Perturbation Theory for the Logarithm of a Positive Operator

In various contexts in mathematical physics one needs to compute the logarithm of a positive unbounded operator. Examples include the von Neumann entropy of a density matrix and the flow of operators with the modular Hamiltonian in the Tomita-Takesaki theory. Often, one encounters the situation where the operator under consideration, that we denote by $\Delta$, can be related by a perturbative series to another operator $\Delta_0$, whose logarithm is known. We set up a perturbation theory for the logarithm $\log \Delta$. It turns out that the terms in the series possess remarkable algebraic structure, which enable us to write them in the form of nested commutators plus some "contact terms."Comment: 30 page

An analytical Model which Determines the Apparent T1 for Modified Look-Locker Inversion Recovery (MOLLI) -- Analysis of the Longitudinal Relaxation under the Influence of Discontinuous Balanced and Spoiled Gradient Echo Readouts

Quantitative nuclear magnetic resonance imaging (MRI) shifts more and more into the focus of clinical research. Especially determination of relaxation times without/and with contrast agents becomes the foundation of tissue characterization, e.g. in cardiac MRI for myocardial fibrosis. Techniques which assess longitudinal relaxation times rely on repetitive application of readout modules, which are interrupted by free relaxation periods, e.g. the Modified Look-Locker Inversion Recovery = MOLLI sequence. These discontinuous sequences reveal an apparent relaxation time, and, by techniques extrapolated from continuous readout sequences, the real T1 is determined. What is missing is a rigorous analysis of the dependence of the apparent relaxation time on its real partner, readout sequence parameters and biological parameters as heart rate. This is provided in this paper for the discontinuous balanced steady state free precession (bSSFP) and spoiled gradient echo readouts. It turns out that the apparente longitudinal relaxation rate is the time average of the relaxation rates during the readout module, and free relaxation period. Knowing the heart rate our results vice versa allow to determine the real T1 from its measured apparent partner.Comment: 1 Figur

Threshold singularities of the spectral shift function for geometric perturbations of magnetic Hamiltonians

We consider the 3D Schr\"odinger operator $H_0$ with constant magnetic field $B$ of scalar intensity $b>0$, and its perturbations $H_+$ (resp., $H_-$) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain $\Omega_{\rm in} \subset {\mathbb R}^3$. We introduce the Krein spectral shift functions $\xi(E;H_\pm,H_0)$, $E \geq 0$, for the operator pairs $(H_\pm,H_0)$, and study their singularities at the Landau levels $\Lambda_q : = b(2q+1)$, $q \in {\mathbb Z}_+$, which play the role of thresholds in the spectrum of $H_0$. We show that $\xi(E;H_+,H_0)$ remains bounded as $E \uparrow \Lambda_q$, $q \in {\mathbb Z}_+$ being fixed, and obtain three asymptotic terms of $\xi(E;H_-,H_0)$ as $E \uparrow \Lambda_q$, and of $\xi(E;H_\pm,H_0)$ as $E \downarrow \Lambda_q$. The first two terms are independent of the perturbation while the third one involves the {\em logarithmic capacity} of the projection of $\Omega_{\rm in}$ onto the plane perpendicular to $B$.Comment: 35 page

Exposed faces of semidefinitely representable sets

A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinite representable sets. Part of the interest in spectrahedra and semidefinite representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming, like one can do on polyhedra by linear programming. It is known that every face of a spectrahedron is exposed. This is also true in the general context of rigidly convex sets. We study the same question for semidefinite representable sets. Lasserre proposed a moment matrix method to construct semidefinite representations for certain sets. Our main result is that this method can only work if all faces of the considered set are exposed. This necessary condition complements sufficient conditions recently proved by Lasserre, Helton and Nie

Optimized Quantification of Spin Relaxation Times in the Hybrid State

Purpose: The analysis of optimized spin ensemble trajectories for relaxometry in the hybrid state. Methods: First, we constructed visual representations to elucidate the differential equation that governs spin dynamics in hybrid state. Subsequently, numerical optimizations were performed to find spin ensemble trajectories that minimize the Cram\'er-Rao bound for $T_1$-encoding, $T_2$-encoding, and their weighted sum, respectively, followed by a comparison of the Cram\'er-Rao bounds obtained with our optimized spin-trajectories, as well as Look-Locker and multi-spin-echo methods. Finally, we experimentally tested our optimized spin trajectories with in vivo scans of the human brain. Results: After a nonrecurring inversion segment on the southern hemisphere of the Bloch sphere, all optimized spin trajectories pursue repetitive loops on the northern half of the sphere in which the beginning of the first and the end of the last loop deviate from the others. The numerical results obtained in this work align well with intuitive insights gleaned directly from the governing equation. Our results suggest that hybrid-state sequences outperform traditional methods. Moreover, hybrid-state sequences that balance $T_1$- and $T_2$-encoding still result in near optimal signal-to-noise efficiency. Thus, the second parameter can be encoded at virtually no extra cost. Conclusion: We provide insights regarding the optimal encoding processes of spin relaxation times in order to guide the design of robust and efficient pulse sequences. We find that joint acquisitions of $T_1$ and $T_2$ in the hybrid state are substantially more efficient than sequential encoding techniques.Comment: 10 pages, 5 figure

On Infinite Words Determined by Indexed Languages

We characterize the infinite words determined by indexed languages. An infinite language $L$ determines an infinite word $\alpha$ if every string in $L$ is a prefix of $\alpha$. If $L$ is regular or context-free, it is known that $\alpha$ must be ultimately periodic. We show that if $L$ is an indexed language, then $\alpha$ is a morphic word, i.e., $\alpha$ can be generated by iterating a morphism under a coding. Since the other direction, that every morphic word is determined by some indexed language, also holds, this implies that the infinite words determined by indexed languages are exactly the morphic words. To obtain this result, we prove a new pumping lemma for the indexed languages, which may be of independent interest.Comment: Full version of paper accepted for publication at MFCS 201

Dephasing Effect in Photon-Assisted Resonant Tunneling through Quantum Dots

We analyze dephasing in single and double quantum dot systems. The decoherence is introduced by the B\"{u}ttiker model with current conserving fictitious voltage leads connected to the dots. By using the non-equilibrium Green function method, we investigate the dephasing effect on the tunneling current. It is shown that a finite dephasing rate leads to observable effects. The result can be used to measure dephasing rates in quantum dots.Comment: 4 pages, 3 figures, to be published in Rapid Communications of Phys. Rev.