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I presume by a "resonance" you mean a scattering resonance, so a pole of the determinant of the scattering matrix. On finite-area hyperbolic surfaces all scattering resonances correspond to eigenvalue …

By specifying $PP^{\dagger}=W$ you are prescribing the singular values $s_n$ ($n=1,2,\ldots N$) of the $N\times N$ matrix $P$. These are just the positive square roots of the eigenvalues of the Hermit …

A unitary transformation $\tilde{H}=e^{-i\mu xy}He^{i\mu xy}$ will transform the Hamiltonian into $$\tilde{H}(\mu,\epsilon)=D_x^2+(D_y+\mu x)^2+\epsilon x+V(x,y),\;\;D_\nu=-i\partial/\partial\nu,$$ s …

Matrices of this form are called "periodic Jacobi matrices", see for example, -- The spectrum of Jacobi matrices. -- The construction of Jacobi and periodic Jacobi matrices with prescribed spectra. …

This is a classic problem solved by Poincaré in La méthode de Neumann et le problème de Dirichlet (1897). You can find the solution explained in section 8.2 of Poincaré's variational problem in potent …

you have to take the real part of $\bar{t}X$ in the exponent, in order for the integral to make sense; if you then decompose $t=t_1+it_2$ and $X=X_1+iX_2$ into real and imaginary parts, you just have …

These are indeed two different definitions, see the discussion in One-parameter Semigroups of Positive Operators. That reference also gives two different names for the two definitions, peripheral spec …

You can find a mathematically precise treatment in On the Energy Levels of the Hydrogen Atom. For $l=0$ the eigenvalues $1-\kappa^2$ of $H$ are given by the square integrable solutions of $$\left(-\fr …

It may be of interest to note that the $\pm E$ symmetry of the spectrum of the Dirac operator holds not only for a constant $m$, but also for a spatially dependent $m(x,y)$, $$H = \begin{pmatrix} m(x, …

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 $n\times n$ 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 …

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 …

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 …

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 …