distinguishing between a collection of quantum states, represented by a set of density operators. We show that the design of the optimal detector can be formulated as a semidefinite programming problem. Based on this formulation, we derive a set of necessary and sufficient cond...