M. A. Naĭmark

Bio: M. A. Naĭmark is an academic researcher. The author has contributed to research in topics: Fourier integral operator & Operator theory. The author has an hindex of 5, co-authored 7 publications receiving 2429 citations.

Making sense of non-Hermitian Hamiltonians

Abstract: The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose +complex conjugate) is replaced by the physically transparent condition of space?time reflection ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive!Does a -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken symmetry, then it has another symmetry represented by a linear operator . In terms of , one can construct a time-independent inner product with a positive-definite norm. Thus, -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution.The Lee model provides an excellent example of a -symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm 'ghost' state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is -symmetric. The operator for the Lee model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.

Fourier Integral Operators. I

Abstract: Pseudo-differential operators have been developed as a tool for the study of elliptic differential equations. Suitably extended versions are also applicable to hypoelliptic equations, but their value is rather limited in genuinely non-elliptic problems. In this paper we shall therefore discuss some more general classes of operators which are adapted to such applications. For these operators we shall develop a calculus which is almost as smooth as that of pseudo-differential operators. It also seems that one gains some more insight into the theory of pseudo-differential operators by considering them from the point of view of the wider classes of operators to be discussed here so we shall take the opportunity to include a short exposition.

The spectrum of positive elliptic operators and periodic bicharacteristics

Abstract: Let X be a compact boundaryless C ∞ manifold and let P be a positive elliptic self-adjoint pseudodifferential operator of order m>0 on X. For technical reasons we will assume that P operates on half-densities rather than functions.

Interpretation of Inaccurate, Insufficient and Inconsistent Data

Abstract: Sumntary Many problems in physical science involve the estimation of a number of unknown parameters which bear a linear or quasi-linear relationship to a set of experimental data. The data may be contaminated by random errors, insufficient to determine the unknowns, redundant, or all of the above. This paper presents a method of optimizing the conclusions from such a data set. The problem is formulated as an ill-posed matrix equation, and general criteria are established for constructing an ‘ inverse ’ matrix. The ‘ solution ’ to the problem is defined in terms of a set of generalized eigenvectors of the matrix, and may be chosen to optimize the resolution provided by the data, the expected error in the solution, the fit to the data, the proximity of the solution to an arbitrary function, or any combination of the above. The classical ‘ least-squares ’ solution is discussed as a special case.