A perturbation theory of resonances.

*(English)*Zbl 0902.47007The article represent an attempt in organizing results which appear in quantum mechanics for Schrödinger operators. The author made an attempt to represent the perturbation theory of holomorphic families of operators based upon the ideal that resonance problems from mathematical physics should be treated in some formal way like eigenvalues. The foundation of this theory is the resonance-eigenvalue connection. We give some definitions first.

A closed linear operator \(P\) in a given Banach space \(B\) is considered, the spectrum \(\sigma(P)\neq\mathbb{C}\). \(D\) is a domain in \(\mathbb{C}\) such that \(D\cap\sigma(P)\subset \sigma_{dis}(P)\). The resolvent \(R(\lambda)= (P-\lambda)^{- 1}\) is a well defined meromorphic operator function in \(D\) with values in \(\zeta(B)\). We also consider spaces \(B_0\), \(B_1\), \(B_0\subset B\subset B_1\) with continuous \(J_0\), \(J\), \(J_0B_0\to B\), \(JB\to B_1\), \(\lambda\in D\setminus\sigma_{dis}(P)\), \(\widetilde R(\lambda)= JR(\lambda)J_0\). The set of all poles of \(\widetilde R(\lambda)\) in \(D_+,D_+\supset D\) is denoted by \(\Lambda(P)\), which is composed of two disjoint sets:

1) resonances of \(P\) in \(D_+\);

2) isolated eigenvalues of \(P\) in \(D_+\).

Let \(\Delta\) be a bounded domain in \(\mathbb{C}\) with boundary \(\Gamma\) of class \(C^1:\overline\Delta\subset D_+\), \(\Gamma\cap\Lambda(P)= \emptyset\). The following theorem (3.3. in the article) shows the resonance-eigenvalue connection.

Theorem. The operator \(P_\Gamma\) has discrete spectrum in \(\Delta\) given by: \(\sigma(P)\cap \Delta= \Lambda(P)\cap\Delta\). In particular, all resonances of \(P\) in \(\Delta\) are eigenvalues of \(P_\Gamma\), \(P_\Gamma\) is a closed linear operator in \(B_\Gamma\) defined as follows: \(\text{Dom}(P_\Gamma)= \text{Ran }R_\Gamma(\lambda_0)\), \(P_\Gamma u=\lambda_0 u+f\), \(u\in R_\Gamma(\lambda_0)f\in \text{Dom}(P_\Gamma)\), \(f\in B_\Gamma\), \(R_\Gamma(\lambda)= (P_\Gamma- \lambda)^{- 1}\), \(\lambda\in\Delta\setminus \Lambda(P)\).

A closed linear operator \(P\) in a given Banach space \(B\) is considered, the spectrum \(\sigma(P)\neq\mathbb{C}\). \(D\) is a domain in \(\mathbb{C}\) such that \(D\cap\sigma(P)\subset \sigma_{dis}(P)\). The resolvent \(R(\lambda)= (P-\lambda)^{- 1}\) is a well defined meromorphic operator function in \(D\) with values in \(\zeta(B)\). We also consider spaces \(B_0\), \(B_1\), \(B_0\subset B\subset B_1\) with continuous \(J_0\), \(J\), \(J_0B_0\to B\), \(JB\to B_1\), \(\lambda\in D\setminus\sigma_{dis}(P)\), \(\widetilde R(\lambda)= JR(\lambda)J_0\). The set of all poles of \(\widetilde R(\lambda)\) in \(D_+,D_+\supset D\) is denoted by \(\Lambda(P)\), which is composed of two disjoint sets:

1) resonances of \(P\) in \(D_+\);

2) isolated eigenvalues of \(P\) in \(D_+\).

Let \(\Delta\) be a bounded domain in \(\mathbb{C}\) with boundary \(\Gamma\) of class \(C^1:\overline\Delta\subset D_+\), \(\Gamma\cap\Lambda(P)= \emptyset\). The following theorem (3.3. in the article) shows the resonance-eigenvalue connection.

Theorem. The operator \(P_\Gamma\) has discrete spectrum in \(\Delta\) given by: \(\sigma(P)\cap \Delta= \Lambda(P)\cap\Delta\). In particular, all resonances of \(P\) in \(\Delta\) are eigenvalues of \(P_\Gamma\), \(P_\Gamma\) is a closed linear operator in \(B_\Gamma\) defined as follows: \(\text{Dom}(P_\Gamma)= \text{Ran }R_\Gamma(\lambda_0)\), \(P_\Gamma u=\lambda_0 u+f\), \(u\in R_\Gamma(\lambda_0)f\in \text{Dom}(P_\Gamma)\), \(f\in B_\Gamma\), \(R_\Gamma(\lambda)= (P_\Gamma- \lambda)^{- 1}\), \(\lambda\in\Delta\setminus \Lambda(P)\).

Reviewer: A.Kondrat’ev (Pensacola)

##### MSC:

47A55 | Perturbation theory of linear operators |

47A75 | Eigenvalue problems for linear operators |

47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |