Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-Hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, C-selfadjoint extensions of C-symmetric unbounded operators, resolvent estimates, reality of spectrum, bases of C-orthonormal vectors and conjugate-linear symmetric operators. The main results are complemented by a variety of natural examples arising in field theory, quantum physics and complex variables.

/pdf/mathematical-and-physical-aspects-of-complex-symmetric-4r85dkbaa9.pdf

Mathematical and physical aspects of complex symmetric operators

Let S be a Riemannian symmetric space of noncompact type, and let G be the group of motions of S. Then the algebra L-~ of G-invariant differential operators on S is commutative, and its spectrum A(S) can be canonically identified with ~/w where ~ is a complex vector space with dimension equal to the rank of S, and to is a finite subgroup of G L ( ~ ) generated by reflexions. Let P be a discrete subgroup of G that acts freely on S and let X = E \\ S . Then the members of 5~ may be regarded as differential operators on X. Let us now assume that X is compact and define the spectrum A of X as the set of those elements of A(S) for which one can find a nonzero eigenfunction defined on X. In this paper we study the relationship of A to the geometry of X and determine the asymptotic growth of A as a subset of A(S). In subsequent papers we plan to study the asymptotic behaviour of the eigenfunctions and to examine the problem of obtaining improvements on the error estimates. It is well-known that G, which is transitive on S, is a connected real semisimple Lie group with trivial center, and that the stabilizers in G of the points of S are the maximal compact subgroups of G. So we can take S = G/K, X =F\\G/K, where K is a fixed maximal compact subgroup of G, and F is a discrete subgroup of G containing no elliptic elements (= elements conjugate to an element of K) other than e, such that F\\G is compact. Let G = K A N be an Iwasawa decomposit ion of G; let o be the Lie algebra of A; and let to be the Weyl group of (G, A). If we take ,~to be the dual of the complexification a c of a, then A ( S ) ~ / w canonically. In what follows we shall commit an abuse of notation and identify A(S) with ,~, but with the proviso that points of ~ in the same w-orbit represent the same element of A(S).

/pdf/spectra-of-compact-locally-symmetric-manifolds-of-negative-dzuuvcw07m.pdf

Spectra of Compact Locally Symmetric Manifolds of Negative Curvature.

In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degenerate. As application we obtain the Bergman kernel asymptotics for adjoint semi-positive line bundles over complete Kaehler manifolds, on the set where the curvature is positive. We also prove the asymptotics for big line bundles endowed with singular Hermitian metrics with strictly positive curvature current. In this case the full asymptotics holds outside the singular locus of the metric.

Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles

In this paper we study the asymptotic behaviour of the spectral function corre- sponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degenerate. As application we obtain the Bergman kernel asymptotics for adjoint semi-positive line bundles over complete Khler manifolds, on the set where the curvature is positive. We also prove the asymptotics for big line bundles endowed with singular Hermitian metrics with strictly positive curvature current. In this case the full asymptotics holds outside the singular locus of the metric.

/pdf/asymptotics-of-spectral-function-of-lower-energy-forms-and-3wd3133mnk.pdf

. Let L be a nonsymmetric operator of type L u = L aij ai 8 j u ai u on an arbitrary subset U c R . We analyse L as an operator on L 1 (U, f1) where f1 is an invariant measure, i.e., a possibly infinite measure f1 satisfying the equation = 0 (in the weak sense). We explicitely construct, under mild regularity assumptions, extensions of L generating sub-Markovian Co-semigroups on L 1 (U, it) as well as associated diffusion processes. We give sufficient conditions on the coefficients so that there exists only one extension of L generating a Co-semigroup and apply the results to prove uniqueness of the invariant measure f1. Our results imply in particular that if ~ E ~ # 0 dx-a.e., the generalized Schrodinger operator (A + V, C’(R d)) has exactly one extension generating a Co-semigroup if and only if the Friedrich’s extension is conservative. We also give existence and uniqueness results for a corresponding class of infinite dimensional operators acting on smooth cylinder functions on a separable real Banach space.

Annali della Scuola normale superiore di Pisa. Classe di scienze. Serie IV

Some examples of hypoelliptic partial differential equations

Pseudo-differential Operators with Double Characteristics.

On the eigenvalues of a class of hypoelliptic operators

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Some examples of hypoelliptic partial differential equations

Pseudo-differential Operators with Double Characteristics.

On the eigenvalues of a class of hypoelliptic operators