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A model independent way to describe a resonance is through the frequency dependent scattering operator $S(\omega)$. Causality requires that this object is analytic in the upper half of the complex $\o …

The full physics problem is complex, the vibrating membrane displaces the air, which causes a backreaction and signifantly modifies the response. Moreover, the response also depends sensitively on whe …

The $n\times n$ imaginary matrix $A$ satisfies $A^\top=-A$, so it is skew-symmetric. The Youla decomposition is $$A=iO\Sigma O^\top,$$ where $O$ is a real orthogonal matrix and $\Sigma$ is a real bloc …

Some pointers to the (extensive) literature on generalized Feyman-Kac formulas: Stochastic Solution of Elliptic and Parabolic Boundary Value Problems for the Spectral Fractional Laplacian Fractional …

Since $O$ is orthogonal, $O^\top=O^{-1}$ commutes with $O$, hence the eigenvalues $\mu_n$ of $O+O^\top$ are related to the eigenvalues $e^{i\phi_n}$ of $O$ by $\mu_n=2\cos\phi_n$. The spectral density …

An overview of threshold effects on a band structure is given by Birman and Suslina, in their paper on Second order periodic differential operators (2004). Generically, this refers to cases where it s …

The answer is no. Here is a counterexample: take $\rho(x)=1$ for all $x$ and choose the eigenfunction $\phi_1(t;x)=\cos(\pi x/2t)$; then the integral $\int_{-1}^{1} f(x) \phi_1(t;x) \,dx =0$ for all o …

I interpret the question in a bit more general terms, as a request to "shed light" on the phenomenon that "oftentimes many eigenvalues lie exactly on the real axis". The key thing to notice is that th …

For this purpose ("beginning grad student") it would make sense to focus on the case that the discrete eigenvalues appear in an energy range that does not overlap with the continuous spectrum (say, $E …

Indeed, this is the result of Davies - Pseudo-Spectra, the Harmonic Oscillator and Complex Resonances (1982): The resolvent operator $(H-zI)^{-1}$ of $$H=-d^2/dx^2+cx^2,\;\;\operatorname{Re}c>0,\;\; \ …

The Schrödinger equation with the $\text{sech}^2$ potential (sometimes called the Pöschl–Teller potential) was first studied by Epstein in 1930 [1]. There is an extensive literature on this exactly so …

Since $\sigma_1^2=\lambda_+$ and $\sigma_2^2=\lambda_-$ are the two eigenvalues of the symmetric matrix product $MM^t$, we have $\lambda_++\lambda_-={\rm tr}\,MM^t=\|m_1\|^2+\|m_2\|^2$. Hence we may w …

The Hamiltonian $$H=\begin{pmatrix} \alpha^2+a^\ast a&\alpha a+\beta a^\ast\\ \alpha a^\ast+\beta a&\beta^2+a^\ast a \end{pmatrix} $$ is known in the physics literature as the anisotropic Rabi Hamilto …

The condition $\lambda+\mu_1>0$ ensures that $M(\lambda)=A+\lambda $ is invertible, and then one can use Jacobi's formula $$\frac{d}{d\lambda} \det M(\lambda) = \det M(\lambda) \operatorname{tr} \left …

Define the unitary and Hermitian matrices $$U=\left( \begin{array}{cccc} 0 & 0 & -i & 0 \\ 0 & 0 & 0 & -i \\ i & 0 & 0 & 0 \\ 0 & i & 0 & 0 \\ \end{array} \right),\;\; V=\left( \begin{array}{cccc} …