This is the second in a series of works devoted to small non-selfadjoint perturbations of selfadjoint semiclassical pseudodifferential operators in dimension 2. As in our previous work, we consider the case when the classical flow of the unperturbed part is periodic. Under the assumption that the flow average of the leading perturbation vanishes identically, we show how to obtain a complete asy...

Krein space operator-theoretic methods are used to prove that the operator (sgn x) d is similar to a selfadjoint operator in the Hilbert space L2 (R) . Let L be a symmetric ordinary differential expression. Spectral properties of the operators associated with the weighted eigenvalue problem Lu = )wu have been studied extensively. When w is positive, this problem leads to a selfadjoint problem i...

Classically, there are many interesting connections between differential operators and the theory of elliptic modular forms and many interesting results have been explored. In particular, it has been known for some time how to obtain an elliptic modular form from the derivatives ofN elliptic modular forms, which has already been studied in detail by R. Rankin in [9] and [10]. When N = 2, as a s...

Distinguished selfadjoint extensions of operators which are not semibounded can be deduced from the positivity of the Schur Complement (as a quadratic form). In practical applications this amounts to proving a Hardy-like inequality. Particular cases are Dirac-Coulomb operators where distinguished selfadjoint extensions are obtained for the optimal range of coupling constants.

For any non-negative integer v we construct explicitly ⌊v2⌋+1 independent covariant bilinear differential operators from Jk,m × Jk′,m′ to Jk+k′+v,m+m′ . As an application we construct a covariant bilinear differential operator mapping S (2) k ×S (2) k′ to S (2) k+k′+v. Here Jk,m denotes the space of Jacobi forms of weight k and index m and S (2) k the space of Siegel modular forms of degree 2 a...

In L 2 ( R 3 ; C ) , we consider a selfadjoint operator ? > 0 given by the differential expression ? ? 1 / curl ? x ? ? div where is constant positive matrix, matrix-valued function and real-valued are periodic with respect to some lattice, definite bounded. We study behavior of operator-valued functions cos ? ? sin for ? small . It shown that these operators converge corresponding in norm acti...

We study spectral asymptotics for small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part possesses several invariant Lagrangian tori enjoying a Diophantine property. We get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases: in the...