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**Determinants of pseudodifferential operators and complex deformations of phase space.**
*(English)*
Zbl 1082.35176

It is known that in the theory of nonselfadjoint operators, determinants play an important role. Furthermore, recent developments in the theory of resonances have used determinants to obtain upper bounds on the density of resonances. The authors obtain estimates and asymptotics of determinants by direct microlocal methods, and this paper represents a very important step in this direction.

The authors consider an \(h\)-pseudodifferential operator \(p(x,hD)\), whose symbol \(p\) extends holomorphically in a tubular neighborhood of the (real) phase space, and approaches \(1\), as \(| (x,\xi)| \to+\infty\), sufficiently fast. Hence \({\det}\,p(x,hD)\) is well defined. They show that \[ {\log}| {\det}\,p(x,hD)| \leq{1\over{(2\pi h)^n}}\Bigl(I(\Lambda,p)+o(1)\Bigr),\,\,\,\,h\to 0+,(1) \] where \[ I(\Lambda,p):={1\over 2}\int_\Lambda{\log}(p(\rho)\overline{p(\rho)})\mu(d\rho), \] where \(\mu(d\rho)=| \sigma^n| /n!\) is the symplectic volume element on \(\Lambda\subset{\mathbb C}^{2n},\) \(\Lambda\) belonging to a suitable class of IR-manifolds (i.e. manifolds of \({\mathbb C}^{2n}\) of real dimension \(2n\) on which the restriction of the symplectic form is real and nondegenerate) contained in a tubular neighborhood of \({\mathbb R}^{2n}\) in \({\mathbb C}^{2n},\) and \(\sigma=\sum d\xi_j\wedge dx_j,\) \((x,\xi)\in{\mathbb C}^{2n}\) is the canonical (complex) symplectic form.

The starting point is to exploit the interesting feature of the determinant of an operator, of being independent of the choice of norm of the Hilbert space on which the operator acts, and sometimes to some extent even of the space itself. Hence, for such a manifold \(\Lambda\), they define an \(h\)-dependent Hilbert space \(H(\Lambda)\) and prove that \(p(x,hD)\) acts as a uniformly bounded operator in \(H(\Lambda)\) and (under certain conditions) that \({\det}\,p(x,hD)\) is well-defined and independent of whether \(p(x,hD)\) acts in \(L^2({\mathbb R}^n)\) or in \(H(\Lambda).\) This approach leads to estimate \((1)\) above. Moreover, when \(p\) is elliptic on \(\Lambda\) (that is, when \(p(\rho)\not=0\) for \(\rho\in\Lambda\)) one has equality in \((1).\)

The next step is to choose \(\Lambda\) to make \(I(\Lambda,p)\) as small as possible. The authors do not solve this crucial and difficult problem, but obtain several remarkable results in this direction, that are of independent interest, by proving, for instance, that \(I(\Lambda,p)\) is a Lipschitz function of \(\Lambda\) (whose differential and sometimes its Hessian they study), and that the critical points of the functional \(I(\cdot,p)\) are infinitesimal minima to infinite order.

The authors consider also the case of (holomorphic) dependence of \(p\) on a spectral parameter, the case of \(p\bigl| _\Lambda\) of principal type, and finally consider two examples of bounds for relative determinants with a spectral parameter.

The authors consider an \(h\)-pseudodifferential operator \(p(x,hD)\), whose symbol \(p\) extends holomorphically in a tubular neighborhood of the (real) phase space, and approaches \(1\), as \(| (x,\xi)| \to+\infty\), sufficiently fast. Hence \({\det}\,p(x,hD)\) is well defined. They show that \[ {\log}| {\det}\,p(x,hD)| \leq{1\over{(2\pi h)^n}}\Bigl(I(\Lambda,p)+o(1)\Bigr),\,\,\,\,h\to 0+,(1) \] where \[ I(\Lambda,p):={1\over 2}\int_\Lambda{\log}(p(\rho)\overline{p(\rho)})\mu(d\rho), \] where \(\mu(d\rho)=| \sigma^n| /n!\) is the symplectic volume element on \(\Lambda\subset{\mathbb C}^{2n},\) \(\Lambda\) belonging to a suitable class of IR-manifolds (i.e. manifolds of \({\mathbb C}^{2n}\) of real dimension \(2n\) on which the restriction of the symplectic form is real and nondegenerate) contained in a tubular neighborhood of \({\mathbb R}^{2n}\) in \({\mathbb C}^{2n},\) and \(\sigma=\sum d\xi_j\wedge dx_j,\) \((x,\xi)\in{\mathbb C}^{2n}\) is the canonical (complex) symplectic form.

The starting point is to exploit the interesting feature of the determinant of an operator, of being independent of the choice of norm of the Hilbert space on which the operator acts, and sometimes to some extent even of the space itself. Hence, for such a manifold \(\Lambda\), they define an \(h\)-dependent Hilbert space \(H(\Lambda)\) and prove that \(p(x,hD)\) acts as a uniformly bounded operator in \(H(\Lambda)\) and (under certain conditions) that \({\det}\,p(x,hD)\) is well-defined and independent of whether \(p(x,hD)\) acts in \(L^2({\mathbb R}^n)\) or in \(H(\Lambda).\) This approach leads to estimate \((1)\) above. Moreover, when \(p\) is elliptic on \(\Lambda\) (that is, when \(p(\rho)\not=0\) for \(\rho\in\Lambda\)) one has equality in \((1).\)

The next step is to choose \(\Lambda\) to make \(I(\Lambda,p)\) as small as possible. The authors do not solve this crucial and difficult problem, but obtain several remarkable results in this direction, that are of independent interest, by proving, for instance, that \(I(\Lambda,p)\) is a Lipschitz function of \(\Lambda\) (whose differential and sometimes its Hessian they study), and that the critical points of the functional \(I(\cdot,p)\) are infinitesimal minima to infinite order.

The authors consider also the case of (holomorphic) dependence of \(p\) on a spectral parameter, the case of \(p\bigl| _\Lambda\) of principal type, and finally consider two examples of bounds for relative determinants with a spectral parameter.

Reviewer: Alberto Parmeggiani (Bologna)

### MSC:

35S05 | Pseudodifferential operators as generalizations of partial differential operators |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

58J52 | Determinants and determinant bundles, analytic torsion |