On singular spectrum of $N$-dimensional perturbations (to the Aronszajn-Donoghue-Kac theory)

1 Jul 2021, 17:05


Mark Malamud (Peoples Friendship University of Russia (RUDN University))


The main results of the Aronszajn-Donoghue-Kac theory are extended to the case of $n$-dimensional (in the resolvent sense) perturbations $\widetilde A$ of an operator $A_0=A^*_0$ defined on a Hilbert space $\frak H$. Applying technique of boundary triplets we describe singular continuous and point spectra of extensions $A_B$ of a simple symmetric operator $A$ acting in $\frak H$ in terms of the Weyl function $M(\cdot)$ of the pair $\{A,A_0\}$ and a boundary $n$-dimensional operator $B = B^*$. Assuming that the multiplicity of singular spectrum of $A_0$ is maximal it is established that the singular parts $E^s_{A_B}$ and $E^s_{A_0}$ of the spectral measures $E_{A_B}$ and $E_{A_0}$ of the operators $A_B$ and $A_0$, respectively, are mutually singular. We also obtain estimates of the multiplicity of point and singular continuous spectra of selfadjoint extensions of $A$.

Applying this result to direct sums $A = A^{(1)}\oplus A^{(2)}$ allow us to generalize and clarify Kac theorem on multiplicity of singular spectrum of Schr\"odinger operator on the line. Applications to differential operators will be also discussed. The talk is based on results announced in [1].

[1] Malamud M.M., Doklady Math., On Singular Spectrum of Finite-Dimensional Perturbations (toward the Aronszajn–Donoghue–Kac Theory), 2019, Vol. 100, No. 1, p. 358–362.

Primary author

Mark Malamud (Peoples Friendship University of Russia (RUDN University))

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