This lecture is a complete introduction to the general theory of operators on Hilbert spaces. We particularly focus on those tools that are essentials in Quantum Mechanics: unbounded operators, multiplication operators, self-adjointness, spectrum, functional calculus, spectral measures and von Neumann’s Spectral Theorem. 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . ...

We extend recent work concerning isospectral deformations for one-dimensional Schrödinger operators to the case of Jacobi operators. We provide a complete spectral characterization of a new method that constructs isospectral deformations of a given Jacobi operator (Hu)(n) = a(n)u(n + 1) + a(n − 1)u(n − 1) − b(n)u(n). Our technique is connected to Dirichlet data, that is, the spectrum of the ope...

Displacement map is a modeling technique that creates shape by mapping the height field of the complicated shape feature on the surface of the target shape, which is preferable in describing sophisticated feature details. This paper investigates the techniques for fast creation of the displacement map based on the rich research achievements of shape analysis and digital signal processing. The p...

We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = ∑m i=1 X i Xi + X0 + f , where the Xj denote first order differential operators, f is a function with at most polynomial growth, and X i denotes the formal adjoint of Xi in L. For any ε > 0 we show that an inequality of the form ‖u‖δ,δ ≤ C(‖u‖0,ε + ‖(K + iy)u‖0,0) holds for suitable δ and C which are...

Every student of numerical linear algebra is familiar with block matrices and vectors. The same ideas can be applied to the continuous analogues of operators, functions, and functionals. It is shown here how the explicit consideration of block structures at the continuous level can be a useful tool. In particular, block operator diagrams lead to templates for spectral discretization of differen...